Recent zbMATH articles in MSC 35Bhttps://zbmath.org/atom/cc/35B2023-05-31T16:32:50.898670ZWerkzeugVariational estimates for the speed propagation of fronts in a nonlinear diffusive Fisher equationhttps://zbmath.org/1508.350022023-05-31T16:32:50.898670Z"Benguria, Rafael D."https://zbmath.org/authors/?q=ai:benguria.rafael-d"Depassier, M. Cristina"https://zbmath.org/authors/?q=ai:depassier.m-cristina"Rica, Sergio"https://zbmath.org/authors/?q=ai:rica.sergio(no abstract)Least energy sign-changing solutions for Kirchhoff-type problems with potential wellhttps://zbmath.org/1508.350112023-05-31T16:32:50.898670Z"Chen, Xiao-Ping"https://zbmath.org/authors/?q=ai:chen.xiaoping"Tang, Chun-Lei"https://zbmath.org/authors/?q=ai:tang.chun-leiSummary: In this paper, we investigate the existence of least energy sign-changing solutions for the Kirchhoff-type problem \(-\left(a + b \int_{\mathbb{R}^3} |\nabla u|^2 \mathrm{d}x\right)\Delta u + V(x) u = f(u)\), \(x\in\mathbb{R}^3\), where \(a, b > 0\) are parameters, \(V\in\mathcal{C}(\mathbb{R}^3, \mathbb{R})\), and \(f\in\mathcal{C}(\mathbb{R}, \mathbb{R})\). Under weaker assumptions on \(V\) and \(f\), by using variational methods with the aid of a new version of global compactness lemma, we prove that this problem has a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.
{\copyright 2022 American Institute of Physics}Existence of multi-bump solutions for a nonlinear Kirchhoff-type systemhttps://zbmath.org/1508.350132023-05-31T16:32:50.898670Z"Liu, Weiming"https://zbmath.org/authors/?q=ai:liu.weimingSummary: In this paper, we use the Lyapunov-Schmidt reduction method to obtain the existence of multi-bump solutions for a nonlinear Kirchhoff-type system with the parameter \(\varepsilon\). As a result, when \(\varepsilon\rightarrow0\), the system has more and more multi-bump positive solutions.
{\copyright 2022 American Institute of Physics}Existence of normalized solutions for semilinear elliptic systems with potentialhttps://zbmath.org/1508.350162023-05-31T16:32:50.898670Z"Liu, Chuangye"https://zbmath.org/authors/?q=ai:liu.chuangye"Yang, Xiaolong"https://zbmath.org/authors/?q=ai:yang.xiaolongSummary: In this paper, we consider the existence of normalized solutions to the following system: \(-\Delta u + V_1(x)u + \lambda u = \mu_1 u^3 + \beta v^2u\) and \(-\Delta v + V_2(x)v + \lambda v = \mu_2 v^3 + \beta u^2v\) in \(\mathbb{R}^3\), under the mass constraint \(\int_{\mathbb{R}^3} u^2 + v^2 = \rho^2\), where \(\rho\) is prescribed, \(\mu_i, \beta > 0\) (\(i = 1, 2\)), and \(\lambda\in\mathbb{R}\) appears as a Lagrange multiplier. Then, by a min-max argument, we show the existence of fully nontrivial normalized solutions under various conditions on the potential \(V_i : \mathbb{R}^3\to\mathbb{R}\) (\(i = 1, 2\)).
{\copyright 2022 American Institute of Physics}Spatiotemporal dynamics on a class of \((n+1)\)-dimensional reaction-diffusion neural networks with discrete delays and a conical structurehttps://zbmath.org/1508.350192023-05-31T16:32:50.898670Z"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.7|chen.jing.5|chen.jing.3|chen.jing.4|chen.jing.2"Xiao, Min"https://zbmath.org/authors/?q=ai:xiao.min"Wu, Xiaoqun"https://zbmath.org/authors/?q=ai:wu.xiaoqun"Wang, Zhengxin"https://zbmath.org/authors/?q=ai:wang.zhengxin"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jinde(no abstract)Stability of high-order delayed Markovian jumping reaction-diffusion HNNs with uncertain transition rateshttps://zbmath.org/1508.350202023-05-31T16:32:50.898670Z"Suriguga"https://zbmath.org/authors/?q=ai:suriguga.ma"Kao, Yonggui"https://zbmath.org/authors/?q=ai:kao.yonggui"Shao, Chuntao"https://zbmath.org/authors/?q=ai:shao.chuntao"Chen, Xiangyong"https://zbmath.org/authors/?q=ai:chen.xiangyongSummary: This paper focuses on mean square exponential stability of high-order Markovian jump reaction-diffusion HNNs (RHNNs) with uncertain transition rates (GUTRs) and time-varying delays by Lyapunov-Krasovskii functional method and linear matrix inequality (LMI). In this GUTR model, only part of the transition rates can be known, namely, its estimate error and estimate value are known, but the others have no useful information. Our models are more comprehensive and some existing results are special cases of ours. Finally, a numerical example illustrates the validity of our findings.Global well-posedness for pseudo-parabolic \(p\)-Laplacian equation with singular potential and logarithmic nonlinearityhttps://zbmath.org/1508.350212023-05-31T16:32:50.898670Z"Yuan, Wen-Shuo"https://zbmath.org/authors/?q=ai:yuan.wenshuo"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.binSummary: The main goal of this work is to investigate the initial boundary value problem for a class of pseudo-parabolic \(p\)-Laplacian equations with singular potential and logarithmic nonlinearity. First of all, we prove the local existence of weak solutions. Second, we show the existence of the global solution and the weak solution converging to the stationary solution when the time tends to infinity, and we show blow-up phenomena of solutions with the initial energy less than the mountain pass level \(d\) by using the potential well method. Finally, we parallelly stretch all the conclusions for the subcritical case to the critical case.
{\copyright 2022 American Institute of Physics}On a semilinear wave equation in anti-de Sitter spacetime: the critical casehttps://zbmath.org/1508.350222023-05-31T16:32:50.898670Z"Palmieri, Alessandro"https://zbmath.org/authors/?q=ai:palmieri.alessandro"Takamura, Hiroyuki"https://zbmath.org/authors/?q=ai:takamura.hiroyukiSummary: In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.
{\copyright 2022 American Institute of Physics}Riemann-Hilbert problems and soliton solutions for a generalized coupled Sasa-Satsuma equationhttps://zbmath.org/1508.350282023-05-31T16:32:50.898670Z"Liu, Yaqing"https://zbmath.org/authors/?q=ai:liu.yaqing"Zhang, Wen-Xin"https://zbmath.org/authors/?q=ai:zhang.wenxin"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiuSummary: This paper studies the multi-component Sasa-Satsuma integrable hierarchies via an arbitrary-order matrix spectral problem, based on the zero curvature formulation. A generalized coupled Sasa-Satsuma equation is derived from the multi-component Sasa-Satsuma integrable hierarchies with a bi-Hamiltonian structure. The inverse scattering transform of the generalized coupled Sasa-Satsuma equation is presented by the spatial matrix spectral problem and the Riemann-Hilbert method, which enables us to obtain the N-soliton solutions. And then the dynamics of one- and two-soliton solutions are discussed and presented graphically. Asymptotic analyses of the presented two-soliton solution are finally analyzed.Riemann-Hilbert method and multi-soliton solutions of an extended modified Korteweg-de Vries equation with \(N\) distinct arbitrary-order poleshttps://zbmath.org/1508.350292023-05-31T16:32:50.898670Z"Yang, Jin-Jie"https://zbmath.org/authors/?q=ai:yang.jinjie"Tian, Shou-Fu"https://zbmath.org/authors/?q=ai:tian.shoufu"Li, Zhi-Qiang"https://zbmath.org/authors/?q=ai:li.zhiqiang.1Summary: The Riemann-Hilbert (RH) method is developed to study the extended modified Korteweg-de Vries (emKdV) equation with zero boundary conditions. The analytical and asymptotic properties of Jost functions are obtained by the direct scattering analysis to establish a suitable RH problem. We consider the singular RH problem of scattering data with \(N\) distinct poles. By using the generalized residue condition to solve the RH problem, we construct the exact solution of emKdV equation under the condition of no reflection. In addition, four kinds of special poles and their corresponding soliton solutions are discussed in detail, including one second-order pole, one third-order pole, three first-order poles and two second-order poles.Long-time asymptotic behavior of the coupled dispersive AB system in low regularity spaceshttps://zbmath.org/1508.350302023-05-31T16:32:50.898670Z"Zhu, Jin-Yan"https://zbmath.org/authors/?q=ai:zhu.jinyan"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yongSummary: In this paper, we mainly investigate the long-time asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann-Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the long-time asymptotic behavior of these solutions. We demonstrate that in any fixed time cone \(\mathcal{C}(x_1, x_2, v_1, v_2) = \{(x, t)\in\mathbb{R}^2 \mid x = x_0 + vt, x_0\in[x_1, x_2], v\in[v_1, v_2]\}\), the long-time asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by \(N(\mathcal{I})\) solitons on the discrete spectrum, the leading order term \(\mathcal{O}(t^{-1/2})\) on the continuous spectrum, and the allowable residual \(\mathcal{O}(t^{-3/4})\).
{\copyright 2022 American Institute of Physics}A revisit of the velocity averaging lemma: on the regularity of stationary Boltzmann equation in a bounded convex domainhttps://zbmath.org/1508.350312023-05-31T16:32:50.898670Z"Chen, I.-Kun"https://zbmath.org/authors/?q=ai:chen.i-kun"Chuang, Ping-Han"https://zbmath.org/authors/?q=ai:chuang.ping-han"Hsia, Chun-Hsiung"https://zbmath.org/authors/?q=ai:hsia.chun-hsiung"Su, Jhe-Kuan"https://zbmath.org/authors/?q=ai:su.jhe-kuanIn this paper the authors consider the velocity averaging lemma to establish regularity for the class of stationary linearized Boltzmann equations in a bounded convex domain. It describes the case when the gas is confined in a bounded convex domain. The authors consider the incoming data with four iterations. The regularity in fractional Sobolev space in space variable up to order \(1^{-}\) is established. It is considered the regularity theory for the equation
\[
v\cdot \nabla f(x,\xi )=L(f),
\]
where \(v\in\mathbb{R}^3\) and \(x\in\Omega \), where \(\Omega\subset\mathbb{R}^3\) is a \(C^2\) bounded strictly convex domain such that \(\partial\Omega \) is of positive Gaussian curvatures, and \(L\) represents the linearization of the collision operator. The collision operator in Boltzmann equation reads
\[
Q(F,G)=\int_{\mathbb{R}^3} \int_{0}^{2\pi } \int_{0}^{\pi /2} (F(v^{\prime })G(\xi_{\ast }^{\prime })- F(v)G(v_{\ast })B(|v_{\ast }-v|,\theta )) d\theta d\epsilon dv_{\ast },
\]
where \(v\), \(v_{\ast }\) and \(v^{\prime }\), \(v_{\ast }^{\prime }\) are pairs of velocities before and after the impact, \(B\) is called the cross section, depending on interaction between particles, \(L\) is obtained by linearizing \(Q\) around the standard Maxwellian \(M(v)=\pi^{-3/2}e^{-|v|^2}\), \(F=M+M^{1/2}f\), and \(L(f)=M^{-1/2}(v) [Q(M^{1/2}f,M)+Q(M,M^{1/2}f]\). The well known angular cutoff potential is a mathematical model introduced by
\textit{H. Grad} [Math. Models Phys. Sci., Proc. Conf. Univ. Notre Dame 1962, 3--16 (1963; Zbl 0178.28002); with \textit{R. van Norton} [Nuclear Fusion 1962, Suppl. 61--65 (1962; Zbl 0115.23104); in: Proceedings of the 5th international conference on ionization phenomena in gases, Munich, 28 Aug.--1 Sept. 1961. Volume 2. Amsterdam: North-Holland Publ. Comp. 1630--1649 (1962; Zbl 0105.42104)]
where \(0\leq B\leq C|v-v_{\ast}|^{\gamma } \cos{\theta}\sin{\theta}\). In the present paper the authors follow Grad's idea, that is, \( B=|v-v_{\ast}|^{\gamma }\beta (\theta )\), \(0\leq\beta (\theta ) \leq C\cos{\theta}\sin{\theta}\), \(0\leq\gamma\leq 1\). The range of \(\gamma \) corresponds to the hard sphere model, cutoff hard potential, and cutoff Maxwellian molecular gases. Considering the incoming data, with four iterations, the authors establish regularity in fractional Sobolev space in space variable up to order \(1^{-}\).
It is assumed that \(\Omega\subset\mathbb{R}^3\) is a bounded convex open domain satisfying the positive curvature condition, linearized collision operator \(L\) satisfies an angular cutoff assumption \(0\leq B (|v-v_{\ast}|,\theta ) \leq C|v-v_{\ast}|^{\gamma } \cos{\theta}\sin{\theta}\), and collision frequency \(\nu \) is bounded by a special inequality. Also it is supposed that the incoming data \(g\) satisfies \(|g(q_1,v)|\leq Ce^{-a|v|^2}\). Then under the above stated assumptions it is proved that any solution \(f\in L^2(\Omega\times\mathbb{R}^3)\) for the stationary linearized Boltzmann equation belongs to the class \(L_{v}^{2}\), i.e. \(f\in L_{v}^{2} (\mathbb{R}^3;H_{x}^{1-\epsilon}(\Omega ))\) for any \(\epsilon\in (0,1)\).
Reviewer: Dimitar A. Kolev (Sofia)Inverse of divergence and homogenization of compressible Navier-Stokes equations in randomly perforated domainshttps://zbmath.org/1508.350332023-05-31T16:32:50.898670Z"Bella, Peter"https://zbmath.org/authors/?q=ai:bella.peter"Oschmann, Florian"https://zbmath.org/authors/?q=ai:oschmann.florianSummary: We analyze the behavior of weak solutions to compressible viscous fluid flows in a bounded domain in \(\mathbb{R}^3\), randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like \(\varepsilon^\alpha\), \(\alpha > 3\), with \(\varepsilon\) denoting the average distance between the balls, the problem homogenize to the same limiting equation. Our main contribution is a construction of the Bogovskiĭ operator, uniformly in \(\varepsilon\), without any assumptions on the minimal distance between the balls.Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtureshttps://zbmath.org/1508.350342023-05-31T16:32:50.898670Z"Bulíček, Miroslav"https://zbmath.org/authors/?q=ai:bulicek.miroslav"Jüngel, Ansgar"https://zbmath.org/authors/?q=ai:jungel.ansgar"Pokorný, Milan"https://zbmath.org/authors/?q=ai:pokorny.milan"Zamponi, Nicola"https://zbmath.org/authors/?q=ai:zamponi.nicolaSummary: The existence of large-data weak solutions to a steady compressible Navier-Stokes-Fourier system for chemically reacting fluid mixtures is proved. General free energies are considered satisfying some structural assumptions, with a pressure containing a \(\gamma\)-power law. The model is thermodynamically consistent and contains the Maxwell-Stefan cross-diffusion equations in the Fick-Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. \textit{A priori} estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the pressure in \(L^p\) with \(p > 1\), the effective viscous flux identity, and uniform bounds related to Feireisl's oscillation defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.
{\copyright 2022 American Institute of Physics}On a toy-model related to the Navier-Stokes equationshttps://zbmath.org/1508.350372023-05-31T16:32:50.898670Z"Hounkpe, F."https://zbmath.org/authors/?q=ai:hounkpe.francisSummary: A parabolic toy-model for the incompressible Navier-Stokes system is considered. This model shares a lot of similar features with the incompressible model, including the energy inequality, the scaling symmetry, and it is also supercritical in 3D. A goal is to establish some regularity results for this toy-model in order to get, if possible, better insight to the standard Navier-Stokes system. A Caffarelli-Kohn-Nirenberg type result for the model is also proved in a direct manner. Finally, the absence of divergence-free constraint allows us to study this model in the radially symmetric setting for which full regularity is established.Zero-Mach limit of the compressible Navier-Stokes-Korteweg equationshttps://zbmath.org/1508.350382023-05-31T16:32:50.898670Z"Ju, Qiangchang"https://zbmath.org/authors/?q=ai:ju.qiangchang"Xu, Jianjun"https://zbmath.org/authors/?q=ai:xu.jianjunSummary: We consider the Cauchy problem for the compressible Navier-Stokes-Korteweg system in three dimensions. Under the assumption of the global existence of strong solutions to incompressible Navier-Stokes equations, we demonstrate that the compressible Navier-Stokes-Korteweg system admits a global unique strong solution without smallness restrictions on initial data when the Mach number is sufficiently small. Furthermore, we derive the uniform convergence of strong solutions for compressible Navier-Stokes-Korteweg equations toward those for incompressible Navier-Stokes equations as long as the solution of the limiting system exists.
{\copyright 2022 American Institute of Physics}Dynamics for 2D incompressible Navier-Stokes flow coupled with time-dependent Darcy flowhttps://zbmath.org/1508.350392023-05-31T16:32:50.898670Z"Liu, Yang"https://zbmath.org/authors/?q=au:Liu, Yang"Ma, Shan"https://zbmath.org/authors/?q=ai:ma.shan"Sun, Chunyou"https://zbmath.org/authors/?q=ai:sun.chunyouSummary: In this paper, we use the method of evolutionary systems introduced by \textit{A. Cheskidov} and \textit{C. Foias} [J. Differ. Equations 231, No. 2, 714--754 (2006; Zbl 1113.35140)] to describe the existence of global attractor for 2D incompressible Navier-Stokes flow coupled with time-dependent Darcy flow. Furthermore, stationary statistical solutions of this system are constructed from the global attractor.
{\copyright 2022 American Institute of Physics}Solving the axisymmetric Navier-Stokes equations in critical spaces (I): The case with small swirl componenthttps://zbmath.org/1508.350402023-05-31T16:32:50.898670Z"Liu, Yanlin"https://zbmath.org/authors/?q=ai:liu.yanlinSummary: In this paper, we prove the global existence of smooth solutions to axisymmetric Navier-Stokes equations with initial data in some critical spaces, provided the swirl part of the initial velocity is sufficiently small.Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolutionhttps://zbmath.org/1508.350412023-05-31T16:32:50.898670Z"Prosviryakov, Evgeniĭ Yur'evich"https://zbmath.org/authors/?q=ai:prosviryakov.evgenii-yurevichSummary: Swirling laminar axisymmetric flows of viscous incompressible fluids in a potential field of body forces are considered. The study of flows is carried out in a cylindrical coordinate system. In the flows, the regions in which the axial derivative of the circumferential velocity cannot take on zero value in some open neighborhood (essentially swirling flows) and the regions in which this derivative is equal to zero (the region with layered swirl) are considered separately. It is shown that a well-known method (the method of viscous vortex domains) developed for non-swirling flows can be used for regions with layered swirling. For substantially swirling flows, a formula is obtained for calculating the radial-axial velocity of an imaginary fluid through the circumferential vorticity component, the circumferential circulation of a real fluid, and the partial derivatives of these functions. The particles of this imaginary fluid ``transfer'' vortex tubes of the radial-axial vorticity component while maintaining the intensity of these tubes, and also ``transfer'' the circumferential circulation and the product of the circular vorticity component by some function of the distance to the axis of symmetry. A non-integral method for reconstructing the velocity field from the vorticity field is proposed. It is reduced to solving a system of linear algebraic equations in two variables. The obtained result is proposed to be used to extend the method of viscous vortex domains to swirling axisymmetric flows.On Type I blowups of suitable weak solutions to the Navier-Stokes equations near boundaryhttps://zbmath.org/1508.350442023-05-31T16:32:50.898670Z"Seregin, G."https://zbmath.org/authors/?q=ai:seregin.gregory-a|seregin.grigorySummary: In this note, boundary Type I blowups of suitable weak solutions to the Navier-Stokes equations are discussed. In particular, it has been shown that, under certain assumptions, the existence of nontrivial mild bounded ancient solutions in half space leads to the existence of suitable weak solutions with Type I blowup on the boundary.Navier-Stokes problems with small parameters in half-space and applicationhttps://zbmath.org/1508.350452023-05-31T16:32:50.898670Z"Shakhmurov, V. B."https://zbmath.org/authors/?q=ai:shakhmurov.veli-bSummary: We derive the existence, uniqueness, and uniform \(L^p\) estimates for the abstract Navier-Stokes problem with small parameters in half-space. The equation involves small parameters and an abstract operator in a Banach space \(E \). Hence, we obtain the singular perturbation property for the Stokes operator depending on a parameter. We can obtain the various classes of Navier-Stokes equations by choosing \(E\) and the linear operators \(A \). These classes occur in a wide variety of physical systems. As application we establish the existence, uniqueness, and uniform \(L^p\) estimates for the solution of the mixed problems for infinitely many Navier-Stokes equations and nonlocal mixed problems for the high order Navier-Stokes equations.Inverse image of precompact sets and regular solutions to the Navier-Stokes equationshttps://zbmath.org/1508.350462023-05-31T16:32:50.898670Z"Shlapunov, Aleksandr Anatol'evich"https://zbmath.org/authors/?q=ai:shlapunov.alexander-a"Tarkhanov, Nikolaĭ Nikolaevich"https://zbmath.org/authors/?q=ai:tarkhanov.nikolai-nikolaevichSummary: We consider the initial value problem for the Navier-Stokes equations over \({\mathbb R}^3 \times [0,T]\) with time \(T>0\) in the spatially periodic setting. We prove that it induces open injective mappings \({\mathcal A}_s\colon B^s_1 \to B^{s-1}_2\) where \(B^s_1\), \(B^{s-1}_2\) are elements from scales of specially constructed function spaces of Bochner-Sobolev type parametrized with the smoothness index \(s \in \mathbb N\). Finally, we prove that a map \({\mathcal A}_s\) is surjective if and only if the inverse image \({\mathcal A}_s^{-1}(K)\) of any precompact set \(K\) from the range of the map \({\mathcal A}_s\) is bounded in the Bochner space \(L^{\mathfrak s} ([0,T], L^{{\mathfrak r}} ({\mathbb T}^3))\) with the Ladyzhenskaya-Prodi-Serrin numbers \({\mathfrak s}, {\mathfrak r}\).Anomalous energy flux in critical \(L^p\)-based spaceshttps://zbmath.org/1508.350472023-05-31T16:32:50.898670Z"Burczak, Jan"https://zbmath.org/authors/?q=ai:burczak.jan"Sattig, Gabriel"https://zbmath.org/authors/?q=ai:sattig.gabrielSummary: We construct a three-dimensional vector field that exhibits positive energy flux at every Littlewood-Paley shell and has the best possible regularity in \(L^p\)-based spaces, \(p \leq 3\); in particular, it belongs to \(H^{(\frac{5}{6})^-}\).Global solutions of the compressible Euler equations with large initial data of spherical symmetry and positive far-field densityhttps://zbmath.org/1508.350482023-05-31T16:32:50.898670Z"Chen, Gui-Qiang G."https://zbmath.org/authors/?q=ai:chen.gui-qiang-g"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.7Summary: We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density so that the total initial-energy is unbounded. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. For the large initial data of positive far-field density, various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at a certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded and the wave propagation is not at finite speed starting initially. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve the key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit for the Navier-Stokes approximations constructed in this paper even when the total initial-energy is unbounded, though the density may blow up near the origin at certain time and the wave propagation is not at finite speed.Partial differential equations with quadratic nonlinearities viewed as matrix-valued optimal ballistic transport problemshttps://zbmath.org/1508.350512023-05-31T16:32:50.898670Z"Vorotnikov, Dmitry"https://zbmath.org/authors/?q=ai:vorotnikov.dmitry-aSummary: We study a rather general class of optimal ``ballistic'' transport problems for matrix-valued measures. These problems naturally arise, in the spirit of \textit{Y. Brenier} [Commun. Math. Phys. 364, No. 2, 579--605 (2018; Zbl 1410.35102)], from a certain dual formulation of nonlinear evolutionary equations with a particular quadratic structure reminiscent both of the incompressible Euler equation and of the quadratic Hamilton-Jacobi equation. The examples include the ideal incompressible MHD, the template matching equation, the multidimensional Camassa-Holm (also known as the \(H(\text{div})\) geodesic equation), EPDiff, Euler-\(\alpha\), KdV and Zakharov-Kuznetsov equations, the equations of motion for the incompressible isotropic elastic fluid and for the damping-free Maxwell's fluid. We prove the existence of the solutions to the optimal ``ballistic'' transport problems. For formally conservative problems, such as the above mentioned examples, a solution to the dual problem determines a ``time-noisy'' version of the solution to the original problem, and the latter one may be retrieved by time-averaging. This yields the existence of a new type of absolutely continuous in time generalized solutions to the initial-value problems for the above mentioned PDE. We also establish a sharp upper bound on the optimal value of the dual problem, and explore the weak-strong uniqueness issue.Estimates for the dimension of attractors of a regularized Euler system with dissipation on the spherehttps://zbmath.org/1508.350522023-05-31T16:32:50.898670Z"Zelik, S. V."https://zbmath.org/authors/?q=ai:zelik.sergey-v"Ilyin, A. A."https://zbmath.org/authors/?q=ai:ilyin.alexei-a"Kostyanko, A. G."https://zbmath.org/authors/?q=ai:kostyanko.a-gSummary: We prove the existence of a global attractor of a regularized Euler-Bardina system with dissipation on the two-dimensional sphere and in arbitrary domains on the sphere. Explicit estimates for the fractal dimension of the attractor in terms of its physical parameters are obtained.Global well-posedness of generalized magnetohydrodynamics equations in variable exponent Fourier-Besov-Morrey spaceshttps://zbmath.org/1508.350542023-05-31T16:32:50.898670Z"Abidin, Muhammad Zainul"https://zbmath.org/authors/?q=ai:abidin.muhammad-zainul"Chen, Jie Cheng"https://zbmath.org/authors/?q=ai:chen.jiechengSummary: A generalized incompressable magnetohydrodynamics system is considered in this paper. Furthermore, results of global well-posednenss are established with the aid of Littlewood-Paley decomposition and Fourier localization method in mentioned system with small initial condition in the variable exponent Fourier-Besov-Morrey spaces. Moreover, the Gevrey class regularity of the solution is also achieved in this paper.Use of Atangana-Baleanu fractional derivative in helical flow of a circular pipehttps://zbmath.org/1508.350552023-05-31T16:32:50.898670Z"Abro, Kashif Ali"https://zbmath.org/authors/?q=ai:abro.kashif-ali"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Sooppy Nisar, Kottakkaran"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana-Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox-\textbf{H} function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.Exact solution and the multidimensional Godunov scheme for the acoustic equationshttps://zbmath.org/1508.350572023-05-31T16:32:50.898670Z"Barsukow, Wasilij"https://zbmath.org/authors/?q=ai:barsukow.wasilij"Klingenberg, Christian"https://zbmath.org/authors/?q=ai:klingenberg.christianSummary: The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.Two-dimensional incompressible micropolar fluid models with singular initial datahttps://zbmath.org/1508.350592023-05-31T16:32:50.898670Z"Béjar-López, Alexis"https://zbmath.org/authors/?q=ai:bejar-lopez.alexis"Cunha, Cleyton"https://zbmath.org/authors/?q=ai:cunha.cleyton"Soler, Juan"https://zbmath.org/authors/?q=ai:soler.juan-sSummary: This paper deals with the interaction between microstructures and the appearance or persistence of singular configurations in the Cauchy problem for the two-dimensional model of incompressible micropolar fluids. We analyze the case of null angular viscosity and singular initial data, including the possibility of vortex sheets or measures as initial data in Morrey spaces. Through integral techniques we establish the existence of weak solutions local or global in time. In addition, the uniqueness and stability of these solutions is analyzed.Lagrangian regularity of the electron magnetohydrodynamics flow on a bounded domainhttps://zbmath.org/1508.350602023-05-31T16:32:50.898670Z"Besse, Nicolas"https://zbmath.org/authors/?q=ai:besse.nicolasSummary: In this paper we investigate the regularity in time of the Lagrangian flow associated with the electron magnetohydrodynamics (e-MHD) equations on a bounded domain with a smooth (ultradifferentiable) boundary. This model is widely used in controlled magnetic fusion, in space and astrophysics plasmas and also in physics of solids. We show that initial data with limited smoothness in Sobolev spaces induce a Lagrangian flow-map \(X\) and a Lagrangian magnetic vector potential \(A\) (viz. the magnetic vector potential evaluated at the Lagrangian spatial point \(X)\), which are ultradifferentiable in time, with the two particular cases of real analytic and Gevrey time regularity. It turns out that the Lagrangian canonical momentum \(P\), the Lagrangian magnetic field \(B\), and the Lagrangian electric field \(E\) inherit this Lagrangian regularity property. Among others, the proof makes crucial use of a novel Lagrangian formulation of the e-MHD in terms of the Lagrangian fields \((X, A, P, B, E)\). A by-product of this Lagrangian and constructive proof is the design of arbitrary high-order semi-Lagrangian schemes to solve the e-MHD equations on a bounded domain.On wave-breaking for the two-component Fornberg-Whitham systemhttps://zbmath.org/1508.350622023-05-31T16:32:50.898670Z"Cheng, Wenguang"https://zbmath.org/authors/?q=ai:cheng.wenguangSummary: This paper is concerned with the Cauchy problem of the two-component Fornberg-Whitham system, which does not have the \(L^2\)-norm conservation law of the component \(u\). We establish two new results of wave-breaking to this system by using the sign preservation of \(\rho \), the \(L^1\)-norm conservation law of \(\rho\) and a priori estimate for the \(L^2\)-norm of \(u\).Global well-posedness for the 2D stable Muskat problem in \(H^{3/2}\)https://zbmath.org/1508.350652023-05-31T16:32:50.898670Z"Córdoba, Diego"https://zbmath.org/authors/?q=ai:cordoba.diego|cordoba.diego-langarica"Lazar, Omar"https://zbmath.org/authors/?q=ai:lazar.omarSummary: We prove a global existence result of a unique strong solution in \(H^{5/2}\) with small \(\dot{H}^{3/2}\) semi-norm for the 2D Muskat problem. Hence, allowing the interface to have arbitrary large finite slopes and finite energy (thanks to the \(L^2\) maximum principle). The proof is based on the use of a new formulation of the Muskat equation that involves oscillatory terms. Then, a careful use of interpolation inequalities in homogeneneous Besov spaces allows us to close the a priori estimates.Blow-up of a dyadic model with intermittency dependence for the Hall MHDhttps://zbmath.org/1508.350662023-05-31T16:32:50.898670Z"Dai, Mimi"https://zbmath.org/authors/?q=ai:dai.mimiSummary: We derive dyadic models for the magnetohydrodynamics with Hall effect by including the intermittency dimension as a parameter. For such dyadic models, existence of global weak solutions is established. In addition, local strong solution is obtained; while global strong solution is obtained in the case of high intermittency dimension. Moreover, we show that positive solution with large initial data develops blow-up in finite time provided the intermittency dimension is lower than a threshold.Existence and exponential behavior for the stochastic 2D Cahn-Hilliard-Oldroyd model of order onehttps://zbmath.org/1508.350672023-05-31T16:32:50.898670Z"Deugoué, G."https://zbmath.org/authors/?q=ai:deugoue.gabriel"Jidjou Moghomye, B."https://zbmath.org/authors/?q=ai:jidjou-moghomye.b"Tachim Medjo, T."https://zbmath.org/authors/?q=ai:tachim-medjo.theodoreThe authors consider the stochastic Cahn-Hilliard-Oldroyd model of order one, for the motion of an incompressible isothermal mixture of two immiscible non-Newtonian fluids written as:
\[
du(t)+[-\nu _{1}\Delta u+(\beta \ast \Delta u)(t)+(u\cdot \nabla )u+\nabla p-\mathcal{K}\mu \nabla \phi ]dt=\sigma _{1}(t,u,\phi )dW_{t}^{1}+\int_{Z}\gamma (t,u(t^{-}),\phi (t),z) \widetilde{\pi }(dt,dz),
\]
\[
d\phi (t)=[\nu _{2}\Delta \mu -u\cdot \nabla \phi ]dt+\sigma _{2}(t,u,\phi )dW_{t}^{2},
\]
\(\mu =-\varepsilon \Delta \phi +\alpha f(\phi )\), \(\operatorname{div}(u)=0\), posed in \((0,T)\times \mathcal{M}\), where \(T>0 \) and \(\mathcal{M}\) is a bounded and open domain in \(\mathbb{R}^{2}\) with a smooth boundary \(\partial \mathcal{M}\). Here \(\beta (t)=\gamma e^{-\delta t}\), with \(\gamma =\frac{1}{\varsigma }(\nu -\frac{\kappa }{\varsigma })>0\), \( \nu _{1}=\frac{\kappa }{\varsigma }\), \(\delta =\frac{1}{\varsigma }>0\), \( \varsigma,\kappa >0\), \(W_{t}^{i}\), \(i=1,2\), are two cylindrical Wiener processes in a separable Hilbert space \(U\) defined on the probability space \( (\Omega,\mathcal{F},\mathbb{P})\), \(Z\) is a measurable subspace of some Hilbert space, \(\widetilde{\pi }(dt,dz)=\pi (dt,dz)-\lambda (dz)dt\) is a compensated Poisson random measure, \(\lambda (dz)\) being a \(\sigma \)-finite Lévy measure on the Hilbert space with an associated Poisson random measure \(\pi (dt,dz)\) such that \(\mathbb{E}[\pi (dt,dz)]=\lambda (dz)dt\), \( \mu \) is the chemical potential of the binary mixture which is given by the variational derivative of the free energy functional \(\mathcal{E}_{0}(\phi )=\int_{D}(\frac{\varepsilon }{2}\left\vert \nabla \phi \right\vert ^{2}+\alpha F(\phi ))dx\), where \(F(r)=\int_{0}^{r}f(\zeta )d\zeta \) is a double-well potential, \(\nu _{2}\) and \(\kappa \) are positive constants that correspond to the mobility constant and capillarity (stress) coefficient, respectively, \(\varepsilon \) and \(\alpha \) are two positive parameters describing the interactions between the two phases. The processes \(W_{t}^{i}\), \(i=1,2\), and \(\widetilde{\pi }\) are mutually independent. The boundary conditions \(\partial _{\eta }\phi =\partial _{\eta }\Delta \phi =0=u\) are imposed on \((0,T)\times \partial \mathcal{M}\), together with the initial conditions \((u,\phi )(0)=(u_{0},\phi _{0})\) in \(\mathcal{M}\). The authors introduce a variational formulation of the problem in appropriate spaces and the notion of global strong solution. Assuming appropriate hypotheses on the data, \(\sigma _{1}(\cdot,0,0)\in L^{p}(\Omega,\mathcal{F},\mathbb{P} ;L^{2}(0,T;\mathcal{L}^{2}(U;H^{1})))\) and \((u_{0},\phi _{0})\in L^{p}(\Omega,\mathcal{F},\mathbb{P},\mathbb{H})\) with \(\mathbb{E}\mathcal{E} ^{p}(u_{0},\phi _{0})<\infty \), for all \(p\geq 2\), they prove that the problem has a unique strong solution. For the proof, the authors introduce a finite dimensional Galerkin approximation on which they prove uniform estimates. For the uniqueness of the strong solution, they argue by contradiction. In the last part of their paper, the authors prove an exponential stability result. Under appropriate hypotheses, the strong solution to a stochastic 2D Cahn-Hilliard-Oldroyd model converges to the unique stationary solution to a stationary equation, which is exponentially stable.
Reviewer: Alain Brillard (Riedisheim)Anomalous dissipation in passive scalar transporthttps://zbmath.org/1508.350692023-05-31T16:32:50.898670Z"Drivas, Theodore D."https://zbmath.org/authors/?q=ai:drivas.theodore-d"Elgindi, Tarek M."https://zbmath.org/authors/?q=ai:elgindi.tarek-mohamed"Iyer, Gautam"https://zbmath.org/authors/?q=ai:iyer.gautam"Jeong, In-Jee"https://zbmath.org/authors/?q=ai:jeong.in-jeeSummary: We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible \(C^\infty([0,T)\times\mathbb{T}^d)\cap L^1([0,T];C^{1-}(\mathbb{T}^d))\) velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows the non-uniqueness of solutions to the transport equation with an incompressible \(L^1([0,T];C^{1-}(\mathbb{T}^d))\) drift, which is smooth except at one point in time. We also give a sufficient condition for anomalous dissipation based on solutions to the inviscid equation becoming singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications.On the maximal \(L_p\)-\(L_q\) regularity theorem for the linearized electro-magnetic field equations with interface conditionshttps://zbmath.org/1508.350702023-05-31T16:32:50.898670Z"Frolova, E."https://zbmath.org/authors/?q=ai:frolova.e-s|frolova.e-v|frolova.elena"Shibata, Y."https://zbmath.org/authors/?q=ai:shibata.yoshiharu|shibata.yuki|shibata.yusuke|shibata.yoshihiro|shibata.yoshitaka|shibata.youhei|shibata.yukioSummary: This paper deals with the maximal \(L_p\)-\(L_q\) regularity theorem for the linearized electro-magnetic field equations with interface conditions and perfect wall condition. This problem is motivated by linearization of the coupled magnetohydrodynamics system which generates two separate problems. The first problem is associated with well studied Stokes system. Another problem related to the magnetic field is studied in this paper. The maximal \(L_p\)-\(L_q\) regularity theorem for the Stokes equations with interface and nonslip boundary conditions has been proved by \textit{J. Prüss} and \textit{G. Simonett} [Moving interfaces and quasilinear parabolic evolution equations. Basel: Birkhäuser/Springer (2016; Zbl 1435.35004)], and \textit{S. Maryani} and \textit{H. Saito} [Differ. Integral Equ. 30, No. 1--2, 1--52 (2017; Zbl 1424.35277)]. Combination of these results and the result obtained in the present paper yields local well-posedness for the MHD problem in the case of two incompressible liquids separated by a closed interface. It is planned to prove it in a forthcoming paper. The main part of the present paper is devoted to proving the existence of \(\mathcal{R} \)-bounded solution operators associated with generalized resolvent problem. The maximal \(L_p\)-\(L_q\) regularity is established by applying the Weis operator-valued Fourier multiplier theorem.Time periodic solutions for 3D quasi-geostrophic modelhttps://zbmath.org/1508.350722023-05-31T16:32:50.898670Z"García, Claudia"https://zbmath.org/authors/?q=ai:garcia.claudia-i"Hmidi, Taoufik"https://zbmath.org/authors/?q=ai:hmidi.taoufik"Mateu, Joan"https://zbmath.org/authors/?q=ai:mateu.joanSummary: This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by \textit{generic} revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator. We are able to implement the bifurcation only from the largest eigenvalues of the operator, which are simple. Additional difficulties generated by the singularities of the poles are solved through the use of suitable function spaces with Dirichlet boundary condition type and refined potential theory with anisotropic kernels.Retraction note to: ``Notes on the result of solutions of the equilibrium equations''https://zbmath.org/1508.350732023-05-31T16:32:50.898670Z"He, Gaiping"https://zbmath.org/authors/?q=ai:he.gaiping"Wang, Lihong"https://zbmath.org/authors/?q=ai:wang.lihong"Rodrigo, Ben"https://zbmath.org/authors/?q=ai:rodrigo.benFrom the text: The Editor-in-Chief has retracted the article [ibid. 2018, Paper No. 57, 3 p. (2018; Zbl 1499.35500)] by \textit{G. He} et al. because the results are invalid. The article also shows evidence of authorship manipulation. Additionally, Universidad de Talca have confirmed that Ben Rodrigo has not been affiliated with their institution. The authors have not responded to any correspondence regarding this retraction.Well-posedness and attractors for a 2D Boussinesq system with partial dissipationhttps://zbmath.org/1508.350742023-05-31T16:32:50.898670Z"He, Jinfang"https://zbmath.org/authors/?q=ai:he.jinfang"Ma, Shan"https://zbmath.org/authors/?q=ai:ma.shan"Sun, Chunyou"https://zbmath.org/authors/?q=ai:sun.chunyouSummary: This article is devoted to the global well-posedness and the long-time behavior of solutions of a 2D Boussinesq equations with partial dissipation. We prove that this system is global well-posed under some weaker assumptions on the initial data and has a weak sigma-attractor which retains some of the common properties of global attractors for the dissipative dynamical system, moreover, the local attractor which is the composition of the weak sigma-attractor is upper semicontinuous under small viscosity perturbations.On inhibition of the Rayleigh-Taylor instability by a horizontal magnetic field in ideal MHD fluids with velocity dampinghttps://zbmath.org/1508.350762023-05-31T16:32:50.898670Z"Jiang, Fei"https://zbmath.org/authors/?q=ai:jiang.fei"Jiang, Song"https://zbmath.org/authors/?q=ai:jiang.song"Zhao, Youyi"https://zbmath.org/authors/?q=ai:zhao.youyiSummary: It is still open whether the inhibition phenomenon of the Rayleigh-Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly verified in the linearized case by \textit{Y. Wang} [J. Math. Phys. 53, No. 7, 073701, 22 p. (2012; Zbl 1277.76124)]. In this paper, we show that this inhibition phenomenon can be rigorously verified in the (nonlinear) inhomogeneous, incompressible, inviscid case with velocity damping. More precisely, we show that there is a critical number \(m_{\operatorname{C}} \), such that if the strength \(| m |\) of a horizontal magnetic field is bigger than \(m_{\operatorname{C}} \), then the small perturbation solution around the magnetic RT equilibrium state is exponentially stable in time. Moreover, we also provide a nonlinear instability result for the case \(| m | \in(0, m_{\operatorname{C}})\). Our instability result reveals that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.Rotational effect on the asymptotic stability of the MHD systemhttps://zbmath.org/1508.350772023-05-31T16:32:50.898670Z"Kim, Junha"https://zbmath.org/authors/?q=ai:kim.junhaSummary: In this paper, we consider the large time behavior of solutions of the incompressible rotational magnetohydrodynamics equations. First, we prove the unique existence of a global smooth solution in the Sobolev spaces \(H^s( \mathbb{R}^3)\) for \(1 / 2 < s < 3 / 2\), when the rotation is sufficiently rapid. Second, we establish the temporal decay estimates for the solution under additional assumptions. It is observed that the rotation affects the decay rate of the velocity field.Waves of maximal height for a class of nonlocal equations with inhomogeneous symbolshttps://zbmath.org/1508.350782023-05-31T16:32:50.898670Z"Le, Hung"https://zbmath.org/authors/?q=ai:le.hungSummary: In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order \(\alpha\) for \(\alpha> 1\). Based on the properties of the convolution operator, we apply a global bifurcation technique to show the existence of a highest, even, \(2 \pi \)-periodic traveling-wave solution. The regularity of this wave is proved to be exactly Lipschitz.The combined non-equilibrium diffusion and low Mach number limits of a model arising in radiation magnetohydrodynamicshttps://zbmath.org/1508.350792023-05-31T16:32:50.898670Z"Li, Fucai"https://zbmath.org/authors/?q=ai:li.fucai"Zhang, Shuxing"https://zbmath.org/authors/?q=ai:zhang.shuxingSummary: We consider the combined non-equilibrium diffusion and low Mach limits of a model arising in radiation magnetohydrodynamics, which is described by the ideal compressible magnetohydrodynamic equations coupled to the radiation transfer equation. We study the case that the temperature has a large variation. In this situation, due to the complex asymmetric singular structure in the model, it is very hard to obtain uniform estimates of solutions in standard Sobolev spaces. To overcome the difficulties caused by the singular structure, we introduce two new weighted norms and construct new auxiliary equations. In the appropriate normed spaces, we show that the contribution of singular terms to the total energy is bounded by \(O(\epsilon)\) with respect to the parameter \(\epsilon\), and then establish the uniform estimates of solutions. Moreover, we rigorously prove that, for the well-prepared initial data, the target system is a coupling of the nonhomogeneous incompressible magnetohydrodynamic equations and a diffusion equation.Global well-posedness to the Cauchy problem of 2D nonhomogeneous Bénard system with large initial data and vacuumhttps://zbmath.org/1508.350802023-05-31T16:32:50.898670Z"Li, Huanyuan"https://zbmath.org/authors/?q=ai:li.huanyuanSummary: This paper establishes the global well-posedness of strong solutions to the nonhomogeneous Bénard system with positive density at infinity in the whole space \(\mathbb{R}^2\). We obtain the global existence and uniqueness of strong solutions for general large initial data. Our method relies on the dedicate energy estimates and a logarithmic interpolation inequality.
{\copyright 2022 American Institute of Physics}Global mild solutions to three-dimensional magnetohydrodynamic equations in Morrey spaceshttps://zbmath.org/1508.350812023-05-31T16:32:50.898670Z"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng"Xi, Shuai"https://zbmath.org/authors/?q=ai:xi.shuai"Zeng, Zirong"https://zbmath.org/authors/?q=ai:zeng.zirong"Zhu, Shengguo"https://zbmath.org/authors/?q=ai:zhu.shengguoSummary: In this paper, we consider the Cauchy problem of three-dimensional incompressible magnetohydrodynamic equations. Some uniform estimates with respect to time for the coupling terms between the fluid and the magnetic field will be presented, under the condition that the initial \(\mathcal{M}^{1 , 1}\) norms of the vorticity and the current density are both sufficiently small. By the above estimates, we can obtain a global-in-time well-posedness of mild solutions in Morrey spaces via some effective arguments. The asymptotic behaviours of the solutions are also obtained.Liouville-type theorems for the stationary inhomogeneous incompressible MHD equationshttps://zbmath.org/1508.350822023-05-31T16:32:50.898670Z"Liu, Pan"https://zbmath.org/authors/?q=ai:liu.panSummary: In this paper, we investigate the Liouville-type problem for the three-dimensional stationary inhomogeneous incompressible MHD equations in the Lorentz spaces without any integrability condition for \((\nabla \boldsymbol{u}, \nabla \boldsymbol{B})\). More precisely, we prove that the velocity and magnetic field \((\boldsymbol{u}, \boldsymbol{B})\), being in some Lorentz spaces, must be zero provided that the density \(\rho\) is essentially bounded.Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentialshttps://zbmath.org/1508.350842023-05-31T16:32:50.898670Z"Matioc, Bogdan-Vasile"https://zbmath.org/authors/?q=ai:matioc.bogdan-vasile"Prokert, Georg"https://zbmath.org/authors/?q=ai:prokert.georgThis paper addresses the following moving boundary problem of Stokes flow driven by the capillarity of the moving interface \(t\to \Gamma(t)\) between two fluid phases \(\Omega^{\pm}(t)\) in \(\mathbb{R}^{2}\):
\[
\begin{cases}
\mu\Delta v^{\pm} - \nabla q^{\pm} = 0 & \text{ in } \Omega^{\pm}(t), \\
\operatorname{div } v^{\pm} = 0 & \text{ in } \Omega^{\pm}(t), \\
v^{+}= v^{-} & \text{ on } \Gamma(t), \\
[T(v,q)]\tilde{\nu}= -\sigma\tilde{\kappa}\tilde{\nu} & \text{ on } \Gamma(t), \\
(v^{\pm},q^{\pm})\rightarrow 0 \text{ for } |x|\to \infty, V_{n}=v\cdot \tilde{\nu} & \text{ on } \Gamma(t),
\end{cases}
\]
where \(v^{\pm}: \Omega^{\pm}(t)\to \mathbb{R}^{2}\) is a vector field representing the velocity of the liquid located in \(\Omega^{\pm}(t)\) and \(q^{\pm}: \Omega^{\pm}(t)\to \mathbb{R}\) its pressure, \(\nu\) is the unit exterior normal to \(\Omega^{-}(t)\) and \(\tilde{\kappa}\) denotes the curvature of the interface, \([T(v,q)]\) denotes the jump of the stress tensor across \(\Gamma(t)\), and the positive constants \(\mu\) and \(\sigma\) denote the viscosity of the liquids and the surface tension coefficient of the interface, respectively.
Based on the the theory of maximal regularity for nonlinear parabolic equations in weighted Hölder spaces of vector-valued functions presented in [\textit{A. Lunardi}, Analytic semigroups and optimal regularity in parabolic problems. Basel: Birkhäuser (1995; Zbl 0816.35001)], the authors established the following results:
\begin{itemize}
\item[1.] existence and uniqueness of maximal solutions with initial data that are arbitrary within our phase space;
\item[2.] a corresponding semiflow property;
\item[3.] parabolic smoothing up to \(C^{\infty}\) of solutions in time and space (away from the initial time);
\item[4.] a criterion for global existence of solutions, or equivalently, a necessary condition for blow-up.
\end{itemize}
Reviewer: Weihua Wang (Beijing)On the generalized Magnetohydrodynamics-\(\alpha\) equations with fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaceshttps://zbmath.org/1508.350852023-05-31T16:32:50.898670Z"Melo, Wilberclay G."https://zbmath.org/authors/?q=ai:melo.wilberclay-g"de Souza, Manassés"https://zbmath.org/authors/?q=ai:de-souza.manasses-x"Rosa Santos, Thyago Souza"https://zbmath.org/authors/?q=ai:rosa-santos.thyago-souzaSummary: This paper is based on determining the well-posedness of the 3D generalized Magnetohydrodynamics-\(\alpha\) equations. More precisely, this work presents results related to the existence and uniqueness of global in time solutions for these equations with the same parameters of the fractional dissipations \(\beta\) in Lei-Lin and Lei-Lin-Gevrey spaces \(\mathcal{X}^s_{a,\sigma}\) by considering that the initial data are small enough, \(a\geq 0\), \(\sigma \geq 1\), \(s\in [-1,0]\) and \(\beta \in [\frac{1}{2},1]\). In addition, we study some large time decay results by applying the analyticity of the global solution \((\boldsymbol{v},\boldsymbol{b})\) and proving, for instance, that this one satisfies the following limit superior:
\[
\limsup_{t\rightarrow \infty} t^{\delta_{\beta} +\frac{\kappa -s}{2\beta}} [\Vert \boldsymbol{v}(t)\Vert_{\mathcal{X}^{\kappa}_{a,\sigma}(\mathbb{R}^3)} +\Vert \boldsymbol{b}(t)\Vert_{\mathcal{X}^{\kappa}_{a,\sigma}(\mathbb{R}^3)}] =0,\quad \forall \kappa >s,
\]
where the initial conditions \((\boldsymbol{v}_0,\boldsymbol{b}_0)\) belong to \(\mathcal{X}^s_{a,\sigma} (\mathbb{R}^3) \cap \mathcal{X}^{1-2\beta}_{a,\sigma} (\mathbb{R}^3)\cap L^2 (\mathbb{R}^3)\) with \((\beta,s) \in \big( (\frac{1}{2},1]\times [1-2\beta, 0)\big) \cup \{ (\frac{1}{2},0)\}\) and \(\delta_{\beta} \in (-\infty, \frac{5-4\beta}{4\beta}]\).Asymptotic analysis of a dynamic flow of the Bingham fluidhttps://zbmath.org/1508.350902023-05-31T16:32:50.898670Z"Saadallah, Abdelkader"https://zbmath.org/authors/?q=ai:saadallah.abdelkader"Benseridi, Hamid"https://zbmath.org/authors/?q=ai:benseridi.hamidSummary: The aim of this paper is to study the asymptotic behavior of an incompressible 3 Bingham fluid in a dynamic regime occupying a bounded domain of \(\mathbb{R}^3\) with nonlinear friction of Tresca type. Firstly, the existence and uniqueness of weak solution is proved. Then we show the estimates for the velocity field and the pressure independently of the parameter \(\varepsilon\). Finally, we give a specific Reynolds equation associated with variational inequalities and prove the uniqueness. The proof uses the asymptotic behavior when the dimension of the domain tends to zero.Exact Riemann solutions for the drift-flux equations of two-phase flow under gravityhttps://zbmath.org/1508.350912023-05-31T16:32:50.898670Z"Shen, Chun"https://zbmath.org/authors/?q=ai:shen.chun"Sun, Meina"https://zbmath.org/authors/?q=ai:sun.meinaThe authors consider the system of one-dimensional drift-flux equations describing a two-phase flow and written as:
\((\alpha_{g}\rho_{g})_{t}+(\alpha_{g}\rho_{g}u_{g})_{x}=0\),
\((\alpha_{l}\rho_{l})_{t}+(\alpha_{l}\rho_{l}u_{l})_{x}=0\),
\((\alpha_{g}\rho_{g}u_{g}+\alpha_{l}\rho_{l}u_{l})_{t}+(\alpha_{g}\rho_{g}u_{g}^{2}+\alpha_{l}\rho_{l}u_{l}^{2}+p)_{x}=-q\),
where \(\alpha_{g}\), \(\alpha_{l}\in \lbrack 0,1]\) are the volume fractions of gas and liquid which satisfy \(\alpha_{g}+\alpha_{l}=1\), \(\rho_{g}\) the gas density, \(u_{g}\) the gas velocity, \(\rho_{l}\) the liquid density, \(u_{l}\) the liquid velocity, \(p\) the pressure for both gas and liquid, and \(q\) the external forces such as gravity and friction. Considering a 1D inviscid, compressible and isentropic liquid-gas two-phase flow on an inclined pipeline under gravity, which leads to special expressions for the external forces \(q\) and pressure \(p\), the authors simplify the above system as the following one written in a conservative form as:
\(m_{t}+(m(v-\mu t))_{x}=0\),
\(n_{t}+(n(v-\mu t))_{x}=0\),
\(((m+n)v)_{t}+((m+n)v(v-\mu t)+(\gamma -1)(m+n)^{\gamma })_{x}=0\),
where \(m\) and \(n\) are the gas and liquid masses defined as \(m=\alpha_{g}\rho_{g}\), \(n=\alpha_{l}\rho_{l}\), and \(v\) is the velocity \(v(x,t)=u(x,t)+\mu t\). Riemann initial conditions \((m,n,v)(x,t=0)=(m_{-},n_{-},u_{-})\), \(x<0\), \((m,n,v)(x,t=0)=(m_{+},n_{+},u_{+})\), \(x>0\), where \(m_{-},n_{-},m_{+},n_{+}>0\) are naturally deduced from physical considerations are imposed. The authors rewrite this last problem in a quasi-linear form for which they derive the associated characteristic equation: \((m+n)(\lambda -v+\mu t)((\lambda -v+\mu t)^{2}-(\gamma -1)\gamma (m+n)^{\gamma -1})=0\) and the three eigenvalues. They draw computations to determine and analyze the properties of the Riemann solutions to the last problem. The first main result proves that for any given Riemann initial conditions, there exists a unique solution to the last problem for fixed parameters \(\gamma\) and \(\mu\). Moving to the system: \(m_{t}+(m(v-\mu t))_{x}=0\), \(n_{t}+(n(v-\mu t))_{x}=0\), \(((m+n)v)_{t}+((m+n)v(v-\mu t))_{x}=0\), with the same initial conditions as above, the authors define the notion of generalized delta shock wave solution and they prove the existence of such solution if \(u_{+}<u_{-}\). They give its expression. In the last part of the paper, the authors describe the limits of Riemann solutions to the second system as \(\gamma \rightarrow 1+\).
Reviewer: Alain Brillard (Riedisheim)Randomized final-state problem for the Zakharov system in dimension threehttps://zbmath.org/1508.350922023-05-31T16:32:50.898670Z"Spitz, Martin"https://zbmath.org/authors/?q=ai:spitz.martinSummary: We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For \((u_+,v_+)\in H^1\times L^2\) without any size restriction, symmetry assumption or additional angular regularity, we perform a physical-space randomization on \(u_+\) and an angular randomization on \(v_+\) yielding random final states \((u^\omega_+,v^\omega_+)\). We obtain that for almost every \(\omega\), there is a unique solution of the Zakharov system scattering to the final state \((u^\omega_+,v^\omega_+)\). The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates which are accessible due to the randomization.Pointwise a priori estimates for solutions to some \(p\)-Laplacian equationshttps://zbmath.org/1508.350932023-05-31T16:32:50.898670Z"Sun, Xiao Qiang"https://zbmath.org/authors/?q=ai:sun.xiaoqiang"Bao, Ji Guang"https://zbmath.org/authors/?q=ai:bao.jiguangSummary: In this article, we apply blow-up analysis to study pointwise a priori estimates for some \(p\)-Laplacian equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the \(p\)-harmonic function, i.e., the solution of \(\Delta_p u = 0\), \(x \in \Omega \). We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type \(p\)-Laplacian equation \(- \Delta_p u\) = |\(u|^{ \lambda - 1}u\), \(x \in \Omega \), which are then extended to the equation with general right hand term \(f(x, u)\) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type \(p\)-Laplacian equation, in a case of \(p = 2\), are also discussed.Thermodynamically consistent modeling for complex fluids and mathematical analysishttps://zbmath.org/1508.350942023-05-31T16:32:50.898670Z"Suzuki, Yukihito"https://zbmath.org/authors/?q=ai:suzuki.yukihito"Ohnawa, Masashi"https://zbmath.org/authors/?q=ai:ohnawa.masashi"Mori, Naofumi"https://zbmath.org/authors/?q=ai:mori.naofumi"Kawashima, Shuichi"https://zbmath.org/authors/?q=ai:kawashima.shuichiSummary: The goal of this paper is to derive governing equations for complex fluids in a thermodynamically consistent way so that the conservation of energy and the increase of entropy is guaranteed. The model is a system of first-order partial differential equations on density, velocity, energy (or equivalently temperature), and conformation tensor. A barotropic model is also derived. In the one-dimensional case, we express the barotropic model in the form of hyperbolic balance laws, and show that it satisfies the stability condition. Consequently, the global existence of solutions around equilibrium states is proved and the convergence rates is obtained.Large time behavior to the 2D micropolar Boussinesq fluidshttps://zbmath.org/1508.350952023-05-31T16:32:50.898670Z"Wang, Xuewen"https://zbmath.org/authors/?q=ai:wang.xuewen"Lei, Keke"https://zbmath.org/authors/?q=ai:lei.keke"Han, Pigong"https://zbmath.org/authors/?q=ai:han.pigongSummary: In this paper, we prove the global existence of classical solutions to the 2D incompressible Boussinesq equations for the micropolar fluid. Furthermore, applying the Fourier splitting methods, we obtain the lager time decay properties.Analysis of Hartmann boundary layer peristaltic flow of Jeffrey fluid: quantitative and qualitative approacheshttps://zbmath.org/1508.350962023-05-31T16:32:50.898670Z"Yasmeen, Shagufta"https://zbmath.org/authors/?q=ai:yasmeen.shagufta"Asghar, Saleem"https://zbmath.org/authors/?q=ai:asghar.saleem"Anjum, Hafiz Junaid"https://zbmath.org/authors/?q=ai:anjum.hafiz-junaid"Ehsan, Tayyaba"https://zbmath.org/authors/?q=ai:ehsan.tayyabaSummary: Peristalsis of Jeffrey fluid in axisymmetric tube is studied when the fluid is subject to strong magnetic field. Mathematical analysis is made under the assumption of small Reynolds number and long wave length approximations. An alternate assumption of lubrication theory can also be applied to the peristalsis problem. Both the approaches lead to same mathematical expressions. The effects of magnetic field are investigated in the boundary layer (Hartman boundary layer) and on the boundary layer thickness. The effects of strong magnetic field, in the boundary layer, are explored using asymptotic analysis to find analytical solution. We notice that this approach facilitates to unveil the effects of strong magnetic field explicitly and determines the boundary layer thickness mathematically. The objective behind this study is: how to control the boundary layer thickness through the application of strong magnetic field. This phenomenon is central from theoretical and applied points of view. While going for the asymptotic analysis of large magnetic field; the mathematical model leads to singular perturbation problem which is an apparent diversion to most of the peristalsis problems -- attempted by regular perturbation method.
The boundary value problem is solved analytically using singular perturbation approach together with higher order matching technique. The important features of peristalsis like stream function, velocity, and pressure rise are calculated. In addition to the analytical solution, we explore the qualitative behavior of the flow using the theory of dynamical systems. The velocity field is determined by the phase plane analysis and the stability of the solution is found through bifurcation diagrams. Qualitative analysis of the solution has been carried out for magnetic parameter, amplitude ratio, flow rate and the Jeffrey fluid parameter. The concomitant change of stability is given through topological flow patterns. Equilibrium plots (bifurcation diagrams) give a complete description of the various flow patterns developed for the complete range of a flow parameters in contrast to most of the studies that describe the flow patterns at some particular value of a parameter. The study helps to analyze the behavior of the fluid at the critical points that represent steady solution. The mix of the analytical and qualitative approach will help to broaden the scope and understanding of peristaltic transport of fluid in channels, tubes and curved tubes (this study).Nowhere-uniform continuity of the data-to-solution map for the two-component Fornberg-Whithamhttps://zbmath.org/1508.350972023-05-31T16:32:50.898670Z"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Tang, Weijie"https://zbmath.org/authors/?q=ai:tang.weijieSummary: In this paper, we present a new method to prove that the data-to-solution map for the two-component Fornberg-Whitham is not uniformly continuous from any bounded subset in \(H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})\) with \(s > 3/2\) and further to show that this data-to-solution map is nowhere-uniform continuity in the same space.Mechanisms of stationary converted waves and their complexes in the multi-component AB systemhttps://zbmath.org/1508.350982023-05-31T16:32:50.898670Z"Zhang, Han-Song"https://zbmath.org/authors/?q=ai:zhang.han-song"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei"Sun, Wen-Rong"https://zbmath.org/authors/?q=ai:sun.wen-rong"Wang, Xin"https://zbmath.org/authors/?q=ai:wang.xin.44"Xu, Tao"https://zbmath.org/authors/?q=ai:xu.taoSummary: Under investigation in this article is a multi-component AB system which models the self-induced transparency phenomenon. By using the modified Darboux transformation, we present the breather solutions of such system. We study the subtle mechanism that converts the breathing state into the solitary and periodic ones, through which we obtain various stationary nonlinear excitations such as the multi-peak solitons, (quasi) periodic waves, (quasi) anti-dark solitons, W-shaped solitons and M-shaped solitons which exhibit stationary feature. According to the analysis of the group velocity difference, we give the corresponding conversion rule and present the explicit correspondence of phase diagram of wave numbers for various converted waves, by which we show the gradient relation among these converted waves. Further, by separating the converted waves into the solitary wave as well as the periodic wave, we classify different kinds of nonlinear waves and indicate the difference of the superposition mechanism among them. We show that the breather and various converted waves are formed by different superposition modes between the solitary wave components with different localities and periodic wave components with different frequencies. By virtue of the second-order solutions, we consider all possible superposition situations of two nonlinear waves and present the corresponding nonlinear wave complexes. In particular, for the hybrid structure made of a breather and a nonlinear wave with variable velocity, we then discover that the nonlinear wave does not change its state under the conversion condition, leading to that an additional breathing structure or a dark structure is contained in the converted waves. Finally, we unveil the underlying relationship between the conversion and modulation instability.Sharp decay estimates for 3D incompressible MHD system with mixed partial dissipation and magnetic diffusionhttps://zbmath.org/1508.350992023-05-31T16:32:50.898670Z"Zheng, Dahao"https://zbmath.org/authors/?q=ai:zheng.dahao"Li, Jingna"https://zbmath.org/authors/?q=ai:li.jingnaSummary: This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. By using the energy methods, we obtain that this system possesses a global solution when the initial data is small in \(H^3( \mathbb{R}^3)\). In addition, the optimal decay rates of this global solution and its first-order derivatives are established by bounding two time-weighted norms and applying a bootstrapping argument.Solvability of the initial-boundary value problem for the high-order Oldroyd modelhttps://zbmath.org/1508.351002023-05-31T16:32:50.898670Z"Zvyagin, V. G."https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich"Orlov, V. P."https://zbmath.org/authors/?q=ai:orlov.vladimir-p"Turbin, M. V."https://zbmath.org/authors/?q=ai:turbin.mikhail-vSummary: In this paper we consider the solvability in the weak sense of the initial-boundary value problem for the high-order Oldroyd model. For the considered model through the Laplace transform, from the rheological relation, the stress tensor is expressed. After its substitution into the motion equations, the initial-boundary value problem is obtained for an integro-differential equation with a memory along trajectories of the velocity field. After that, through the approximating-topological approach to the study of hydrodynamic problems, the existence of a weak solution is proved. In the proof of the assertions, properties of regular Lagrangian flows are essentially used.Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equationhttps://zbmath.org/1508.351032023-05-31T16:32:50.898670Z"Bao, Weizhu"https://zbmath.org/authors/?q=ai:bao.weizhu"Cai, Yongyong"https://zbmath.org/authors/?q=ai:cai.yongyong"Feng, Yue"https://zbmath.org/authors/?q=ai:feng.yueSummary: We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger equation with small potential and the nonlinear Schrödinger equation (NLSE) with weak nonlinearity. For the Schrödinger equation with small potential characterized by a dimensionless parameter \(\varepsilon \in (0, 1]\), we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in \(L^2\)-norm to prove a uniform error bound at time \(t_\varepsilon =t/\varepsilon\) as \(C(t)\widetilde{C}(T)(h^m +\tau^2)\) up to \(t_\varepsilon \leq T_\varepsilon = T/\varepsilon\) for any \(T>0\) and uniformly for \(\varepsilon \in (0,1]\), while \(h\) is the mesh size, \(\tau\) is the time step, \(m \geq 2\) and \(\tilde{C}(T)\) (the local error bound) depend on the regularity of the exact solution, and \(C(t) = C_0 + C_1t\) grows at most linearly with respect to \(t\) with \(C_0\) and \(C_1\) two positive constants independent of \(T\), \(\varepsilon\), \(h\) and \(\tau\). Then by introducing a new technique of regularity compensation oscillation (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform (w.r.t \(\varepsilon)\) error bound at \(O(h^{m-1} + \varepsilon \tau^2)\) is established in \(H^1\)-norm for the long-time dynamics up to the time at \(O(1/\varepsilon)\) of the Schrödinger equation with \(O(\varepsilon)\)-potential with \(m \geq 3\). Moreover, the RCO technique is extended to prove an improved uniform error bound at \(O(h^{m-1} + \varepsilon^2 \tau^2)\) in \(H^1\)-norm for the long-time dynamics up to the time at \(O(1/\varepsilon^2)\) of the cubic NLSE with \(O(\varepsilon^2)\)-nonlinearity strength. Extensions to the first-order and fourth-order time-splitting methods are discussed. Numerical results are reported to validate our error estimates and to demonstrate that they are sharp.Resonance, fusion and fission dynamics of bifurcation solitons and hybrid rogue wave structures of Sawada-Kotera equationhttps://zbmath.org/1508.351052023-05-31T16:32:50.898670Z"Ahmad, Shabir"https://zbmath.org/authors/?q=ai:ahmad.shabir"Saifullah, Sayed"https://zbmath.org/authors/?q=ai:saifullah.sayed"Khan, Arshad"https://zbmath.org/authors/?q=ai:khan.arshad-ali|khan.arshad-ahmad|khan.arshad-alam|khan.arshad-m"Wazwaz, Abdul Majid"https://zbmath.org/authors/?q=ai:wazwaz.abdul-majidSummary: The goal of this article is to investigate brand-new exact solutions to the Sawada-Kotera (SK) equation. To get at a general form solution to the SK equation, the Hirota bilinear method is used. In order to study the novel soliton solutions, several dispersion coefficients are adopted. For the first time in the literature, we analyze novel multiple-bifurcated soliton and novel rogue wave solutions of the SK equation. We study the interaction of dark soliton with rogue waves and its resonance. Another type of hybrid novel rogue wave solution is investigated, which is the fusion of bright X-shaped and rogue wave. All results are validated and displayed through graphs by using MATLAB-2020.Lie symmetries and traveling wave solutions of the 3D Benney-Roskes/Zakharov-Rubenchik systemhttps://zbmath.org/1508.351102023-05-31T16:32:50.898670Z"Gönül, Şeyma"https://zbmath.org/authors/?q=ai:gonul.seyma"Özemir, Cihangir"https://zbmath.org/authors/?q=ai:ozemir.cihangir(no abstract)Families of fundamental and vortex solitons under competing cubic-quintic nonlinearity with complex potentialshttps://zbmath.org/1508.351132023-05-31T16:32:50.898670Z"Weng, Yuanhang"https://zbmath.org/authors/?q=ai:weng.yuanhang"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.11|wang.hong.5|wang.hong.3|wang.hong.4"Huang, Jing"https://zbmath.org/authors/?q=ai:huang.jingSummary: We investigate the properties of two-dimensional (2D) fundamental and vortex solitons propagating in PT symmetric lattices with the competing cubic-quintic nonlinearity. We discuss the influence of the competing nonlinearity and the gain-loss coefficient on the existence and stability of both 2D fundamental solitons and vortex solitons, and obtain the existence and stability ranges of them. The stability of solitons under different competing CQ nonlinearity is analyzed by using the VK criterion or the anti-VK criterion. We demonstrate that the whole nonlinearity is determined by the coefficients of nonlinear terms and the propagation constants of solitons. In particular, the power curve of vortex soliton leap and move back near the Bloch band instead of bifurcating from it under the defocusing cubic and focusing quintic nonlinearity.The IVP for a periodic generalized ZK equationhttps://zbmath.org/1508.351152023-05-31T16:32:50.898670Z"Albarracin, Carolina"https://zbmath.org/authors/?q=ai:albarracin.carolina"Rodriguez-Blanco, Guillermo"https://zbmath.org/authors/?q=ai:rodriguez-blanco.guillermoSummary: We establish one result over local well-posedness for the Cauchy problem associated to the dispersion generalized Zakharov-Kuznetsov equation
\[
\partial_t u - \partial_x \Big(D_x^{1 + \alpha} \pm D_y^{1 + \beta}\Big) u + uu_x = 0,
\]
in bi-periodic Sobolev spaces \(H^s \Big( \mathbb{T}^2 \Big)\), \(s > \frac{3}{2} - \frac{1}{2^{\alpha + 2}} + \frac{ \lceil \beta \rceil}{2 (1 + \beta)}\).On boundary layers for the Burgers equations in a bounded domainhttps://zbmath.org/1508.351182023-05-31T16:32:50.898670Z"Choi, Junho"https://zbmath.org/authors/?q=ai:choi.junho"Jung, Chang-Yeol"https://zbmath.org/authors/?q=ai:jung.changyeol"Lee, Hoyeon"https://zbmath.org/authors/?q=ai:lee.hoyeonSummary: As a simplified model derived from the Navier-Stokes equations, we consider the viscous Burgers equations in a bounded domain with two-point boundary conditions. We investigate the singular behaviors of their solutions \(u^\varepsilon\) as the viscosity parameter \(\varepsilon\) gets smaller. The idea is constructing the asymptotic expansions in the order of the \(\varepsilon\) and validating the convergence of the expansions to the solutions \(u^\varepsilon\) as \(\varepsilon\rightarrow 0\). In this article, we consider the case where sharp transitions occur at the boundaries, i.e. boundary layers, and we fully analyze the convergence at any order of \(\varepsilon\) using the so-called boundary layer correctors. We also numerically verify the convergences.On 1d quadratic Klein-Gordon equations with a potential and symmetrieshttps://zbmath.org/1508.351202023-05-31T16:32:50.898670Z"Germain, Pierre"https://zbmath.org/authors/?q=ai:germain.pierre"Pusateri, Fabio"https://zbmath.org/authors/?q=ai:pusateri.fabio"Zhang, Katherine Zhiyuan"https://zbmath.org/authors/?q=ai:zhang.katherine-zhiyuanSummary: This paper is a continuation of the previous work [Forum Math. Pi 10, Paper No. e17, 172 p. (2022; Zbl 1495.35126)] by the first two authors. We focus on 1-dimensional quadratic Klein-Gordon equations with a potential, under some assumptions that are less general than (Pusateri, in: Forum of mathematics, Cambridge University Press), but that allow us to present some simplifications in the proof of the global existence with decay for small solutions. In particular, we can propagate a stronger control on a basic \(L^2\)-weighted-type norm while providing some shorter and less technical proofs for some of the arguments.Modulation equations near the Eckhaus boundary: the KdV equationhttps://zbmath.org/1508.351222023-05-31T16:32:50.898670Z"Haas, Tobias"https://zbmath.org/authors/?q=ai:haas.tobias"de Rijk, Björn"https://zbmath.org/authors/?q=ai:de-rijk.bjorn"Schneider, Guido"https://zbmath.org/authors/?q=ai:schneider.guidoSummary: We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation \(\partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta) \Psi |\Psi|^2\) near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters \(\alpha, \beta \), a number of modulation equations can be derived, such as the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau based amplitude equations. Here we establish error estimates showing that the Korteweg-de Vries (KdV) approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping, we work in spaces of analytic functions.Smoothing properties for a two-dimensional Kawahara equationhttps://zbmath.org/1508.351242023-05-31T16:32:50.898670Z"Levandosky, Julie L."https://zbmath.org/authors/?q=ai:levandosky.julie-lSummary: In this paper we study smoothness properties of solutions to a two-dimensional Kawahara equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data \(\phi\) possesses certain regularity and sufficient decay as \(x \to \infty \), then the solution \(u(t)\) will be smoother than \(\phi\) for \(0 < t \leq T\) where \(T\) is the existence time of the solution.Breather-soliton molecules and breather-positons for the extended complex modified KdV equationhttps://zbmath.org/1508.351282023-05-31T16:32:50.898670Z"Lv, Nannan"https://zbmath.org/authors/?q=ai:lv.nannan"Huang, Lin"https://zbmath.org/authors/?q=ai:huang.lin.1|huang.linSummary: In this paper, we investigate an extended complex modified Korteweg-de Vries equation. On zero background, based on Darboux transformation, breather solutions, breather molecules, breather-soliton molecules and breather-positons for extended complex modified Korteweg-de Vries equation are obtained by module resonance, velocity resonance and degenerate Darboux transformation. On nonzero background, breather solutions and breather-positons for extended complex modified Korteweg-de Vries equation are generated through Darboux transformation and degenerate Darboux transformation, respectively.Properties of synchronous collisions of solitons in the Korteweg-de Vries equationhttps://zbmath.org/1508.351342023-05-31T16:32:50.898670Z"Tarasova, Tatiana V."https://zbmath.org/authors/?q=ai:tarasova.tatiana-v"Slunyaev, Alexey V."https://zbmath.org/authors/?q=ai:slunyaev.alexey-vSummary: Synchronous collisions of solitons of the Korteweg-de Vries equation are considered as a representative example of the interaction of a large number of solitons in a soliton gas. Statistical properties of the soliton field are examined for a model distribution of soliton amplitudes according to a power law. \(N\)-soliton solutions \((N\leq 50)\) are constructed with the help of a numerical procedure using the Darboux transformation and 100-digits arithmetic. It is shown that there exist qualitatively different patterns of evolving multisoliton solutions depending on the amplitude distribution. Collisions of a large number of solitons lead to the decrease of values of statistical moments (the orders from 3 to 7 have been considered). The statistical moments are shown to exhibit long intervals of quasi-stationary behavior in the case of a sufficiently large number of interacting solitons with close amplitudes. These intervals can be characterized by the maximum value of the soliton gas density and by ``smoothing'' of the wave fields in integral sense. The analytical estimates describing these degenerate states of interacting solitons are obtained.Behaviour of the extended modified Volterra lattice-reductions to generalised mKdV and NLS equationshttps://zbmath.org/1508.351362023-05-31T16:32:50.898670Z"Wattis, Jonathan A. D."https://zbmath.org/authors/?q=ai:wattis.jonathan-a-d"Gordoa, Pilar R."https://zbmath.org/authors/?q=ai:gordoa.pilar-ruiz"Pickering, Andrew"https://zbmath.org/authors/?q=ai:pickering.andrewSummary: We consider the first member of an extended modified Volterra lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the modified Korteweg-de Vries and nonlinear Schrödinger equations.Soliton solution and asymptotic analysis of the three-component Hirota-Satsuma coupled KdV equationhttps://zbmath.org/1508.351372023-05-31T16:32:50.898670Z"Zhang, Ling-Ling"https://zbmath.org/authors/?q=ai:zhang.lingling"Wang, Xin"https://zbmath.org/authors/?q=ai:wang.xin.44Summary: In this paper, we study a class of Hirota-Satsuma coupled KdV equations that can be used to describe the interaction of two classes of long waves. By using the Hirota bilinear method, the 1, 2, 3-soliton solutions are obtained. On this basis, the asymptotic analysis of soliton solutions proves that the collisions between solitons are elastic, and a set of visual figure is given to illustrate the results.Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaceshttps://zbmath.org/1508.351392023-05-31T16:32:50.898670Z"An, JinMyong"https://zbmath.org/authors/?q=ai:an.jinmyong"Kim, JinMyong"https://zbmath.org/authors/?q=ai:kim.jinmyong"Ryu, PyongJo"https://zbmath.org/authors/?q=ai:ryu.pyongjoThis paper studies the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation
\begin{align*}
&iu_t+\Delta^2 u=\lambda |x|^{-b}|u|^\sigma u,\ (t,x)\in\mathbb{R}\times\mathbb{R}^d\\
&u(0,x)=u_0(x)\in H^s(\mathbb{R}^d),
\end{align*}
where \(d\in \mathbb{N}\), \(s \geq 0\), \(0<b<4\), \(\sigma>0\) and \(\lambda\in \mathbb{R}\). This paper investigates the well-posedness for the IBNLS equation in Sobolev spaces \(H^s(\mathbb{R}^d)\) by establishing the various delicate nonlinear estimates and using the contraction mapping principle combined with the Strichartz estimates. This type of local well-posedness result improves the ones of \textit{C. M. Guzmán} and \textit{A. Pastor} [Nonlinear Anal., Real World Appl. 56, Article ID 103174, 35 p. (2020; Zbl 1451.35185)] and \textit{X. Liu} and \textit{T. Zhang} [J. Differ. Equations 296, 335--368 (2021; Zbl 1476.35065)] by extending the validity of \(s\) and \(b\).
Reviewer: Shuangjie Peng (Wuhan)Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial casehttps://zbmath.org/1508.351402023-05-31T16:32:50.898670Z"Cardoso, Mykael"https://zbmath.org/authors/?q=ai:cardoso.mykael"Farah, Luiz Gustavo"https://zbmath.org/authors/?q=ai:farah.luiz-gustavoSummary: In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation
\[
i \partial_t u +\Delta u +|x|^{-b} |u|^{2\sigma }u = 0, \quad x \in{\mathbb{R}}^N
\]
with \(N\ge 3\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\) satisfies \(0<s_c<1\). In a previous work [J. Funct. Anal. 281, No. 8, Article ID 109134, 38 p. (2021; Zbl 1473.35498)], for radial initial data in \(\dot{H}^{s_c}\cap \dot{H}^1\), we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in \(H^1\).Well-posedness of dispersion managed nonlinear Schrödinger equationshttps://zbmath.org/1508.351412023-05-31T16:32:50.898670Z"Choi, Mi-Ran"https://zbmath.org/authors/?q=ai:choi.mi-ran"Hundertmark, Dirk"https://zbmath.org/authors/?q=ai:hundertmark.dirk"Lee, Young-Ran"https://zbmath.org/authors/?q=ai:lee.young-ranThe authors consider the following dispersion managed nonlinear Schrödinger equation, also known as Gabitov-Turitsyn equation,
\[
i\partial_t u +d_{\mathrm{av}}\partial_x^2 u +\int_{\mathbb R}T_r^{-1}\left( P(T_ru)\right)\psi(r)dr,\quad u_{\mid t=0}=u_0,\tag{1} \]
where \(x\in \mathbb R\), \(d_{\mathrm{av}}\in \mathbb R\) is a constant which may or may not be zero according to the origin of the model in optics, \(T_r=e^{ir\partial_x^2}\) is the Schrödinger evolution group at time \(r\), \(P(z)=h(|z|)z\) is the nonlinearity, and \(\psi\ge 0\) is a probability density (hence \(\psi\in L^1(\mathbb R)\)), connected to the local periodic dispersion profile in optics. According to various assumptions on the nonlinearity \(h\), the average dispersion \(d_{\mathrm{av}}\) (\(d_{\mathrm{av}}=0\) or \(d_{\mathrm{av}}\not=0\)), and possibly extra integrability conditions on \(\psi\), the authors first show several results regarding the local and the global Cauchy problem (1), for \(u_0\in L^2(\mathbb R)\) or \(u_0\in H^1(\mathbb R)\). The nonlinear term is estimated thanks to (homogeneous) Strichartz estimates, and Hölder inequality. Based on the nonlinear estimates, the Cauchy problem is studied via a fixed point argument based on Duhamel's formula. Globalization is based on the conservation of the mass (\(\|u(t)\|_{L^2}^2\)) and of the energy
\begin{align*}
E(u(t))&=\frac{d_{\mathrm{av}}}{2}\|\partial_x u(t)\|_{L^2}^2- \iint_{\mathbb R^2} V\left(|T_r(u(t)|\right)dx\psi(r)dr,\\
&\text{where } V(a)=\int_0^aP(s)ds=\int_0^ah(s)sds.
\end{align*}
In a final section, the authors analyze the set of energy minimizers,
\[
S_\lambda^{d_{\mathrm{av}}}=\{f\in X,\quad E(f) = E_\lambda^{d_{\mathrm{av}}},\ \|f\|_{L^2}^2=\lambda\},
\]
where
\[
E_\lambda^{d_{\mathrm{av}}}=\inf\{ E(f),\ f\in X,\ \|f\|_{L^2}^2=\lambda\},
\]
with \(X=L^2\) in the absence of dispersion \(d_{\mathrm{av}}=0\), and \(X=H^1\) otherwise (in which case the signs of \(d_{\mathrm{av}}\) and the nonlinearity \(P\) must be related adequately so the above variational problem is nontrivial). Under yet different assumptions on the nonlinearity and the weight \(\psi\), the authors establish the existence of minimizers (ground states), and their orbital stability under the flow of (1), following the general strategy of Cazenave and Lions.
Reviewer: Rémi Carles (Rennes)Small energy stabilization for 1D nonlinear Klein Gordon equationshttps://zbmath.org/1508.351422023-05-31T16:32:50.898670Z"Cuccagna, Scipio"https://zbmath.org/authors/?q=ai:cuccagna.scipio"Maeda, Masaya"https://zbmath.org/authors/?q=ai:maeda.masaya"Scrobogna, Stefano"https://zbmath.org/authors/?q=ai:scrobogna.stefanoSummary: We give a partial extension to dimension 1 of the result proved by \textit{D. Bambusi} and \textit{S. Cuccagna} [Am. J. Math. 133, No. 5, 1421--1468 (2011; Zbl 1237.35115)] on the absence of small energy real valued periodic solutions for the NLKG in dimension 3. We combine the framework in [\textit{M. Kowalczyk} and \textit{Y. Martel}, ``Kink dynamics under odd perturbations for \((1 + 1)\)-scalar field models with one internal mode'', Preprint, \url{arXiv:2203.04143}] with the notion of ``refined profile''.On the instability of standing waves for 3D dipolar Bose-Einstein condensateshttps://zbmath.org/1508.351442023-05-31T16:32:50.898670Z"Dinh, Van Duong"https://zbmath.org/authors/?q=ai:dinh.van-duongSummary: We consider the nonlinear Schrödinger equation describing the dipolar Bose-Einstein condensates with and without harmonic trapping potential. We first show the existence and strong instability of ground state standing waves for the equation in the case of no external potential. We next prove the strong instability of ground state standing waves for the equation in the presence of a general harmonic potential, under a suitable assumption. These results are improvements of our recent results [with \textit{L. Forcella} and \textit{H. Hajaiej}, Commun. Math. Sci. 20, No. 1, 165--200 (2022; Zbl 1479.35775)] whose proofs require an additional condition. Finally we prove the existence and orbital stability of prescribed mass standing waves for the equation with harmonic potential. The proof of the latter result contains some improvements of previous literature.The low energy scattering for nonlinear Schrödinger equationhttps://zbmath.org/1508.351462023-05-31T16:32:50.898670Z"Fang, Conghui"https://zbmath.org/authors/?q=ai:fang.conghui"Han, Zheng"https://zbmath.org/authors/?q=ai:han.zhengSummary: In this article, we consider the nonlinear Schrödinger equation
\[
\begin{cases}
iu_t +\Delta u+\lambda |u|^{\alpha}u=0, \\
u(0,x)=u_0.
\end{cases}
\]
on \(\mathbb{R}^n\) with \(n\geq 4\) and \(\lambda \in\mathbb{C}\). We prove that if \(\frac{8}{n+\sqrt{n^2 +16n+32}}<\alpha <\frac{4}{n}\), then there exists a unique, global solution and it scatters as \(t\rightarrow \infty\).The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superpositionhttps://zbmath.org/1508.351472023-05-31T16:32:50.898670Z"Ferriere, Guillaume"https://zbmath.org/authors/?q=ai:ferriere.guillaumeSummary: We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson -- a time-independent Gaussian function -- is an orbitally stable solution. In the general case in dimension \(d=1\), the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in \(L^2\)) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions.On threshold solutions of the equivariant Chern-Simons-Schrödinger equationhttps://zbmath.org/1508.351552023-05-31T16:32:50.898670Z"Li, Zexing"https://zbmath.org/authors/?q=ai:li.zexing"Liu, Baoping"https://zbmath.org/authors/?q=ai:liu.baopingSummary: We consider the self-dual Chern-Simons-Schrödinger model in two spatial dimensions. This problem is \(L^2\)-critical. Under the equivariant setting, global well-posedness and scattering were proved in [\textit{B. Liu} and \textit{P. Smith}, Rev. Mat. Iberoam. 32, No. 3, 751--794 (2016; Zbl 1352.35161)] for a solution with initial charge below a certain threshold given by the ground state. In this work, we show that the only nonscattering solutions with threshold charge are exactly the ground state up to scaling, phase rotation and the pseudoconformal transformation. We also obtain a partial result for the non-self-dual system.The focusing NLS equation with step-like oscillating background: the genus \(3\) sectorhttps://zbmath.org/1508.351562023-05-31T16:32:50.898670Z"Monvel, Anne Boutet de"https://zbmath.org/authors/?q=ai:boutet-de-monvel.anne-marie"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatan"Shepelsky, Dmitry"https://zbmath.org/authors/?q=ai:shepelsky.dmitryIn this work, the authors investigate the long-time behavior of the solutions of the focusing nonlinear Schrödinger equation on the real line, generated from a Cauchy data close to plane waves at infinity. Their main goal consists to perform a complete asymptotic analysis according to \(\xi= x/t\) of the solutions of the Cauchy problem
\[
\left\{ \begin{array}{rcl} i q_t + q_{xx} + 2 |q|^ {2} q& = &0\\
{ q}{}_{|t=0} &= &q_0, \end{array} \right.
\]
with \(q_0\) exhibiting the following asymptotic at infinity
\[
q_0(x) \sim \left\{ \begin{array}{rcl} A_1 e^{i \varphi_1} e^{-2i B_1 x}, & x \to + \infty& \\
A_2 e^{i \varphi_2} e^{-2i B_2 x}, & x \to - \infty.& \end{array} \right.
\]
More precisely, the authors provide uniformly in a sector \(\xi_1\leq \xi \leq \xi_2\) the long-time asymptotic of \(q\), up to a remainder term, with a leading term explicitly expressed by means of hyperbolic theta functions and a sub-leading term involving parabolic cylinder and Airy functions.
Reviewer: Hajer Bahouri (Paris)Logarithmic Schrödinger equations in infinite dimensionshttps://zbmath.org/1508.351582023-05-31T16:32:50.898670Z"Read, Larry"https://zbmath.org/authors/?q=ai:read.larry"Zegarliński, Bogusław"https://zbmath.org/authors/?q=ai:zegarlinski.boguslaw"Zhang, Mengchun"https://zbmath.org/authors/?q=ai:zhang.mengchunSummary: We study the logarithmic Schrödinger equation with a finite range potential on \(\mathbb{R}^{\mathbb{Z}^d}\). Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.
{\copyright 2022 American Institute of Physics}Three-component coupled nonlinear Schrödinger system in a multimode optical fiber: Darboux transformation induced via a rank-two projection matrixhttps://zbmath.org/1508.351602023-05-31T16:32:50.898670Z"Tian, He-Yuan"https://zbmath.org/authors/?q=ai:tian.he-yuan"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Sun, Yan"https://zbmath.org/authors/?q=ai:sun.yan"Zhang, Chen-Rong"https://zbmath.org/authors/?q=ai:zhang.chen-rongSummary: In this paper, the investigation is on a three-component coupled nonlinear Schrödinger (NLS) system which governs the wave evolution in a multimode optical fiber. We construct a Darboux transformation (DT) induced via a rank-two projection matrix, and then derive an \((N,m)\)-generalized DT and the \(N\)th-order solution, where the positive integers \(N\) and \(m\) denote the numbers of iterative times and distinct spectral parameters, respectively. Focusing on the \(N\)th-order solution on the nonzero-zero-zero background, we derive two kinds of waves which could not be derived by the DT induced via a rank-one projection matrix, that is, the so-called fundamental nonlinear wave \((N=1)\) and degenerate fundamental nonlinear wave \((N=2\) and \(m=1)\). Via the asymptotic analysis, we find that the fundamental nonlinear wave is the nonlinear superposition of two dark-bright-bright solitons and a breather/Kuznetsov-Ma breather/rogue wave; and the degenerate fundamental nonlinear wave is the nonlinear superposition of four dark-bright-bright solitons and two breathers/two Kuznetsov-Ma breathers/a second-order rogue wave. Since such phenomena are not admitted for the one-component NLS equation and two-component coupled NLS system, they are more useful to understand the three-component coupled NLS system than what the latter two models admit. For other three-component coupled systems, more phenomena may be expected when the rank of projection matrix used to construct a DT is two rather than one, because our study presents an example.Defocusing \(\dot{H}^{\frac{ 1}{ 2}} \)-critical inhomogeneous nonlinear Schrödinger equationshttps://zbmath.org/1508.351612023-05-31T16:32:50.898670Z"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.16|wang.ying.30|wang.ying.2|wang.ying.9|wang.ying.8|wang.ying.12|wang.ying.7"Xu, Chengbin"https://zbmath.org/authors/?q=ai:xu.chengbinSummary: In this paper, we study the critical norm problem for the defocusing inhomogeneous nonlinear Schrödinger equations (INLS). As a first attempt, we consider the defocusing \(\dot{H}^{\frac{1}{2}} \)-critical (INLS) \(i u_t + \Delta u = | x |^{- b} | u |^{p - 1} u\) with \(p = 1 + \frac{2 (2 - b)}{d - 1}\) and \(0 < b < \frac{ 1}{ 2}\) in dimensions \(d \geq 5\). We utilize the concentration-compactness/rigidity method as in [\textit{C. E. Kenig} and \textit{F. Merle}, Trans. Am. Math. Soc. 362, No. 4, 1937--1962 (2010; Zbl 1188.35180)] and [\textit{J. Murphy}, Discrete Contin. Dyn. Syst. 34, No. 2, 733--748 (2014; Zbl 1295.35368)] to show that if a solution \(u\) is uniformly bounded in the critical space \(\dot{H}_x^{\frac{1}{2}}( \mathbb{R}^d)\) throughout its lifespan, then the solution \(u\) must be global and scatter. The key ingredients of the proof include Lin-Strauss Morawetz estimate and inhomogeneous Strichartz estimate.Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiberhttps://zbmath.org/1508.351642023-05-31T16:32:50.898670Z"Zhang, Chen-Rong"https://zbmath.org/authors/?q=ai:zhang.chen-rong"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Qu, Qi-Xing"https://zbmath.org/authors/?q=ai:qu.qixing"Liu, Lei"https://zbmath.org/authors/?q=ai:liu.lei.2"Tian, He-Yuan"https://zbmath.org/authors/?q=ai:tian.he-yuanSummary: In the fiber communication domain, people are facing the challenges due to the rapidly growing requirement on the capacity from new functions and services. Multi-hump solitons are therefore noticed and studied on the feasibility of improving the capacity of the optical fiber communication. In this paper, we study the vector bright solitons and their interaction properties of the coupled Fokas-Lenells system, which models the femtosecond optical pulses in a birefringent optical fiber. We derive the so-called degenerate and nondegenerate vector solitons associated with the one and two eigenvalues, respectively, and the latter admits the symmetric profile. Asymptotically and graphically, interaction patterns of such solitons are classified as follows: Interactions between the degenerate solitons can be elastic or inelastic, reflecting the intensity redistribution between the two components; Interactions between the degenerate and nondegenerate solitons are inelastic, which make the nondegenerate solitons maintaining or losing the profiles in the different situations; Interactions between the nondegenerate solitons do not cause the intensity redistribution, while their shapes change slightly or remain unchanged.On the blow-up solutions for the nonlinear radial Schrödinger equations with spatial variable coefficientshttps://zbmath.org/1508.351652023-05-31T16:32:50.898670Z"Zheng, Bowen"https://zbmath.org/authors/?q=ai:zheng.bowenSummary: We study a generalized nonlinear Schrödinger equations with spatial variable coefficients, which models the remarkable inhomogeneous Schrödinger maps (ISM). A new weighted Sobolev space \(\mathcal{W}^{1,q}(\mathbb{R}^+)\) is introduced and the existence of blow-up solutions of this equations, including the integrable radial ISM, with the initial data in \(\mathcal{W}^{1,2}(\mathbb{R}^+)\) is proved.On quasilinear Maxwell equations in two dimensionshttps://zbmath.org/1508.351772023-05-31T16:32:50.898670Z"Schippa, Robert"https://zbmath.org/authors/?q=ai:schippa.robert"Schnaubelt, Roland"https://zbmath.org/authors/?q=ai:schnaubelt.rolandThe authors establish new Strichartz estimates for the Maxwell equations in two dimensions with rough permittivity. After localizing in space and frequency and using the Fourier-Bros-Iagolnitzer transform to transfer the problem to phase space, they reduce the estimates to prove to dyadic estimates for the half-wave equation. The latter are proved by following the approach proposed by \textit{D. Tataru} [Am. J. Math. 122, No. 2, 349--376 (2000; Zbl 0959.35125); ibid. 123, No. 3, 385--423 (2001; Zbl 0988.35037); J. Am. Math. Soc. 15, No. 2, 419--442 (2002; Zbl 0990.35027)] for the derivation of Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. The authors next use the so obtained Strichartz estimates for proving an improved version of the local well-posedness for quasilinear Maxwell equations in two dimensions. (Previously, well-posedness for hyperbolic systems was obtained by using energy methods.)
Reviewer: Catalin Popa (Iaşi)Linearly stable KAM tori for higher dimensional Kirchhoff equationshttps://zbmath.org/1508.351802023-05-31T16:32:50.898670Z"Chen, Yin"https://zbmath.org/authors/?q=ai:chen.yin|chen.yin.5|chen.yin.2|chen.yin.3|chen.yin.1|chen.yin.4"Geng, Jiansheng"https://zbmath.org/authors/?q=ai:geng.jianshengSummary: We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for higher dimensional Kirchhoff equations with periodic boundary conditions. The proof is based on an improved Kuksin lemma and the special form of the nonlinear term.Correction to: ``The effect of residual stress on the stability of a circular cylindrical tube''https://zbmath.org/1508.351812023-05-31T16:32:50.898670Z"Dorfmann, Luis"https://zbmath.org/authors/?q=ai:dorfmann.luis"Ogden, Ray W."https://zbmath.org/authors/?q=ai:ogden.raymond-wFrom the text: In this note we wish to correct an error in our paper [ibid. 127, Paper No. 9, 20 p. (2021; Zbl 1497.35457)]. The code used in the computation of Figs. 7 and 8 in [loc. cit.] had a small error in computing the values of \(\bar{\nu}\) and \(\bar{a}\) for which bifurcation is possible. This affected the accuracy of the figures, which are replaced with Fig. 1 herein. The difference is not noticeable for a thin walled tube with \(\bar{B} = 1.1\), but becomes more evident with increasing wall thickness. The panels in the first and second columns of Fig. 1 correspond to Figs. 7 and 8 in [loc. cit.], respectively.
The discussion of the results in [loc. cit.] remains largely valid except that now there are no bifurcation curves in the region where \(\bar{a}\geq1\). Figure 1 shows that all bifurcation curves in the strongly elliptic region are in the region defined by \(\bar{a}<1\), which is consistent with the situation where there is no residual stress [\textit{D. M. Haughton} and \textit{R. W. Ogden}, J. Mech. Phys. Solids 27, 489--512 (1979; Zbl 0442.73067)]]. In particular, within the strongly elliptic region bifurcation for an unloaded cylinder is not possible.
We also clarify that Eqs. (75) and (76) in our paper [loc. cit.] show the explicit forms of the first derivative of \(\alpha\) and the first and second derivatives of \(\gamma\) for the case \(\kappa = 0\) only. The corresponding expressions for \(\bar{\kappa}\neq 0\) were not included, but are easily obtained.Dynamics for a plate equation with nonlinear damping on time-dependent spacehttps://zbmath.org/1508.351852023-05-31T16:32:50.898670Z"Zhang, Penghui"https://zbmath.org/authors/?q=ai:zhang.penghui"Yang, Lu"https://zbmath.org/authors/?q=ai:yang.luSummary: In this paper, we study the long-time behavior of the following plate equation
\[
\varepsilon (t) u_{tt} + g (u_t) + \Delta^2 u + \lambda u + f (u) = h,
\] where the coefficient \(\varepsilon\) depends explicitly on time, the nonlinear damping and the nonlinearity both have critical growths.Long-time behavior of the higher-order anisotropic Caginalp phase-field systems based on the Cattaneo lawhttps://zbmath.org/1508.351862023-05-31T16:32:50.898670Z"Ntsokongo, Armel Judice"https://zbmath.org/authors/?q=ai:ntsokongo.armel-judice"Tathy, Christian"https://zbmath.org/authors/?q=ai:tathy.christianSummary: The aim of this paper is to study higher-order Caginalp phase-field systems based on the Maxwell-Cattaneo law, instead of the classical Fourier law. More precisely, one obtains well-posedness results, as well as the existence of finite-dimensional attractors.Uniform convergence to equilibrium for a family of drift-diffusion models with trap-assisted recombination and the limiting Shockley-Read-Hall modelhttps://zbmath.org/1508.351872023-05-31T16:32:50.898670Z"Fellner, Klemens"https://zbmath.org/authors/?q=ai:fellner.klemens"Kniely, Michael"https://zbmath.org/authors/?q=ai:kniely.michaelSummary: In this paper, we establish convergence to equilibrium for a drift-diffusion-recombination system modelling the charge transport within certain semiconductor devices. More precisely, we consider a two-level system for electrons and holes which is augmented by an intermediate energy level for electrons in so-called trapped states. The recombination dynamics use the mass action principle by taking into account this additional trap level. The main part of the paper is concerned with the derivation of an entropy-entropy production inequality, which entails exponential convergence to the equilibrium via the so-called entropy method. The novelty of our approach lies in the fact that the entropy method is applied uniformly in a fast-reaction parameter which governs the lifetime of electrons on the trap level. Thus, the resulting decay estimate for the densities of electrons and holes extends to the corresponding quasi-steady-state approximation.Steady-states solutions of the Vlasov-Maxwell-Fokker-Planck system of proton channeling in crystalshttps://zbmath.org/1508.351892023-05-31T16:32:50.898670Z"Bobrovskiy, V. S."https://zbmath.org/authors/?q=ai:bobrovskiy.v-s"Kazakov, A. L."https://zbmath.org/authors/?q=ai:kazakov.aleksandr-leonidovich"Rojas, E. M."https://zbmath.org/authors/?q=ai:rojas.edixon-m"Sinitsyn, A. V."https://zbmath.org/authors/?q=ai:sinitsyn.aleksandr-vladimirovich"Spevak, L. F."https://zbmath.org/authors/?q=ai:spevak.lev-fridrikhovichSummary: The paper studies special classes of the stationary solutions of the generalized Vlasov-Maxwell-Fokker-Planck (VMFP) system. We reduce the VMFP equations to a nonlinear elliptic system with exponential nonlinearities. For the Vlasov-Poisson-Fokker-Planck system a new form of stationary states is obtained which generalizes the known ones from the works of \textit{K. Dressler} [Math. Methods Appl. Sci. 9, 169--176 (1987; Zbl 0632.35066)] and \textit{R. Glassey} et al. [J. Math. Anal. Appl. 202, No. 3, 1058--1075 (1996; Zbl 0867.35026)]. We consider the one-dimensional case of the elliptic equations, corresponding to the axial symmetry of a crystal. For the associated boundary value problem, the existence of at least one solution is proved by the lower-upper solution method. Besides, we propose an iterative algorithm and perform illustrative numerical calculations. The numerical results are compared with our upper-lower solutions.Propagation of regularity and long time behavior of the \(3D\) Massive relativistic transport equation. II: Vlasov-Maxwell systemhttps://zbmath.org/1508.351902023-05-31T16:32:50.898670Z"Wang, Xuecheng"https://zbmath.org/authors/?q=ai:wang.xuechengSummary: Given any smooth, suitably small initial data, which decays polynomially at infinity, we prove global regularity for the \(3D\) relativistic massive Vlasov-Maxwell system. In particular, the compact support assumption, which was widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector field method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov-Maxwell system.Correction to: ``Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials''https://zbmath.org/1508.351922023-05-31T16:32:50.898670Z"Colli, Pierluigi"https://zbmath.org/authors/?q=ai:colli.pierluigi"Signori, Andrea"https://zbmath.org/authors/?q=ai:signori.andrea"Sprekels, Jürgen"https://zbmath.org/authors/?q=ai:sprekels.jurgenCorrects several typos and misorderings in the authors' paper [ibid. 83, No. 3, 2017--2049 (2021; Zbl 1486.35392)].Well-posedness and singularity formation for inviscid Keller-Segel-fluid system of consumption typehttps://zbmath.org/1508.351942023-05-31T16:32:50.898670Z"Jeong, In-Jee"https://zbmath.org/authors/?q=ai:jeong.in-jee"Kang, Kyungkeun"https://zbmath.org/authors/?q=ai:kang.kyungkeunSummary: We consider the Keller-Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its \(C^2\)-norm with certain initial data.The efficient fractional order based approach to analyze chemical reaction associated with pattern formationhttps://zbmath.org/1508.351972023-05-31T16:32:50.898670Z"Veeresha, P."https://zbmath.org/authors/?q=ai:veeresha.pundikala(no abstract)Turing-Hopf bifurcation co-induced by cross-diffusion and delay in Schnakenberg systemhttps://zbmath.org/1508.351982023-05-31T16:32:50.898670Z"Yang, Rui"https://zbmath.org/authors/?q=ai:yang.rui.1|yang.rui(no abstract)Stability of a fractional advection-diffusion system with conformable derivativehttps://zbmath.org/1508.352002023-05-31T16:32:50.898670Z"Arfaoui, Hassen"https://zbmath.org/authors/?q=ai:arfaoui.hassen"Ben Makhlouf, Abdellatif"https://zbmath.org/authors/?q=ai:ben-makhlouf.abdellatif(no abstract)Nonlinear dispersive wave propagation pattern in optical fiber systemhttps://zbmath.org/1508.352032023-05-31T16:32:50.898670Z"Uddin, M. Hafiz"https://zbmath.org/authors/?q=ai:hafiz-uddin.m"Zaman, U. H. M."https://zbmath.org/authors/?q=ai:zaman.u-h-m"Arefin, Mohammad Asif"https://zbmath.org/authors/?q=ai:arefin.mohammad-asif"Akbar, M. Ali"https://zbmath.org/authors/?q=ai:ali-akbar.m(no abstract)Nonautonomous perturbations of Morse-Smale semigroups: stability of the phase diagramhttps://zbmath.org/1508.370422023-05-31T16:32:50.898670Z"Bortolan, M. C."https://zbmath.org/authors/?q=ai:bortolan.m-c"Carvalho, A. N."https://zbmath.org/authors/?q=ai:nolasco-de-carvalho.alexandre"Langa, J. A."https://zbmath.org/authors/?q=ai:langa.jose-antonio"Raugel, G."https://zbmath.org/authors/?q=ai:raugel.genevieveSummary: In this work we study Morse-Smale semigroups under nonautonomous perturbations, which leads us to introduce the concept of \textit{Morse-Smale evolution processes of hyperbolic type}, associated to nonautonomous evolutionary equations. They are amongst the dynamically gradient evolution processes (in the sense of \textit{A. N. Carvalho} et al. [Attractors for infinite-dimensional non-autonomous dynamical systems. Berlin: Springer (2013; Zbl 1263.37002)]) with a finite number of hyperbolic global solutions, for which the stable and unstable manifolds intersect transversally. We prove the stability of the phase diagram of the attractors for a small continuously differentiable nonautonomous perturbation \(\{T_{\eta}(t,s):(t,s)\in{\mathscr{P}}\}\) of a Morse-Smale semigroup \(\{T(t):t\geqslant 0\}\) with a finite number of hyperbolic equilibria. We present the complete proofs of the local and global \(\lambda\)-lemmas in the infinite-dimensional case. Such results are due to \textit{D. Henry}, presented in his handwritten notes [Invariant manifolds near a fixed point; Manuscripts IME-USP], and are included here for completeness.Symmetries and conservation laws for a generalization of Kawahara equationhttps://zbmath.org/1508.370872023-05-31T16:32:50.898670Z"Vašíček, Jakub"https://zbmath.org/authors/?q=ai:vasicek.jakubSummary: We give a complete description of generalized and formal symmetries for a nonlinear evolution equation which generalizes the Kawahara equation having important applications in the study of plasma waves and capillary-gravity water waves. Using these results and the presence of Hamiltonian structure we also give a complete description of local conservation laws for the equation under study. In particular, we show that the equation in question admits no genuinely generalized symmetries and has only finitely many nontrivial linearly independent local conservation laws, and thus this equation is not symmetry integrable.Continuity of dynamical behaviors for strongly damped wave equations with perturbationhttps://zbmath.org/1508.371022023-05-31T16:32:50.898670Z"Chang, Qingquan"https://zbmath.org/authors/?q=ai:chang.qingquan"Li, Dandan"https://zbmath.org/authors/?q=ai:li.dandanSummary: We explore the convergence of the global attractors for a class of perturbed severely damped wave equations with the Dirichlet boundary condition in the 3D bounded domain. With respect to the perturbation parameter, it is shown that the global attractors are both upper and lower semicontinuous.
{\copyright 2022 American Institute of Physics}Asymptotically autonomous dynamics for non-autonomous stochastic 2D \(g\)-Navier-Stokes equation in regular spaceshttps://zbmath.org/1508.371032023-05-31T16:32:50.898670Z"Xu, Dongmei"https://zbmath.org/authors/?q=ai:xu.dongmei"Li, Fuzhi"https://zbmath.org/authors/?q=ai:li.fuzhiSummary: This work is a continuation of our previous work [\textit{F. Li} et al., Commun. Pure Appl. Anal. 19, No. 6, 3137--3157 (2020; Zbl 1441.37087)] on the regular backward compact random attractor. We prove that under certain conditions, the components of the random attractor of a non-autonomous dynamical system can converge in time to those of the random attractor of the limiting autonomous dynamical system in more regular spaces rather than the basic phase space. As an application of the abstract theory, we show that the backward compact random attractors [\(\cup_{s \leq \tau}\mathcal{A}(s, \omega)\) is precompact for each \(\tau\in\mathbb{R}\)] for the non-autonomous stochastic \(g\)-Navier-Stokes (\(g\)-NS) equation is backward asymptotically autonomous to a random attractor of the autonomous \(g\)-NS equation under the topology of \(H_{0, g}^1(\mathcal{O})^2\).
{\copyright 2022 American Institute of Physics}Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on \(\mathbb{S}^n\)https://zbmath.org/1508.460252023-05-31T16:32:50.898670Z"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhen"Shen, Yansheng"https://zbmath.org/authors/?q=ai:shen.yanshengSummary: In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on \(\mathbb{S}^n\). Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in \(\mathbb{S}^n\), we first employ the Möbius transform between \(\mathbb{S}^n\) and \(\mathbb{R}^n\), poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on \(\mathbb{S}^n\) is equivalent to some integral equation in \(\mathbb{R}^n\). Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on \(\mathbb{S}^n\). As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on \(\mathbb{S}^n\). Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on \(\mathbb{S}^n\).On Dyson-Phillips type approach to differential equations with variable operators in a Banach spacehttps://zbmath.org/1508.470922023-05-31T16:32:50.898670Z"Gil', Michael"https://zbmath.org/authors/?q=ai:gil.michael-iosifSummary: Let \(A(t)\), \(t\ge 0\), be an unbounded variable operator on a Banach space \({\mathcal{X}}\) with a constant dense domain, and \(B(t)\) be a bounded operator in \({\mathcal{X}} \). Assuming that the evolution operator \(U(t, s)\), \(t\ge s\), of the equation \(\text{d}x(t)/\text{d}t=A(t)x(t)\) is known, we built the evolution operator \(\tilde{U}(t,s)\) of the equation \(\text{d}y(t)/\text{d}t=(A(t)+B(t))y(t)\). Besides, we obtain \(C\)-norm estimates for the difference \(\tilde{U}(t,s)-U(t,s)\). We also discuss applications of the obtained estimates to stability of the considered equations. Our results can be considered as a generalization of the well-known Dyson-Phillips theorem for operator semigroups.Existence of conformal metrics with prescribed Q-curvature on manifoldshttps://zbmath.org/1508.530412023-05-31T16:32:50.898670Z"Alghanemi, Azeb"https://zbmath.org/authors/?q=ai:alghanemi.azeb"Chtioui, Hichem"https://zbmath.org/authors/?q=ai:chtioui.hichem"Gdarat, Mohamed"https://zbmath.org/authors/?q=ai:gdarat.mohamedSummary: We consider the problem of existence of conformal metrics with prescribed Q-curvature on riemannian n-dimensional manifolds \(( M^n, g_0), 5 \leq n \leq 7\), not conformally diffeomorphic to the standard sphere \(S^n\). Under the assumptions that \(\operatorname{Q}_{\operatorname{g}_0}\) is semi-positive, \(R_{g_0}\) is non-negative and the prescribed function is flat near its critical points, we study the loss of compactness of the problem and we prove existence results through Euler-Hopf type criteria.Nonlocal flow driven by the radius of curvature with fixed curvature integralhttps://zbmath.org/1508.530962023-05-31T16:32:50.898670Z"Gao, Laiyuan"https://zbmath.org/authors/?q=ai:gao.laiyuan"Pan, Shengliang"https://zbmath.org/authors/?q=ai:pan.shengliang"Tsai, Dong-Ho"https://zbmath.org/authors/?q=ai:tsai.dong-hoSummary: This paper deals with a \(1/\kappa \)-type nonlocal flow for an initial convex closed curve \(\gamma_0\subset{\mathbb{R}}^2\) which preserves the convexity and the integral \(\int_{X\left( \cdot ,t\right) }\kappa^{\alpha +1}ds, \alpha \in \left( -\infty ,\infty \right) ,\) of the evolving curve \(X\left( \cdot ,t\right) \). For \(\alpha \in [1,\infty ),\) it is proved that this flow exists for all time \(t\in [0,\infty) \) and \(X(\cdot ,t)\) converges to a round circle in \(C^{\infty }\) norm as \(t\rightarrow \infty \). For \(\alpha \in \left( -\infty ,1\right) \), a discussion on the possible asymptotic behavior of the flow is also given.Ergodicity for a class of semilinear stochastic partial differential equationshttps://zbmath.org/1508.600662023-05-31T16:32:50.898670Z"Dong, Zhao"https://zbmath.org/authors/?q=ai:dong.zhao"Zhang, Rangrang"https://zbmath.org/authors/?q=ai:zhang.rangrangSummary: In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.Noise effects in some stochastic evolution equations: global existence and dependence on initial datahttps://zbmath.org/1508.600702023-05-31T16:32:50.898670Z"Tang, Hao"https://zbmath.org/authors/?q=ai:tang.hao"Yang, Anita"https://zbmath.org/authors/?q=ai:yang.anitaSummary: In this paper, we consider the noise effects on a class of stochastic evolution equations including the stochastic Camassa-Holm equations with or without rotation. We first obtain the existence, uniqueness and a blow-up criterion of pathwise solutions in Sobolev space \(H^s\) with \(s>3/2\). Then we prove that strong enough noise can prevent blow-up with probability 1, which justifies the regularization effect of strong nonlinear noise in preventing singularities. Besides, such strengths of noise are estimated in different examples. Finally, for the interplay between regularization effect induced by the noise and the dependence on initial conditions, we introduce and investigate the stability of the exiting time and construct an example to show that the multiplicative noise cannot improve both the stability of the exiting time and the continuity of the dependence on initial data simultaneously.An energy-preserving finite difference scheme with fourth-order accuracy for the generalized Camassa-Holm equationhttps://zbmath.org/1508.651122023-05-31T16:32:50.898670Z"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.2|wang.xiaofeng.3|wang.xiaofeng|wang.xiaofeng.1|wang.xiaofeng.4Summary: In this article, an energy-preserving finite difference scheme for solving the generalized Camassa-Holm (gCH) equation with the dual-power law nonlinearities is proposed. We first show that the solution of the initial-boundary-value gCH equation is unique and continuously dependent on the initial condition, then we construct a linear energy-preserving difference scheme for the gCH equation. The proposed difference scheme is three-level implicit, and the numerical convergence order is \(O(\tau^2+h^4)\). The energy conservation, unique solvability, convergence and stability of the finite difference scheme are rigorously proved by using the discrete energy method. Finally, some numerical examples show that the proposed numerical scheme is efficient and reliable.On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factorhttps://zbmath.org/1508.651172023-05-31T16:32:50.898670Z"Karimi, Milad"https://zbmath.org/authors/?q=ai:karimi.milad"Moradlou, Fridoun"https://zbmath.org/authors/?q=ai:moradlou.fridoun"Hajipour, Mojtaba"https://zbmath.org/authors/?q=ai:hajipour.mojtabaSummary: This paper is concerned with a backward heat conduction problem with time-dependent thermal diffusivity factor in an infinite ``strip''. This problem is drastically ill-posed which is caused by the amplified infinitely growth in the frequency components. A new regularization method based on the Meyer wavelet technique is developed to solve the considered problem. Using the Meyer wavelet technique, some new stable estimates are proposed in the Hölder and Logarithmic types which are optimal in the sense of given by Tautenhahn. The stability and convergence rate of the proposed regularization technique are proved. The good performance and the high-accuracy of this technique is demonstrated through various one and two dimensional examples. Numerical simulations and some comparative results are presented.Asymptotic analysis for different partitionings of RLC transmission lineshttps://zbmath.org/1508.651202023-05-31T16:32:50.898670Z"Gander, Martin J."https://zbmath.org/authors/?q=ai:gander.martin-j"Kumbhar, Pratik M."https://zbmath.org/authors/?q=ai:kumbhar.pratik-m"Ruehli, Albert E."https://zbmath.org/authors/?q=ai:ruehli.albert-eFor the entire collection see [Zbl 1485.65003].Abundant exact solutions to the discrete complex mKdV equation by Darboux transformationhttps://zbmath.org/1508.651502023-05-31T16:32:50.898670Z"Ma, Li-Yuan"https://zbmath.org/authors/?q=ai:ma.liyuan"Zhao, Hai-Qiong"https://zbmath.org/authors/?q=ai:zhao.haiqiong"Shen, Shou-Feng"https://zbmath.org/authors/?q=ai:shen.shoufeng"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiuSummary: In this paper, an \(N\)-fold Darboux transformation is constructed for the discrete complex modified Korteweg-de Vries equation of focusing type, in terms of determinants. Through the obtained one-fold and two-fold Darboux transformations, a variety of new exact solutions, including an anti-dark soliton solution, a breather solution, a periodic solution, and a two-soliton solution, are derived from a nonzero constant and plane-wave seed solution. Via numerical simulation, a new kind of dynamical behavior of the two-soliton solution is exhibited, which tells that the two-soliton solution includes an anti-dark solitary wave and a w-shaped solitary wave.Random sampling and efficient algorithms for multiscale PDEshttps://zbmath.org/1508.651562023-05-31T16:32:50.898670Z"Chen, Ke"https://zbmath.org/authors/?q=ai:chen.ke"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin"Lu, Jianfeng"https://zbmath.org/authors/?q=ai:lu.jianfeng"Wright, Stephen J."https://zbmath.org/authors/?q=ai:wright.stephen-jSummary: We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on detailed analytical understanding of specific multiscale PDEs, in particular, their asymptotic limits. We present the application of the framework on two examples-a linear kinetic equation and an elliptic equation with rough media. On these two examples, this framework achieves the asymptotic preserving property for the kinetic equations and numerical homogenization for the elliptic equations.A viscoelastic Timoshenko beam: model development, analysis, and investigationhttps://zbmath.org/1508.740052023-05-31T16:32:50.898670Z"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Li, Yiqun"https://zbmath.org/authors/?q=ai:li.yiqun.1"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: Vibrations are ubiquitous in mechanical or biological systems, and they are ruinous in numerous circumstances. We develop a viscoelastic Timoshenko beam model, which naturally captures distinctive power-law responses arising from a broad distribution of time-scales presented in the complex internal structures of viscoelastic materials and so provides a very competitive description of the mechanical responses of viscoelastic beams, thick beams, and beams subject to high-frequency excitations. We, then, prove the well-posedness and regularity of the viscoelastic Timoshenko beam model. We finally investigate the performance of the model, in comparison with the widely used Euler-Bernoulli and Timoshenko beam models, which shows the utility of the new model.
{\copyright 2022 American Institute of Physics}Global existence and decay for the semilinear thermoelastic contact problemhttps://zbmath.org/1508.740112023-05-31T16:32:50.898670Z"Gao, Hongjun"https://zbmath.org/authors/?q=ai:gao.hongjun"Muñoz Rivera, Jaime E."https://zbmath.org/authors/?q=ai:munoz-rivera.jaime-edilbertoSummary: In this paper a class of semilinear thermoelastic contact problems is considered and the existence and exponential decay of the weak solutions are obtained.Uniform stability in structural acoustic models with flexible curved wallshttps://zbmath.org/1508.740452023-05-31T16:32:50.898670Z"Cagnol, John"https://zbmath.org/authors/?q=ai:cagnol.john"Lasiecka, Irena"https://zbmath.org/authors/?q=ai:lasiecka.irena"Lebiedzik, Catherine"https://zbmath.org/authors/?q=ai:lebiedzik.catherine"Zolésio, Jean-Paul"https://zbmath.org/authors/?q=ai:zolesio.jean-paulSummary: The aim of this paper is twofold. First, we develop an explicit extension of the Kirchhoff model for thin shells, based on the model developed by Michel Delfour and Jean-Paul Zolésio. This model relies heavily on the oriented distance function which describes the geometry. Once this model is established, we investigate the uniform stability of a structural acoustic model with structural damping. The result no longer requires that the active wall be a plate. It can be virtually any shell, provided that the shell is thin enough to accommodate the curvatures.Carreau law for non-Newtonian fluid flow through a thin porous mediahttps://zbmath.org/1508.760062023-05-31T16:32:50.898670Z"Anguiano, María"https://zbmath.org/authors/?q=ai:anguiano.maria"Bonnivard, Matthieu"https://zbmath.org/authors/?q=ai:bonnivard.matthieu"Suárez-Grau, Francisco J."https://zbmath.org/authors/?q=ai:suarez-grau.francisco-javierThe authors perform the homogenization and dimension reduction asymptotics for the case of a generalized Newtonian fluid through a thin porous media. Specifically, the nonlinear viscosity of the fluid is assumed to follow a Careau-type constitutive relationship. Numerical simulations illustrate the behavior of the limit model.
Reviewer: Adrian Muntean (Karlstad)Wave front tracing and asymptotic stability of planar travelling waves for a two-dimensional shallow river modelhttps://zbmath.org/1508.760202023-05-31T16:32:50.898670Z"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Yu, Shih-Hsien"https://zbmath.org/authors/?q=ai:yu.shih-hsienSummary: The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcy's law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations.Vortex patch problem for steady lake equationhttps://zbmath.org/1508.760262023-05-31T16:32:50.898670Z"Cao, Daomin"https://zbmath.org/authors/?q=ai:cao.daomin"Qin, Guolin"https://zbmath.org/authors/?q=ai:qin.guolin"Zou, Changjun"https://zbmath.org/authors/?q=ai:zou.changjunSummary: We study the vortex patch problem for the steady lake equation in a bounded domain and construct three different kinds of solutions where the vorticity concentrates in the domain or near the boundary. Our approach is based on the Lyapunov-Schmidt reduction, which transforms the construction into a problem of seeking critical points for a function related to the kinetic energy. The method in this paper has a wide applicability and can be used to deal with general elliptic equations in divergence form with Heaviside nonlinearity.
{\copyright 2022 American Institute of Physics}Correction to: ``MHD flow through a perturbed channel filled with a porous medium''https://zbmath.org/1508.761272023-05-31T16:32:50.898670Z"Marušić-Paloka, Eduard"https://zbmath.org/authors/?q=ai:marusic-paloka.eduard"Pažanin, Igor"https://zbmath.org/authors/?q=ai:pazanin.igor"Radulović, Marko"https://zbmath.org/authors/?q=ai:radulovic.markoCorrects Equation 4.22 in the authors' paper [ibid. 45, No. 5, 2441--2471 (2022; Zbl 1505.76116)].Estimation of decay rates to large-solutions of 3D compressible magnetohydrodynamic systemhttps://zbmath.org/1508.761302023-05-31T16:32:50.898670Z"Wang, Shuai"https://zbmath.org/authors/?q=ai:wang.shuai"Chen, Fei"https://zbmath.org/authors/?q=ai:chen.fei"Zhao, Yongye"https://zbmath.org/authors/?q=ai:zhao.yongye"Wang, Chuanbao"https://zbmath.org/authors/?q=ai:wang.chuanbaoSummary: The aim of this paper is to get an estimation of decay rates to first-order and second-order derivatives of space for large-solutions to 3D compressible magnetohydrodynamic system. While the condition \((\sigma_0 - 1, \mathbf{u}_0, \mathbf{Q}_0)\in L^1\cap H^2\) is satisfied via a classical energy method and Fourier splitting method, first-order and second-order derivatives of space for large-solutions tending to 0 by \(L^2\)-rate \((1 + t)^{-\frac{5}{4}}\) are shown. It is a necessary supplement to the result of \textit{J. Gao} et al. [Appl. Math. Lett. 102, Article ID 106100, 7 p. (2020; Zbl 1440.35268)] in which they only obtained an estimation of decay rates to magnetic fields. Meanwhile, compared with the work of \textit{J. Gao} et al. [Physica D 406, Article ID 132506, 9 p. (2020; Zbl 1490.76175)], we find that the appearance of magnetic fields does not have any bad effect on the estimation of decay rates to both the velocity field and density.
{\copyright 2022 American Institute of Physics}Problem of radiation heat exchange with boundary conditions of the Cauchy typehttps://zbmath.org/1508.800032023-05-31T16:32:50.898670Z"Chebotarev, Alexander Yu."https://zbmath.org/authors/?q=ai:chebotarev.alexander-yurievich"Kovtanyuk, Andrey E."https://zbmath.org/authors/?q=ai:kovtanyuk.andrey-egorovich"Botkin, Nikolai D."https://zbmath.org/authors/?q=ai:botkin.nikolai-dSummary: In this paper, a quasi-static problem of radiative-conductive heat exchange in a three-dimensional domain is considered in the framework of diffusion \(P_1\) approximation of the radiation transfer equation. The peculiarity of the problem statement is that the boundary values for the radiation intensity are not prescribed. Instead of that, the heat flux is additionally prescribed for the temperature field on the boundary. The unique, nonlocal in time, solvability of the problem is proven. Theoretical results are illustrated by numerical examples.Quantum oscillations. A simple principle underlying important aspects of physicshttps://zbmath.org/1508.810252023-05-31T16:32:50.898670Z"Suekane, Fumihiko"https://zbmath.org/authors/?q=ai:suekane.fumihikoPublisher's description: This book addresses various aspects of physics, using Quantum oscillation (QO) as a common denominator. QO plays an important role in many aspects of physics, such as the Weinberg angle, Caribbo angle, neutrino oscillation, \(K^0\) oscillation and CP violation, mass generation by the Higgs field, hadron mass pattern, lepton anomalous magnetic moment, spin precession, hydrogen HI line, etc.
Usually, these subjects are taught separately. As such, this book allows readers to learn about a wide range of physics subjects in a unified manner and to gain farther-reaching perspectives. The readers may be surprised at the fact that different looking physics are actually closely related with each other. They will also find essential information on quantum mechanics at the heart from many concrete examples. Though the book is mainly intended for graduate students of particle, nuclear and astrophysics, undergraduate students and researchers will also benefit from the content.TCSA and the finite volume boundary statehttps://zbmath.org/1508.810662023-05-31T16:32:50.898670Z"Bajnok, Zoltan"https://zbmath.org/authors/?q=ai:bajnok.zoltan"Tompa, Tamas Lajos"https://zbmath.org/authors/?q=ai:tompa.tamas-lajosSummary: We develop a new way to calculate the overlap of a boundary state with a finite volume bulk state in the truncated conformal space approach. We check this method in the thermally perturbed Ising model analytically, while in the scaling Lee-Yang model numerically by comparing our results to excited state g-functions, which we obtained by the analytical continuation method. We also give a simple argument for the structure of the asymptotic overlap between the finite volume boundary state and a periodic multiparticle state, which includes the ratio of Gaudin type determinants.Geometrothermodynamic description of real gases using the law of corresponding stateshttps://zbmath.org/1508.820452023-05-31T16:32:50.898670Z"Quevedo, Hernando"https://zbmath.org/authors/?q=ai:quevedo.hernando"Quevedo, María N."https://zbmath.org/authors/?q=ai:quevedo.maria-n"Sánchez, Alberto"https://zbmath.org/authors/?q=ai:rivadulla-sanchez.albertoSummary: We propose a geometric model for the description of the equilibrium space of real gases based on the Legendre invariant formalism of geometrothermodynamics. We investigate the curvature of three different Legendre invariant metrics and show that the corresponding singularities are related to the critical points of the response functions, the isotherms in a pressure-volume diagram, and the stability conditions. This implies that it is necessary to consider all Legendre invariant metrics to completely describe the critical behavior of real gases in terms of curvature singularities.Parallel kinetic scheme for transport equations in complex toroidal geometryhttps://zbmath.org/1508.820482023-05-31T16:32:50.898670Z"Boileau, Matthieu"https://zbmath.org/authors/?q=ai:boileau.matthieu"Bramas, Bérenger"https://zbmath.org/authors/?q=ai:bramas.berenger"Franck, Emmanuel"https://zbmath.org/authors/?q=ai:franck.emmanuel"Hélie, Romane"https://zbmath.org/authors/?q=ai:helie.romane"Helluy, Philippe"https://zbmath.org/authors/?q=ai:helluy.philippe"Navoret, Laurent"https://zbmath.org/authors/?q=ai:navoret.laurentA new numerical method (based on a kinetic formulation resembling the Lattice-Boltzmann approach) for solving nonhomogeneous conservative transport equations in toroidal geometries (as is in tokamaks) is proposed. This permits to handle unstructured meshes of the poloidal plane, and allow a parallelization. The algorithm has been tested in a few model problems including the diocotron case (the slipping stream plasma instability).
Reviewer: Piotr Biler (Wrocław)Erratum to: ``Cosmology and gravitational waves in consistent \(D\rightarrow 4\) Einstein-Gauss-Bonnet gravity''https://zbmath.org/1508.830082023-05-31T16:32:50.898670Z"Aoki, Katsuki"https://zbmath.org/authors/?q=ai:aoki.katsuki"Gorji, Mohammad Ali"https://zbmath.org/authors/?q=ai:gorji.mohammad-ali"Mukohyama, Shinji"https://zbmath.org/authors/?q=ai:mukohyama.shinjiIn the authors' paper [ibid. 2020, No. 9, Paper No. 14, 18 p. (2020; Zbl 1493.83005)], the discussion about the observational bounds on the rescaled Gauss-Bonnet parameter in fifth and sixth (last) paragraphs in section 6, page 12 and 13 is changed.Stability of a non-local kinetic model for cell migration with density-dependent speedhttps://zbmath.org/1508.920292023-05-31T16:32:50.898670Z"Loy, Nadia"https://zbmath.org/authors/?q=ai:loy.nadia"Preziosi, Luigi"https://zbmath.org/authors/?q=ai:preziosi.luigiA theoretical and numerical study of the linear stability of equilibria is performed for a kinetic model of cell migration featuring nonlocal sensing and volume filling. Denoting the cell distribution at time \(t\ge 0\), position \(\mathbf{x}\in\Omega\subset \mathbb{R}^d\), velocity \(\nu\in [0,U]\), and direction \(\mathbf{\hat{v}}\in \mathbb{S}^{d-1}\) by \(p(t,\mathbf{x},\nu,\mathbf{\hat{v}})\), its dynamics is described by the semilinear transport equation
\begin{align*}
\partial_t p(t,\mathbf{x},\nu,\mathbf{\hat{v}}) & + \nu \mathbf{\hat{v}} \cdot \nabla_{\mathbf{x}} p(t,\mathbf{x},\nu,\mathbf{\hat{v}}) \\
& = \mu(\mathbf{x}) \left[ \rho(t,\mathbf{x}) T[\rho](t,\mathbf{x},\nu,\mathbf{\hat{v}}) - p(t,\mathbf{x},\nu,\mathbf{\hat{v}}) \right],
\end{align*}
where
\begin{align*}
\rho(t,\mathbf{x}) &= \int_{\mathbb{S}^{d-1}} \int_0^U p(t,\mathbf{x},\nu,\mathbf{\hat{v}})\ d\nu d\mathbf{\hat{v}}, \\
T[\rho](t,\mathbf{x},\nu,\mathbf{\hat{v}}) & = c(t,\mathbf{x}) \int_0^{R_\rho(t,\mathbf{x},\mathbf{\hat{v}})} \gamma(\lambda) \psi(\nu|\rho(t,\mathbf{x} + \lambda \mathbf{\hat{v}}))\ d\lambda , \\
\frac{1}{c(t,\mathbf{x})} & = \int_{\mathbb{S}^{d-1}} \int_0^{R_\rho(t,\mathbf{x},\mathbf{\hat{v}})} \gamma(\lambda)\ d\lambda d\mathbf{\hat{v}},
\end{align*}
with \[ \int_0^U \psi(\nu|r) = 1, \qquad \mathrm{supp}\,\gamma = [0,R], \] and \[ R_\rho(t,\mathbf{x},\mathbf{\hat{v}}) = \inf\left\{ \lambda\in [0,R]\ :\ \rho(t,\mathbf{x} + \lambda \mathbf{\hat{v}}) > \rho_{th} \right\}, \] setting \(R_\rho(t,\mathbf{x},\mathbf{\hat{v}}) =R\) when \(\sup_{\lambda\in [0,R)}\{\rho(t,\mathbf{x} + \lambda \mathbf{\hat{v}})\} \le \rho_{th}\). Besides moving freely in space, the cells change their velocity orientation according to the turning operator \(T[\rho]\) which takes into account the local density of neighbouring cells. Equilibria \(p_\infty\) then solve \(p_\infty = \rho_\infty T[\rho_\infty] \) and their stability is investigated for those satisfying \(\rho_\infty < \rho_{th}\).
Reviewer: Philippe Laurençot (Toulouse)The impact of phenotypic heterogeneity on chemotactic self-organisationhttps://zbmath.org/1508.920302023-05-31T16:32:50.898670Z"Macfarlane, Fiona R."https://zbmath.org/authors/?q=ai:macfarlane.fiona-r"Lorenzi, Tommaso"https://zbmath.org/authors/?q=ai:lorenzi.tommaso"Painter, Kevin J."https://zbmath.org/authors/?q=ai:painter.kevin-jThe authors introduce a model of chemotactic self-organisation with phenotypic heterogeneity,
\begin{align*}
\frac{\partial n_0}{\partial t}&=D\nabla^2n_0-\mu_{01}(\rho, s)n_0+\mu_{10}(\rho,s)n_ 1\\
\frac{\partial n_1}{\partial t}&=D\nabla^2n_1-\chi \nabla\cdot (n_1\nabla s)+\mu_{01}(\rho,s)n_0-\mu_{10}(\rho,s)n_1 \\
\frac{\partial s}{\partial t}&=\nabla^2s+n_0-s
\end{align*}
with
\[
\rho=n_0+n_1, \quad \left.( u\cdot \nabla n_0, u\cdot \nabla n_1, u\cdot \nabla s)\right\vert_{\partial \Omega}=0,
\]
where \(\Omega \subset \mathbb{R}^d\) is a bounded domain with smooth boundary, \(u\) is the unit outer normal vector, and \(\mu_{01}\), \(\mu_{10}\) are switching functions of phenotype. Linear stability of positive uniform steady states is examined for \(d=1\) in accordance with switching functions, the sensitivity \(\chi\), and the size of \(\Omega\). Then numerical simulations are provided, where autoaggregation and oscillating patterns with extinction are observed for \(d=1\), and emergence of patterns is noticed for \(d=2\).
Reviewer: Takashi Suzuki (Osaka)Stability of a class of nonlinear hierarchical size-structured population modelhttps://zbmath.org/1508.921992023-05-31T16:32:50.898670Z"Chen, Weicheng"https://zbmath.org/authors/?q=ai:chen.weicheng"Wang, Zhanping"https://zbmath.org/authors/?q=ai:wang.zhanping(no abstract)Optimal strategies for controlling the outbreak of COVID-19: reducing its cost and durationhttps://zbmath.org/1508.922472023-05-31T16:32:50.898670Z"Dashtbali, Mohammadali"https://zbmath.org/authors/?q=ai:dashtbali.mohammadaliSummary: Social distancing plays an essential role in controlling the spread of an epidemic, but changing the behavior of individuals regarding social distancing is costly. In order to make a rational decision, individuals must compare the cost of social distancing and the cost of infection. People are typically more likely to change their behavior if they are aware that the government is willing to incur additional cost to shorten the duration of an epidemic. I extend an optimal control problem of social distancing by integrating with the SIR model which describes the disease process. I present an optimal control problem to consider the behavior of susceptible individuals and the government in investment as control strategies and compute the equilibrium strategies under the potency of investment, using relative risk functions according to the investment that is made by susceptible individuals and the government. The equilibrium of this problem represents the optimal control strategies for minimizing the cost and duration of controlling an epidemic. Additionally, the model is evaluated using COVID-19 data from Egypt, Japan, Italy, Belgium, Nigeria, and Germany. The findings extracted from this model could be valuable in developing public health policy in the event of an epidemic.On uniform controllability of 1D transport equations in the vanishing viscosity limithttps://zbmath.org/1508.930442023-05-31T16:32:50.898670Z"Laurent, Camille"https://zbmath.org/authors/?q=ai:laurent.camille"Léautaud, Matthieu"https://zbmath.org/authors/?q=ai:leautaud.matthieuAuthors' abstract: We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit. We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [\textit{C. Laurent} and \textit{M. Léautaud}, J. Éc. Polytech., Math. 8, 439--506 (2021; Zbl 1461.93052)]. The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [\textit{C. Laurent} and \textit{M. Léautaud}, ``Uniform observation of semiclassical Schrödinger eigenfunctions on an interval'' (2022)], together with estimates on the spectral gap [\textit{B. Allibert}, Commun. Partial Differ. Equations 23, No. 9--10, 1493--1556 (1998; Zbl 0954.35028); \textit{B. Helffer} and \textit{J. Sjöstrand}, Commun. Partial Differ. Equations 9, 337--408 (1984; Zbl 0546.35053)]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.
Reviewer: Kaïs Ammari (Monastir)