Recent zbMATH articles in MSC 76Bhttps://zbmath.org/atom/cc/76B2023-05-31T16:32:50.898670ZWerkzeugGlobal 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.Stratified equatorial flows in the \(\beta \)-plane approximation with a free surfacehttps://zbmath.org/1508.350492023-05-31T16:32:50.898670Z"Miao, Fahe"https://zbmath.org/authors/?q=ai:miao.fahe"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongSummary: We investigate the exact solutions to the governing equations for the equatorial flows with the associated free surface and rigid bottom boundary conditions in the \(\beta \)-plane approximation which incorporates two considerations of the density stratification. Compared to the spherical coordinates and the cylindrical coordinates, the employment of the \(\beta \)-plane approximation admits that the density can be provided generally. Utilizing the implicit theorem, we present the Bernoulli relation between the pressure imposed on the free surface and the resulting distortion of the surface and we obtain that this relation exhibits the expected monotonicity properties. Finally, we prove that certain flows established by the exact solutions are stable via the short-wavelength stability method and the specific assumption of the density distribution.The global regularity of vortex patches revisitedhttps://zbmath.org/1508.350502023-05-31T16:32:50.898670Z"Verdera, Joan"https://zbmath.org/authors/?q=ai:verdera.joanSummary: We prove persistence of the regularity of the boundary of vortex patches for a large class of transport equations in the plane. The velocity field is given by convolution of the vorticity with an odd kernel, homogeneous of degree \(-1\) and of class \(C^2\) off the origin.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.On Moffatt's magnetic relaxation equationshttps://zbmath.org/1508.350582023-05-31T16:32:50.898670Z"Beekie, Rajendra"https://zbmath.org/authors/?q=ai:beekie.rajendra"Friedlander, Susan"https://zbmath.org/authors/?q=ai:friedlander.susan-j"Vicol, Vlad"https://zbmath.org/authors/?q=ai:vicol.vlad-cSummary: We investigate the stability properties for a family of equations introduced by Moffatt to model magnetic relaxation. These models preserve the topology of magnetic streamlines, contain a cubic nonlinearity, and yet have a favorable \(L^2\) energy structure. We consider the local and global in time well-posedness of these models and establish a difference between the behavior as \(t\rightarrow\infty\) with respect to weak and strong norms.Non conservative products in fluid dynamicshttps://zbmath.org/1508.350642023-05-31T16:32:50.898670Z"Colombo, Rinaldo M."https://zbmath.org/authors/?q=ai:colombo.rinaldo-m"Guerra, Graziano"https://zbmath.org/authors/?q=ai:guerra.graziano"Holle, Yannick"https://zbmath.org/authors/?q=ai:holle.yannickThe authors justify the approximation of solutions of problems in fluid dynamics with nonconservative products in sources. Motivations stem from fluid flows in pipes with discontinuous cross section. Applications of the existence results shown are not limited, however, to fluid dynamics. They apply to abstract balance laws with nonconservative source terms in the nonresonant case, in general BV geometry.
Reviewer: Piotr Biler (Wrocław)On solitary-wave solutions of Boussinesq/Boussinesq systems for internal waveshttps://zbmath.org/1508.350682023-05-31T16:32:50.898670Z"Dougalis, Vassilios A."https://zbmath.org/authors/?q=ai:dougalis.vassilios-a"Durán, Angel"https://zbmath.org/authors/?q=ai:duran.angel"Saridaki, Leetha"https://zbmath.org/authors/?q=ai:saridaki.leethaSummary: In this paper we consider a three-parameter system of Boussinesq/Boussinesq type, modeling the propagation of internal waves. Some theoretical and numerical properties of the systems were previously analyzed in our works [``On the numerical approximation of Boussinesq/Boussinesq systems for internal waves'' (submitted); ``Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves'', Preprint, \url{arXiv:2012.07992}]. As a second part of the study, the present paper is concerned with the analysis of existence and the numerical simulation of some issues of the dynamics of solitary-wave solutions. Standard theories are used to derive several results of existence of classical and generalized solitary waves, depending on the parameters of the models. A numerical procedure based on a Fourier collocation approximation for the ode system of the solitary wave profiles with periodic boundary conditions, and on the iterative solution of the resulting fixed-point equations with the Petviashvili scheme combined with vector extrapolation techniques, is used to generate numerically approximations of solitary waves. These are an essential part of a computational study of the dynamics of the solitary waves, both classical and generalized. Using a full discretization based on spectral approximation in space of the corresponding periodic initial-value problem for the systems, and a fourth-order Runge-Kutta method of composition type as time integrator, we explore the evolution of small and large perturbations of the computed solitary-wave profiles, and we study computationally the collisions of solitary waves as well as the resolution of initial data into trains of solitary waves.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.Global solutions to the dissipative quasi-geostrophic equation with dispersive forcinghttps://zbmath.org/1508.350712023-05-31T16:32:50.898670Z"Fujii, Mikihiro"https://zbmath.org/authors/?q=ai:fujii.mikihiroSummary: We consider the initial value problem for the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing and prove the global existence of a unique solution in the scaling subcritical Sobolev spaces \(H^s(\mathbb{R}^2)\) (\(s > 2 - \alpha\)) and the scaling critical space \(H^{2-\alpha}(\mathbb{R}^2)\). More precisely, for the scaling subcritical case, we establish a unique global solution for a given initial data \(\theta_0 \in H^s(\mathbb{R}^2)\) (\(s > 2 - \alpha\)) if the size of dispersion parameter is sufficiently large and also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. For the scaling critical case, we find that the size of dispersion parameter to ensure the global existence is determined by each subset \(K \subset H^{2-\alpha}(\mathbb{R}^2)\), which is precompact in some homogeneous Sobolev spaces.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.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.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.Long time well-posedness of Whitham-Boussinesq systemshttps://zbmath.org/1508.350892023-05-31T16:32:50.898670Z"Paulsen, Martin Oen"https://zbmath.org/authors/?q=ai:paulsen.martin-oenThis article is concerned with the analyses of different asymptotic systems for weakly nonlinear shallow water surface waves, namely the KdV equation, the Boussinesq systems, and the Whitham-Boussinesq (W-B) systems as an extension of the unidirectional equation conjectured by Whitham. A lot of literature is available on these systems; however, it is expected that the W-B systems are expected to provide a more accurate description of the full water wave system. This paper aims at establishing the long time well-posedness of W-B systems (in one and two dimensions) with uniform bounds taking into account the small parameters, related to the level of dispersion and nonlinearity, and the standard non-cavitation condition in certain cases. The paper uses certain technical results along with commutator estimates for Fourier multipliers to treat the nonlinear terms for estimating the energy. The proofs depend on appropriate symmetrizers. It is an excellent paper with several details which would be useful for someone working on nonlinear water waves.
Reviewer: Vishnu Dutt Sharma (Ghandinagar)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.Two-dimensional rogue wave clusters in self-focusing Kerr-mediahttps://zbmath.org/1508.351662023-05-31T16:32:50.898670Z"Zhong, WenYe"https://zbmath.org/authors/?q=ai:zhong.wenye"Qin, Pei"https://zbmath.org/authors/?q=ai:qin.pei"Zhong, Wei-Ping"https://zbmath.org/authors/?q=ai:zhong.weiping"Belić, Milivoj"https://zbmath.org/authors/?q=ai:belic.milivoj-r(no abstract)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.Some classes of shapes of the rotating liquid drophttps://zbmath.org/1508.530122023-05-31T16:32:50.898670Z"Pulov, Vladimir I."https://zbmath.org/authors/?q=ai:pulov.vladimir-i"Mladenov, Ivaïlo M."https://zbmath.org/authors/?q=ai:mladenov.ivailo-mSummary: The problem of a fluid body rotating with a constant angular velocity and subjected to uniform external pressure is of real interest in both fluid dynamics and nuclear theory. Besides, from the geometrical viewpoint the sought equilibrium configuration of such system turns out to be equivalent to the problem of determining the surface of revolution with a prescribed mean curvature. In the simply connected case, the equilibrium surface can be parameterized explicitly via elliptic integrals of the first and second kind.Periodic perturbation of motion of an unbalanced circular foil in the presence of point vortices in an ideal fluidhttps://zbmath.org/1508.700082023-05-31T16:32:50.898670Z"Vetchanin, Evgeniĭ Vladimirovich"https://zbmath.org/authors/?q=ai:vetchanin.evgenii-vladimirovich"Mamaev, Ivan Sergeevich"https://zbmath.org/authors/?q=ai:mamaev.ivan-sSummary: The dynamics of a system governing the controlled motion of an unbalanced circular foil in the presence of point vortices is considered. The foil motion is controlled by periodically changing the position of the center of mass, the gyrostatic momentum, and the moment of inertia of the system. A derivation of the equations of motion based on Sedov's approach is proposed, the equations of motion are presented in the Hamiltonian form. A periodic perturbation of the known integrable case is considered.On the existence of weak solutions for the 2D incompressible Euler equations with in-out flow and source and sink pointshttps://zbmath.org/1508.760132023-05-31T16:32:50.898670Z"Bravin, Marco"https://zbmath.org/authors/?q=ai:bravin.marcoThe authors show the existence of solutions for a system that describes the flow of a two-dimensional incompressible inviscid fluid which is allowed to enter in and exit from the boundary at some points called source and sink. The method extends the existence results for the two-dimensional Euler system with in-out flow to time-dependent domains.
Reviewer: Fatma Gamze Duzgun (Ankara)Global well-posedness of 2D Euler-\(\alpha\) equation in exterior domainhttps://zbmath.org/1508.760142023-05-31T16:32:50.898670Z"You, Xiaoguang"https://zbmath.org/authors/?q=ai:you.xiaoguang"Zang, Aibin"https://zbmath.org/authors/?q=ai:zang.aibin"Li, Yin"https://zbmath.org/authors/?q=ai:li.yin.1The authors recast the Euler-\(\alpha\) equations into vorticity-stream function formulation and obtain some useful estimates from the vorticity formula in exterior domain. Such results imply the global existence and uniqueness of the solutions to Euler-\(\alpha\) equations in 2D exterior domain provided that the initial data is regular enough.
Reviewer: Fatma Gamze Duzgun (Ankara)A comparative study of integral and coupled approaches for modeling hydraulic exchange processes across a rippled streambedhttps://zbmath.org/1508.760152023-05-31T16:32:50.898670Z"Gollo, Vahid Sobhi"https://zbmath.org/authors/?q=ai:gollo.vahid-sobhi"Broecker, Tabea"https://zbmath.org/authors/?q=ai:broecker.tabea"Marx, Christian"https://zbmath.org/authors/?q=ai:marx.christian"Lewandowski, Jörg"https://zbmath.org/authors/?q=ai:lewandowski.jorg"Nützmann, Gunnar"https://zbmath.org/authors/?q=ai:nutzmann.gunnar"Hinkelmann, Reinhard"https://zbmath.org/authors/?q=ai:hinkelmann.reinhardSummary: Although both are crucial parts of the hydrological cycle, groundwater and surface water had traditionally been addressed separately. In recent decades, considering them as a single hydrological continuum in light of their continuous interaction has become well established in the scientific community through the development of numerous measurement and experimental techniques. Nevertheless, numerical models, as necessary tools to study a wide range of scenarios and future event predictions, are still based on outdated concepts that consider groundwater and surface water separately. This study compares these ``coupled models'', which result from the successive execution of a surface water model and a groundwater model, to a recently developed ``integral model''. The integral model uses a single set of equations to model both groundwater and surface water simultaneously, and can account for the continuous interaction at their interface. For comparison, we investigated small-scale flow across a rippled porous streambed. Although we applied identical model domain details and flow conditions, which resulted in very similar water tables and pressure distributions, comparing the integral and coupled models yielded very dissimilar velocity values across the groundwater-surface water interface. These differences highlight the impact of continuous exchange across the interface in the integral model, which imitates such flow processes more realistically than the coupled model. A few decimeters away from the interface, modeled velocity fields are very similar. Since the integral model and the surface water component of the coupled model are both CFD-based (computational fluid dynamics), they require very similar computational resources, namely access to cluster computers. Unfortunately, replacing the surface water component of the coupled model with the widely used shallow water equations model, which indeed would reduce the computational resources required, produces inaccuracy.Almost extreme waveshttps://zbmath.org/1508.760162023-05-31T16:32:50.898670Z"Dyachenko, Sergey A."https://zbmath.org/authors/?q=ai:dyachenko.sergey-a"Hur, Vera Mikyoung"https://zbmath.org/authors/?q=ai:hur.vera-mikyoung"Silantyev, Denis A."https://zbmath.org/authors/?q=ai:silantyev.denis-aSummary: Numerically computed with high accuracy are periodic travelling waves at the free surface of a two-dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are the following. (i) There is a boundary layer where the angle rises sharply from \(0^\circ\) at the crest to a local maximum, which converges to \(30.3787\dots^\circ \), independently of the vorticity, as the amplitude increases towards that of the extreme wave, which displays a corner at the crest with a \(30^\circ\) angle. (ii) There is an outer region where the angle descends to \(0^\circ\) at the trough for negative vorticity, while it rises to a maximum, greater than \(30^\circ\), and then falls sharply to \(0^\circ\) at the trough for large positive vorticity. (iii) There is a transition region where the angle oscillates about \(30^\circ \), resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.Very high order well-balanced schemes for non-prismatic one-dimensional channels with arbitrary shapehttps://zbmath.org/1508.760172023-05-31T16:32:50.898670Z"Escalante, C."https://zbmath.org/authors/?q=ai:escalante.cipriano"Castro, M. J."https://zbmath.org/authors/?q=ai:castro.manuel-j"Semplice, M."https://zbmath.org/authors/?q=ai:semplice.matteoSummary: Accurate modelling of shallow water flows in canals for realistic scenarios cannot neglect the variations of the size and shape of the cross-section along the canal. In typical situations, especially in view of applications to flows in networks of canals, a 2D model would be too costly, while the standard 1D Saint-Venant model for a rectangular channel with constant breadth would be too coarse. In this paper we derive efficient, high order accurate and robust numerical schemes for a 1.5D model, in which the canal can have an arbitrary cross-section: the shape can vary along the channel and is described by a depth-dependent breadth function \(\sigma(x,z)\), where \(x\) is the coordinate along the channel and \(z\) represents the vertical direction. Contrary to all previous schemes for this model, we reformulate the equations in a way that avoids the appearance of the moments of \(\sigma\) and of \(\partial\sigma/\partial x\) in the source terms. Our numerical schemes are based on the path-conservative approach for dealing with non-conservative products, are well-balanced on the lake at rest solution and can treat wet-try transitions. They can be implemented with any order of accuracy. Schemes up to third order are explicitly constructed and tested, thanks to the \(\mathsf{CWENO}\) reconstruction technique. Through a large set of numerical tests we show the performance of the new schemes and compare the results obtained with different orders of accuracy.Considering the shallow water of a wide channel or an open sea through a generalized \((2+1)\)-dimensional dispersive long-wave systemhttps://zbmath.org/1508.760182023-05-31T16:32:50.898670Z"Gao, Xiao-Tian"https://zbmath.org/authors/?q=ai:gao.xiao-tian"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Shen, Yuan"https://zbmath.org/authors/?q=ai:shen.yuan.1|shen.yuan"Feng, Chun-Hui"https://zbmath.org/authors/?q=ai:feng.chunhuiSummary: Under investigation in this paper is a generalized \((2+1)\)-dimensional dispersive long-wave system, describing the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea. Via symbolic computation, we derive the same bilinear forms as those reported, but through a different method. Four sets of the similarity reductions are obtained, each of which leads to a known ordinary differential equation. The results rely on the coefficients in the original system, with respect to the horizontal velocity and wave elevation above the undisturbed water surface.Thinking about the oceanic shallow water via a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt systemhttps://zbmath.org/1508.760192023-05-31T16:32:50.898670Z"Gao, Xin-Yi"https://zbmath.org/authors/?q=ai:gao.xinyi"Guo, Yong-Jiang"https://zbmath.org/authors/?q=ai:guo.yongjiang"Shan, Wen-Rui"https://zbmath.org/authors/?q=ai:shan.wenrui(no abstract)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.An experimental and numerical study of the resonant flow between a hull and a wallhttps://zbmath.org/1508.760212023-05-31T16:32:50.898670Z"Milne, I. A."https://zbmath.org/authors/?q=ai:milne.i-a"Kimmoun, O."https://zbmath.org/authors/?q=ai:kimmoun.olivier"Graham, J. M. R."https://zbmath.org/authors/?q=ai:graham.j-michael-r"Molin, B."https://zbmath.org/authors/?q=ai:molin.bernardSummary: The wave-induced resonant flow in a narrow gap between a stationary hull and a vertical wall is studied experimentally and numerically. Vortex shedding from the sharp bilge edge of the hull gives rise to a quadratically damped free surface response in the gap, where the damping coefficient is approximately independent of wave steepness and frequency. Particle image velocimetry and direct numerical simulations were used to characterise the shedding dynamics and explore the influence of discretisation in the measurements and computations. Secondary separation was identified as a particular feature which occurred at the hull bilge in these gap flows. This can result in the generation of a system with multiple vortical regions and asymmetries between the inflow and outflow. The shedding dynamics was found to exhibit a high degree of invariance to the amplitude in the gap and the spanwise position of the barge. The new measurements and the evaluation of numerical models of varying fidelity can assist in informing offshore operations such as the side by side offloading from floating liquefied natural gas facilities.Water wave interaction with a circular arc shaped porous barrier submerged in a water of finite depthhttps://zbmath.org/1508.760222023-05-31T16:32:50.898670Z"Samanta, Anushree"https://zbmath.org/authors/?q=ai:samanta.anushree"Mondal, Dibakar"https://zbmath.org/authors/?q=ai:mondal.dibakar"Banerjea, Sudeshna"https://zbmath.org/authors/?q=ai:banerjea.sudeshnaSummary: In this paper, we study the problem of scattering of water waves by a thin circular arc-shaped porous barrier submerged in ocean of finite depth. By judicious application of Green's integral theorem, the problem is formulated in terms of a hypersingular integral equation of second kind where the unknown function represents the difference of potential function across the curved barrier. The hypersingular integral equation is then solved by two methods. The first method is Boundary Element method where the domain and range of the integral equation are discretised into small line segments. Assuming the unknown function satisfying the integral equation to be constant in each small line segment, the hypersingular integral equation is reduced to a system of algebraic equations. This system of equations is then solved to obtain the unknown function in each subinterval. Making the subinterval finer, the process is continued till the solution converges to a desired degree of accuracy. The second method is based on using collocation method where the unknown function is expanded in terms of Chebyshev polynomials of second kind. Choosing the collocation points suitably, the integral equation is reduced to a system of algebraic equations which is then solved to obtain the unknown function satisfying the hypersingular integral equation. Using the solution of the hypersingular integral equation, obtained by both the methods, the reflection coefficient, transmission coefficient and energy dissipation coefficient are computed and depicted graphically against the wave number. It was observed that the reflection, transmission and energy dissipation coefficients obtained by using the solution of hypersingular integral equation by the two methods are in good agreement. In addition, the reflection coefficient obtained by the present method found to match with the known results in the literature. From the graphs, the effect of the porous barrier on the reflected and transmitted waves and energy dissipation are studied. It was observed that the porosity of the barrier has some effect on the wave propagation.The front runner in roll waves produced by local disturbanceshttps://zbmath.org/1508.760232023-05-31T16:32:50.898670Z"Yu, Boyuan"https://zbmath.org/authors/?q=ai:yu.boyuan"Chu, Vincent H."https://zbmath.org/authors/?q=ai:chu.vincent-hSummary: Roll waves produced by a local disturbance comprise a group of shock waves with steep fronts. We used a robust and accurate numerical scheme to capture the steep fronts in a shallow-water hydraulic model of the waves. Our simulations of the waves in clear water revealed the existence of a front runner with an exceedingly large amplitude -- much greater than those of all other shock waves in the wave group. The trailing waves at the back remained periodic. Waves were produced continuously within the group due to nonlinear instability. The celerity depended on the wave amplitude. Over time, the instability produced an increasing number of shock waves in an ever-expanding wave group. We conducted simulations for three types of local disturbances of very different duration over a range of amplitudes. We interpreted the simulation results for the front runner and the trailing waves, guided by an analytical solution and the laboratory data available for the smaller waves in the trailing end of the wave group.Getting the ducks in a rowhttps://zbmath.org/1508.760242023-05-31T16:32:50.898670Z"Ellingsen, Simen Å."https://zbmath.org/authors/?q=ai:ellingsen.simen-aSummary: Vessels -- in the widest sense -- travelling on a water surface continuously do work the water surrounding it, causing energy to be radiated in the form of surface waves. The concomitant resistance force, the wave resistance, can account for as much as half the total drag on the vessel, so reducing it to a minimum has been a major part of ship design research for many decades. Whether the `vessel' is an ocean-going ship or a swimming duckling, the physics governing the V-shaped pattern of radiated waves behind it is in essence the same, and just as fuel economy is important for commercial vessels, it is reasonable to assume that also swimming waterfowl seek to minimise their energy expenditure. Using theory and methods from classic marine hydrodynamics, \textit{Z.-M. Yuan} et al. [J. Fluid Mech. 928, Paper No. R2, 11 p. (2021; Zbl 1496.76171)] consider whether, by organising themselves optimally, ducklings in a row behind a mother duck can reduce, eliminate or even reverse their individual wave resistance. They describe two mechanisms which they term `wave riding' and `wave passing.' The former is intuitive: the ducklings closest to the mother can receive a forward push by riding its mother's stern waves. The latter is perhaps a more striking phenomenon: when the interduckling distance is precisely right, every duckling in the row can, in principle, swim without wave resistance due to destructive wave interference. The phenomenon appears to be the same as motivates the recent US military research project Sea Train, a row of unmanned vehicles travelling in row formation.A Gardner evolution equation for topographic Rossby waves and its mechanical analysishttps://zbmath.org/1508.760252023-05-31T16:32:50.898670Z"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie.5"Zhang, Ruigang"https://zbmath.org/authors/?q=ai:zhang.ruigang"Yang, Liangui"https://zbmath.org/authors/?q=ai:yang.lianguiSummary: Topography plays an important role in the excitation and propagation of nonlinear Rossby solitary waves to atmospheres and oceans. In the present study, we investigate the effect of topography from the approach to topographic Rossby waves, not to the geostrophic viewpoint. It is the first time that a new evolution equation, called Gardner equation, is derived to simulate the propagation of nonlinear Rossby waves amplitude by using the methods of multiple scales and weak nonlinearity. In order to investigate the physical mechanisms of topographic Rossby wave, the shooting method is adopted to solve the Sturm-Liouville model equation with fixed boundary conditions and the Fourier spectral method is used to solve the nonlinear Gardner equation. Numerical results reveal that the magnitude of the meridional topography is more important compared to its meridional frequency on the evolution of nonlinear Rossby solitary waves, also, the variation of planetary vorticity is essential for the propagation of Rossby solitary waves.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}Helical vortices with small cross-section for 3D incompressible Euler equationhttps://zbmath.org/1508.760272023-05-31T16:32:50.898670Z"Cao, Daomin"https://zbmath.org/authors/?q=ai:cao.daomin"Wan, Jie"https://zbmath.org/authors/?q=ai:wan.jieSummary: In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form
\[
\begin{cases}
- \varepsilon^2 \text{div} ( K ( x ) \nabla u ) = ( u - q | \ln \varepsilon | )_+^p , \quad & x \in \Omega, \\
u = 0, & x \in \partial \Omega,
\end{cases}
\]
for small values of \(\varepsilon \). Helical vortex solutions concentrating near several helical filaments with polygonal symmetry are also constructed.On singular vortex patches. I: Well-posedness issueshttps://zbmath.org/1508.760282023-05-31T16:32:50.898670Z"Elgindi, Tarek M."https://zbmath.org/authors/?q=ai:elgindi.tarek-mohamed"Jeong, In-Jee"https://zbmath.org/authors/?q=ai:jeong.in-jeeSummary: The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally \(m\)-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as \(m\geq 3. In\) this case, all of the angles involved solve a \textit{closed} ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that \emph{any} other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle \(\frac{\pi}{2}\) for all time. Even in the case of vortex patches with corners of angle \(\frac{\pi}{2}\) or with corners which are only locally \(m\)-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches [the authors, Trans. Am. Math. Soc. 373, No. 9, 6757--6775 (2020; Zbl 1454.35265)], we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on \(\mathbb{R}^2\) with interesting dynamical behavior such as cusping and spiral formation in infinite time.Quantized point vortex equilibria in a one-way interaction model with a Liouville-type background vorticity on a curved torushttps://zbmath.org/1508.760292023-05-31T16:32:50.898670Z"Sakajo, Takashi"https://zbmath.org/authors/?q=ai:sakajo.takashi"Krishnamurthy, Vikas S."https://zbmath.org/authors/?q=ai:krishnamurthy.vikas-sSummary: We construct point vortex equilibria with strengths quantized by multiples of \(2\pi\) in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with \(4\widehat{n}\) point vortices of identical strengths, \(\widehat{n}\in\mathbb{N}\). The quantized point vortices are stationary in the sense that they are equilibria of a ``one-way interaction'' model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes.
{\copyright 2022 American Institute of Physics}Hamiltonian model for coupled surface and internal waves over currents and uneven bottomhttps://zbmath.org/1508.760302023-05-31T16:32:50.898670Z"Fan, Lili"https://zbmath.org/authors/?q=ai:fan.lili"Liu, Ruonan"https://zbmath.org/authors/?q=ai:liu.ruonan"Gao, Hongjun"https://zbmath.org/authors/?q=ai:gao.hongjunSummary: A Hamiltonian model for the propagation of internal water waves interacting with surface waves, a current and an uneven bottom is examined. Using the so-called Dirichlet-Neumann operators, the water wave system is expressed in the Hamiltonian form, and thus the motions of the internal waves and surface waves are determined by the Hamiltonian formulation. Choosing an appropriate scaling of the variables and employing the Hamiltonian perturbation theory from Hamiltonian formulation of the dynamics, we derive a KdV-type equation with variable coefficients depending on the bottom topography to describe the internal waves.On the interaction between oncoming internal waves and a dense gravity current in a two-layer stratificationhttps://zbmath.org/1508.760312023-05-31T16:32:50.898670Z"Tanimoto, Yukinobu"https://zbmath.org/authors/?q=ai:tanimoto.yukinobu"Ouellette, Nicholas T."https://zbmath.org/authors/?q=ai:ouellette.nicholas-t"Koseff, Jeffrey R."https://zbmath.org/authors/?q=ai:koseff.jeffrey-rSummary: A series of laboratory experiments was conducted to investigate the dynamics of a dense gravity current flowing down an inclined slope into a two-layer stratification in the presence of oncoming internal interfacial waves. The experiment is set up such that the gravity current propagates towards a wave maker emitting interfacial waves such that the current and waves propagate in opposite directions. The results were compared with the case of gravity current without oncoming waves. The gravity current splits into a portion that inserts itself into the pycnocline as an interflow and another that propagates down the slope as an underflow, with the proportionality depending on the characteristics of the gravity current and the oncoming waves when they are present. The interflow is shown to arise from a combination of detrainment and the preferential insertion of fluid with density greater than the upper layer and less than lower layer along the pycnocline. The mass flux of the interflow is observed to be reduced by the oncoming waves, as waves act to decrease the interflow velocity. The internal waves also increase the path length that the interflow must travel. A combination of reduced velocities and increased path length explains the observed reduction in cumulative flux. The trend of the final cumulative flux is consistent with the mass change observed by comparing density profiles obtained before and after the experiment.Triadic resonant instability in confined and unconfined axisymmetric geometrieshttps://zbmath.org/1508.760472023-05-31T16:32:50.898670Z"Boury, S."https://zbmath.org/authors/?q=ai:boury.samuel"Maurer, P."https://zbmath.org/authors/?q=ai:maurer.peter-m"Joubaud, S."https://zbmath.org/authors/?q=ai:joubaud.sylvain"Peacock, T."https://zbmath.org/authors/?q=ai:peacock.thomas"Odier, P."https://zbmath.org/authors/?q=ai:odier.philippeSummary: We present an investigation of the resonance conditions governing triad interactions of cylindrical internal waves, i.e. Kelvin modes, described by Bessel functions. Our analytical study, supported by experimental measurements, is performed both in confined and unconfined axisymmetric domains. We are interested in two conceptual questions: can we find resonance conditions for a triad of Kelvin modes? What is the impact of the boundary conditions on such resonances? In both the confined and unconfined cases, we show that sub-harmonics can be spontaneously generated from a primary wave field if they satisfy at least a resonance condition on their frequencies of the form \(\omega_0 = \pm\omega_1\pm\omega_2\). We demonstrate that the resulting triad is also spatially resonant, but that the resonance in the radial direction may not be exact in confined geometries due to the prevalence of boundary conditions -- a key difference compared with Cartesian plane waves.Onsager's conjecture for subgrid scale \(\alpha\)-models of turbulencehttps://zbmath.org/1508.760672023-05-31T16:32:50.898670Z"Boutros, Daniel W."https://zbmath.org/authors/?q=ai:boutros.daniel-w"Titi, Edriss S."https://zbmath.org/authors/?q=ai:titi.edriss-salehSummary: The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if \(u(\cdot, t)\in C^{0, \theta}(\mathbb{T}^3)\) with \(\theta > \frac{1}{3}\). In this paper, we prove an analogue of Onsager's conjecture for several subgrid scale \(\alpha\)-models of turbulence. In particular we find the required Hölder regularity of the solutions that ensures the conservation of energy-like quantities (either the \(H^1(\mathbb{T}^3)\) or \(L^2(\mathbb{T}^3)\) norms) for these models.
We establish such results for the Leray-\(\alpha\) model, the Euler-\(\alpha\) equations (also known as the inviscid Camassa-Holm equations or Lagrangian averaged Euler equations), the modified Leray-\(\alpha\) model, the Clark-\(\alpha\) model and finally the magnetohydrodynamic Leray-\(\alpha\) model. In a sense, all these models are inviscid regularisations of the Euler equations; and formally converge to the Euler equations as the regularisation length scale \(\alpha\to0^+\).
Different Hölder exponents, smaller than 1/3, are found for the regularity of solutions of these models (they are also formulated in terms of Besov and Sobolev spaces) that guarantee the conservation of the corresponding energy-like quantity. This is expected due to the smoother nonlinearity compared to the Euler equations. These results form a contrast to the universality of the 1/3 Onsager exponent found for general systems of conservation laws by \textit{P. Gwiazda} et al. [Arch. Ration. Mech. Anal. 229, No. 3, 1223--1238 (2018; Zbl 1398.35168)] and \textit{C. Bardos} et al. [Commun. Math. Phys. 370, No. 1, 291--310 (2019; Zbl 1420.35203); J. Nonlinear Sci. 29, No. 2, 501--510 (2019; Zbl 1420.35202)].