Recent zbMATH articles in MSC 35K55https://zbmath.org/atom/cc/35K552021-05-28T16:06:00+00:00WerkzeugEfficient, non-iterative, and decoupled numerical scheme for a new modified binary phase-field surfactant system.https://zbmath.org/1459.651622021-05-28T16:06:00+00:00"Xu, Chen"https://zbmath.org/authors/?q=ai:xu.chen"Chen, Chuanjun"https://zbmath.org/authors/?q=ai:chen.chuanjun"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: We consider in this paper numerical approximations of a Cahn-Hilliard binary phase-field fluid-surfactant model. By adding a quartic form of the gradient potential, we first modify the commonly used total free energy into a form which is bounded from below and establish the energy law for the new system. Then we develop a stabilized-SAV scheme that combines the SAV approach with the stabilization technique, where a crucial linear stabilization term is added to enhance the stability thus allowing large time steps. With many desired properties such as a second-order in time, totally decoupled, linear, and non-iterative, this scheme is unconditionally energy stable and requires solving only four decoupled and linear biharmonic equations with constant coefficients at each time step. We further prove the energy stability and present numerous 2D and 3D numerical simulations to demonstrate the accuracy and stability of the developed schemeNonstandard finite difference schemes. Methodology and applications. 2nd edition.https://zbmath.org/1459.650032021-05-28T16:06:00+00:00"Mickens, Ronald E."https://zbmath.org/authors/?q=ai:mickens.ronald-ePublisher's description: This second edition provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.
Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the ``principle of dynamic consistency'' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.
See the review of the first edition in [Zbl 0810.65083].Transport distances for PDEs: the coupling method.https://zbmath.org/1459.350372021-05-28T16:06:00+00:00"Fournier, Nicolas"https://zbmath.org/authors/?q=ai:fournier.nicolas-g"Perthame, Benoît"https://zbmath.org/authors/?q=ai:perthame.benoitSummary: We informally review a few PDEs for which some transport cost between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This means that we double the variables and solve, globally in time, a well-chosen PDE posed on the Euclidean square of the physical space. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.Boundedness and stabilization enforced by mild saturation of taxis in a producer-scrounger model.https://zbmath.org/1459.920182021-05-28T16:06:00+00:00"Cao, Xinru"https://zbmath.org/authors/?q=ai:cao.xinru"Tao, Youshan"https://zbmath.org/authors/?q=ai:tao.youshanThe authors study the taxis system
\[\begin{cases} u_t = \Delta u - \nabla \cdot (u(u+1)^{\beta -1} \nabla w), & \quad (x,t) \in \Omega \times (0,\infty), \\
v_t = \Delta v - \nabla \cdot (v(v+1)^{\alpha -1} \nabla u), & \quad (x,t) \in \Omega \times (0,\infty), \\
w_t = \Delta w -(u+v)w - \lambda w + g(x,t), & \quad (x,t) \in \Omega \times (0,\infty),
\end{cases} \]
endowed with homogeneous Neumann boundary conditions and nonnegative initial conditions \(u_0, v_0, w_0 \in W^{2,\infty} (\Omega)\) such that \(u_0 \not\equiv 0\) and \(v_0 \not\equiv 0\), where \(\Omega \subset \mathbb{R}^n\), \(n \ge 2\), is a bounded domain with smooth boundary, \(\lambda >0\), \(\beta <1 \), \(\alpha < \frac{n+2}{2n}\), and \(g \in C^1(\overline{\Omega} \times[0,\infty)) \cap L^\infty (\Omega \times (0,\infty))\).
The system describes a foreager-prey model, where \(w\) denotes the prey density and the foreagers consist of producers with density \(u\) and scroungers with density \(v\). The producers move toward increasing prey densities, while the scroungers move toward increasing producer densities.
Assuming that the above conditions are satisfied, the authors prove the existence of a global classical solution. If in addition \(\lim_{t \to \infty} \int_t^{t+1}\int_\Omega g^2(x,s) \, dx ds =0\), then \((u,v,w)(t) \to (\overline{u_0}, \overline{v_0}, 0)\) as \(t \to \infty\) in \((L^\infty (\Omega))^3\), where \(\overline{u_0} := \frac{1}{|\Omega|}\int_\Omega u_0(x) \, dx\) is the spatial average.
The proof of the global existence relies on proving a series of estimates for the solution components. Therein, a uniform estimate for \(u(t)\) in \(L^p(\Omega)\) for any \(p \in (1,\infty)\) is proved with a weighted integral approach and weighted maximal Sobolev regularity. The most difficult estimate for \(v(t)\) in \(L^p(\Omega)\) for any \(p \in (1,\infty)\) is based on a loop of weighted estimates based on weighted maximal Sobolev regularity and is reduced to an estimate of \(\Delta u\), which in turn is reduced to estimating \(\Delta w\). The latter is finally reduced to an estimate of \(v\) which can be absorbed by a diffusion term. Such a loop seems to be new in the context of analysis of foreager-prey models.
Reviewer: Christian Stinner (Darmstadt)Bernstein type theorem for the generalized parabolic 2-Hessian equation under weaker conditions.https://zbmath.org/1459.350152021-05-28T16:06:00+00:00"Takimoto, Kazuhiro"https://zbmath.org/authors/?q=ai:takimoto.kazuhiroSummary: We deal with the characterization of entire solutions to the generalized parabolic 2-Hessian equation of the form \(u_t = \mu( F_2 ( D^2 u )^{1 / 2})\) in \(\mathbb{R}^n \times(- \infty, 0]\). We prove that any strictly 2-convex-monotone solution \(u = u(x, t) \in C^{4 , 2}( \mathbb{R}^n \times(- \infty, 0])\) must be a linear function of \(t\) plus a quadratic polynomial of \(x\), under some assumptions on \(\mu :(0, \infty) \to \mathbb{R} \), some growth conditions on \(u\) and the boundedness of the 3-Hessian of \(u\) from below.Global existence for the \(p\)-Sobolev flow.https://zbmath.org/1459.352622021-05-28T16:06:00+00:00"Kuusi, Tuomo"https://zbmath.org/authors/?q=ai:kuusi.tuomo"Misawa, Masashi"https://zbmath.org/authors/?q=ai:misawa.masashi"Nakamura, Kenta"https://zbmath.org/authors/?q=ai:nakamura.kentaThe authors study a doubly nonlinear parabolic equation arising from the gradient flow for a $p$-Sobolev-type inequality, referred to as $p$-Sobolev flow. In the special case $p=2$, their results include the classical Yamabe flow on a bounded domain in the Euclidean space. They prove the global existence of the $p$-Sobolev flow that is established by applying a nonlinear intrinsic scaling transformation and a deep and subtle argument of expansion of positivity.
Reviewer: Vincenzo Vespri (Firenze)Energy conservation for weak solutions of a surface growth model.https://zbmath.org/1459.352322021-05-28T16:06:00+00:00"Yang, Jiaqi"https://zbmath.org/authors/?q=ai:yang.jiaqiSummary: We are concerned with one-dimensional scalar surface growth model arising from the physical process of molecular epitaxy. The mathematical theory of the surface growth model is known to share a number of striking similarities with the Navier-Stokes equations, including the results regarding existence and uniqueness of solutions. In this paper, we shall investigate an important subject in mathematical physics: the energy conservation for weak solutions of the surface growth model. As an analogue of the Navier-Stokes equations, we find some sufficient integral conditions that guarantee the validity of energy equality. As far as we know, this is the first result in this aspect.Optimal control of stochastic phase-field models related to tumor growth.https://zbmath.org/1459.354152021-05-28T16:06:00+00:00"Orrieri, Carlo"https://zbmath.org/authors/?q=ai:orrieri.carlo"Rocca, Elisabetta"https://zbmath.org/authors/?q=ai:rocca.elisabetta"Scarpa, Luca"https://zbmath.org/authors/?q=ai:scarpa.lucaSummary: We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.Improved regularity for the inhomogeneous porous medium equation.https://zbmath.org/1459.350632021-05-28T16:06:00+00:00"Diehl, Nicolau M. L."https://zbmath.org/authors/?q=ai:diehl.nicolau-m-lThe author studies the sharp regularity for the solutions of \[u_t-\operatorname{div}(m|u|^{m-1}\nabla u) = f \in L^{q,r},\] with \(m> 1\), in function of the optimal Hölder exponent for solutions of the homogeneous case. This result extends the recent one obtained by Araújo, Maia and Urbano \textit{D. J. Araújo} et al. [J. Anal. Math. 140, No. 2, 395--407 (2020; Zbl 1437.35131)]. For the porous medium equation, unlike other known examples, the main difficulty is that the intrinsic parabolic geometry of the equation and the available regularity do not match. Despite this difficulty, the author is able to combine the known regularity results and the optimal local regularity of the associated homogeneous equation to prove this interesting optimal regularity result.
Reviewer: Vincenzo Vespri (Firenze)Local boundedness and Hölder continuity for the parabolic fractional \(p\)-Laplace equations.https://zbmath.org/1459.350532021-05-28T16:06:00+00:00"Ding, Mengyao"https://zbmath.org/authors/?q=ai:ding.mengyao"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.1"Zhou, Shulin"https://zbmath.org/authors/?q=ai:zhou.shulinThe authors study the boundedness and Hölder continuity of local weak solutions to the
fractional parabolic $p$-Laplace operator of $s$-order. Under suitable some structural conditions, they use the De Giorgi-Nash-Moser iteration to establish the a priori boundedness of local weak solutions. Thanks to this preliminary result, they obtain Hölder
continuity of bounded solutions in the superquadratic case. These results extend the elliptic ones due to Di Castro, Kuusi and Palatucci.
Reviewer: Vincenzo Vespri (Firenze)Existence of positive solutions and asymptotic behavior for evolutionary \(q(x)\)-Laplacian equations.https://zbmath.org/1459.352632021-05-28T16:06:00+00:00"Marcos, Aboubacar"https://zbmath.org/authors/?q=ai:marcos.aboubacar"Soglo, Ambroise"https://zbmath.org/authors/?q=ai:soglo.ambroiseSummary: In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving \(q(x)\)-Laplacian parabolic equation \(\left( \partial \rho \left( t, x\right) / \partial t\right)= \operatorname{div}_x\left( \left. \rho \left( t, x\right)\right| \nabla_x \left. \left( G^\prime \left( \rho\right) + V\right)\right|^{q \left( x\right) - 2} \nabla_x \left( G^\prime \left( \rho\right) + V\right)\right)\). The potential \(V\) is not necessarily smooth but belongs to a Sobolev space \(W^{1, \infty}\left( \Omega\right)\). Given the initial datum \(\rho_0\) as a probability density on \(\Omega \), we use a descent algorithm in the probability space to discretize the \(q(x)\)-Laplacian parabolic equation in time. Then, we use compact embedding \(W^{1, q \left( .\right)}\left( \Omega\right) \hookrightarrow \hookrightarrow L^{q \left( .\right)}\left( \Omega\right)\) established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the \(q(x)\)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the \(q(x)\)-Laplacian parabolic equation to equilibrium in the \(p(.)\)-variable exponent Wasserstein space.Exact and approximate solutions of a problem with a singularity for a convection-diffusion equation.https://zbmath.org/1459.352422021-05-28T16:06:00+00:00"Kazakov, A. L."https://zbmath.org/authors/?q=ai:kazakov.aleksandr-leonidovich"Spevak, L. F."https://zbmath.org/authors/?q=ai:spevak.lev-fridrikhovichSummary: Solutions to a nonlinear parabolic convection-diffusion equation are constructed in the form of a diffusion wave that propagates over a zero background at a finite velocity. The theorem of existence and uniqueness of the solution is proven. The solution is constructed in the form of a characteristic series whose coefficients are determined using a recurrent procedure. Exact solutions of the considered type and their characteristics, including the domain of existence, are determined, and the behavior of these solutions on the boundaries of this domain of existence is studied. The boundary element method and the dual reciprocity method are used to develop, implement, and test an algorithm for constructing approximate solutions.Uniqueness of two-convex closed ancient solutions to the mean curvature flow.https://zbmath.org/1459.530802021-05-28T16:06:00+00:00"Angenent, Sigurd"https://zbmath.org/authors/?q=ai:angenent.sigurd-bernardus"Daskalopoulos, Panagiota"https://zbmath.org/authors/?q=ai:daskalopoulos.panagiota"Sesum, Natasa"https://zbmath.org/authors/?q=ai:sesum.natasaThe authors study the classification of closed non-collapsed ancient solutions of mean curvature flow (\(n\ge 2\)) that are uniformly two-convex. They prove either that they are contracting spheres or they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution first constructed by \textit{B. White} [J. Am. Math. Soc. 13, No. 3, 665--695 (2000; Zbl 0961.53039); J. Am. Math. Soc. 16, No. 1, 123--138 (2003; Zbl 1027.53078)] and later by \textit{R. Haslhofer} and \textit{O. Hershkovits} [Commun. Anal. Geom. 24, No. 3, 593--604 (2016; Zbl 1345.53068)].
Reviewer: Shu-Yu Hsu (Chiayi)Propagation acceleration in reaction diffusion equations with anomalous diffusions.https://zbmath.org/1459.350682021-05-28T16:06:00+00:00"Coville, Jérôme"https://zbmath.org/authors/?q=ai:coville.jerome"Gui, Changfeng"https://zbmath.org/authors/?q=ai:gui.changfeng"Zhao, Mingfeng"https://zbmath.org/authors/?q=ai:zhao.mingfengModels of collective movements with negative degenerate diffusivities.https://zbmath.org/1459.352592021-05-28T16:06:00+00:00"Corli, Andrea"https://zbmath.org/authors/?q=ai:corli.andrea"Malaguti, Luisa"https://zbmath.org/authors/?q=ai:malaguti.luisaSummary: We consider an advection-diffusion equation whose diffusivity can be negative. This equation arises in the modeling of collective movements, where the negative diffusivity simulates an aggregation behavior. Under suitable conditions we prove the existence, uniqueness and qualitative properties of traveling-wave solutions connecting states where the diffusivity has opposite signs. These results are extended to end states where the diffusivity is positive but is negative in between. The vanishing-viscosity limit is also considered. Examples from real-world models are provided.
For the entire collection see [Zbl 1453.35003].The Evans-Krylov theorem for nonlocal parabolic fully nonlinear equations.https://zbmath.org/1459.470182021-05-28T16:06:00+00:00"Kim, Yong-Cheol"https://zbmath.org/authors/?q=ai:kim.yong-cheol"Lee, Ki-Ahm"https://zbmath.org/authors/?q=ai:lee.ki-ahmThe authors prove a parabolic version of the nonlocal elliptic result of \textit{L. Caffarelli} and \textit{L. Silvestre} [Ann. Math. (2) 174, No. 2, 1163--1187 (2011; Zbl 1232.49043)]. As main techniques, they use Cordes-Nirenberg-type estimates (and an approximation method) for their nonlocal parabolic concave equation, as well as Harnack-type inequalities (to derive local uniform upper boundedness of consequent viscosity subsolutions).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Quasi-stability and upper semicontinuity for coupled parabolic equations with memory.https://zbmath.org/1459.370652021-05-28T16:06:00+00:00"Aouadi, Moncef"https://zbmath.org/authors/?q=ai:aouadi.moncefThe present work is devoted to the study of the long-time dynamics of a nonlinear system of coupled parabolic equations with memory. This system describes a thermodiffusion phenomenon, where the fluxes of mass diffusion and heat are depending on the past history of the chemical potential and the temperature gradients, respectively, according to Gurtin-Pipkin law. Inspired by the works of \textit{I. Chueshov} and \textit{I. Lasiecka} [Von Karman evolution equations. Well-posedness and long-time dynamics. New York, NY: Springer (2010; Zbl 1298.35001)]
on the property of quasistability of dynamic systems, the author proves this property for the considered problem. The advantage of the quasi-stability approach is that, in contrast to the standard decomposition method, this method assures the possibility of representing the difference of two trajectories as a sum of compact and exponentially stable parts. This guarantees the finite dimensionality of attractors. Using the multiplier functional technique to construct a suitable Lyapunov functional, the author establishes a stabilizability inequality, leading to the asymptotic smoothness property of the dynamical system associated with the considered equations. This allows one to deduce the existence and finite dimensionality of global and exponential attractors. Furthermore, the author derives the upper semicontinuity of global attractors as the damping terms disappear, that is, either \(\alpha_{1}\) and \(\alpha_{2}\) only depends on \(\varepsilon \) or \(\alpha_{1}(t)\) and \(\alpha_{2}(t)\) are bounded. Note that the upper semicontinuity problem of global attractors with arbitrary time-dependent perturbation is still open.
Reviewer: Andrey Zahariev (Plovdiv)