Vladimir V. Chepyzhov, Mark Vishik,
We consider a non-autonomous reaction-diffusion system of two equations having in one equation a diffusion coefficient depending on time ($\delta =\delta (t)\geq 0,t\geq 0$) such that $\delta (t)\rightarrow 0$ as $t\rightarrow +\infty $. The corresponding Cauchy problem has global weak solutions, however these solutions are not necessarily unique. We also study the corresponding "limit'' autonomous system for $\delta =0.$ This reaction-diffusion system is partly dissipative. We construct the trajectory attractor A ...
Tópico(s): Nonlinear Dynamics and Pattern Formation
2010 - American Institute of Mathematical Sciences | Discrete and Continuous Dynamical Systems
Vladimir V. Chepyzhov, Edriss S. Titi, Mark Vishik,
We study the relations between the global dynamics of the 3DLeray-$\alpha $ model and the 3D Navier-Stokes system. We provethat time shifts of bounded sets of solutions of the Leray-$\alpha$ model converges to the trajectoryattractor of the 3D Navier-Stokes system as time tends to infinity and $\alpha $ approaches zero. In particular, we show that thetrajectory attractor of the Leray-$\alpha $ model converges to thetrajectory attractor of the 3D Navier-Stokes system when $\alpha\rightarrow 0\+.$
Tópico(s): Fluid Dynamics and Turbulent Flows
2007 - American Institute of Mathematical Sciences | Discrete and Continuous Dynamical Systems
For rapidly spatially oscillating nonlinearities f and inhomogeneities g we compare solutions u ε of reaction–diffusion systems ∂ t u ε =aΔu ε −f(ε,x,x/ε,u)+g(ε,x,x/ε) with solutions u 0 of the formally homogenized, spatially averaged system ∂ t u 0 =aΔu 0 −f 0 (x,u 0 )+g 0 (x,u 0 ). We consider a smooth bounded domain x∈Ω⊆ $\mathbb{R}^{n}$ , n≤3, with Dirichlet boundary conditions. We also impose sufficient regularity and dissipation conditions, such that solutions exist globally in time and, in fact, converge to ...
Tópico(s): Nonlinear Dynamics and Pattern Formation
2003 - IOS Press | Asymptotic Analysis
Based on an analytic semigroup setting, we first consider semilinear reaction--diffusion equations with spatially quasiperiodic coefficients in the nonlinearity, rapidly varying on spatial scale $\varepsilon$. Under periodic boundary conditions, we derive quantitative homogenization estimates of order $\varepsilon^\gamma$ on strong Sobolev spaces $H^\sigma$ in the triangle $$0 <\gamma < \min (\sigma -n/2,2-\sigma).$$ Here $n$ denotes spatial dimension. The estimates measure the distance to a solution of the homogenized ...
Tópico(s): Fractional Differential Equations Solutions
2001 - | Advances in Differential Equations
Vladimir V. Chepyzhov, Mark Vishik,
Introduction Attractors of autonomous equations: Attractors of autonomous ordinary differential equations Attractors of autonomous partial differential equations Dimension of attractors Attractors of non-autonomous equations: Processes and attractors Translation compact functions Attractors of non-autonomous partial differential equations Semiprocesses and attractors Kernels of processes Kolmogorov $\varepsilon$-entropy of attractors Trajectory attractors: Trajectory attractors of autonomous ordinary ...
Tópico(s): advanced mathematical theories
2001 - American Mathematical Society | Colloquium Publications - American Mathematical Society/Colloquium Publications
Vittorino Pata, Giovanni Prouse, Mark Vishik,
A dissipative nonlinear nonautonomous equation of hyperbolic type with translation-compact forcing term is studied, with particular reference to the problem of traveling waves in a strip and related attractor.
Tópico(s): Nonlinear Dynamics and Pattern Formation
1998 - | Advances in Differential Equations
Bernold Fiedler, Arnd Scheel, Mark Vishik,
We consider systems of elliptic equations ∂t2u + Δxu + γ∂tu + f(u) = 0, u(t, x) ∈ RN in unbounded cylinders (t, x) ∈ R × Ω with bounded cross-section Ω ⊂ Rn and Dirichlet boundary conditions. We establish existence of bounded solutions u(t, x) with non-trivial dependence on t ∈ R, ∂tu(t, x) n= 0. Our main assumptions are dissipativity of the nonlinearity f and the existence of at least two t-independent solutions w1(x), w2(x) which solve Δxwj + f(wj) = 0, j = 1, 2. The proof exploits the dynamical systems structure of the ...
Tópico(s): Nonlinear Partial Differential Equations
1998 - Elsevier BV | Journal de Mathématiques Pures et Appliquées
Vladimir V. Chepyzhov, Mark Vishik,
Tópico(s): Nonlinear Dynamics and Pattern Formation
1997 - Elsevier BV | Journal de Mathématiques Pures et Appliquées
Vladimir V. Chepyzhov, Mark Vishik,
(1) ∂tu+ νLu+B(u) = g(x, t), (∇, u) = 0, u|∂Ω = 0, x ∈ Ω b R, t ≥ 0, u = u(x, t) = (u, u) ≡ u(t), g = g(x, t) = (g, g) ≡ g(t). Here Lu = −P∆u is the Stokes operator, ν > 0, B(u) = P ∑2 i=1 ui∂xiu; P is the orthogonal projector onto the space of divergence-free vector fields (see Section 1). Consider the autonomous case: g(x, t) ≡ g(x), g ∈ H, to begin with. Suppose for t = 0 we are given the initial condition
Tópico(s): Advanced Mathematical Modeling in Engineering
1996 - Juliusz Schauder University Center for Nonlinear Studies | Topological Methods in Nonlinear Analysis
Vladimir V. Chepyzhov, Mark Vishik,
(1) ∂tu = a∆u− f0(u, t) + g0(x, t), u|∂Ω = 0 (or ∂u/∂ν|∂Ω = 0) where u = u(x, t) = (u, . . . , u ), x ∈ Ω b R, t ≥ 0, f0(v, s) = (f 0 , . . . , f 0 ), (v, s) ∈ R × R+, g0(x, s) = (g 0 , . . . , g 0 ), x ∈ Ω, s ≥ 0. We assume that the matrix a and the functions f0, g0 satisfy some general conditions (see Section 2). These conditions provide the existence of a solution u of the Cauchy problem for the system (1) (u|t=0 = u0, u0 ∈ H = (L2(Ω)) ). However, this solution can be non-unique because we do not suppose any Lipschitz conditions for f0(v, s) with respect to v. The pair of functions ( ...
Tópico(s): Nonlinear Dynamics and Pattern Formation
1996 - Juliusz Schauder University Center for Nonlinear Studies | Topological Methods in Nonlinear Analysis
Vladimir V. Chepyzhov, Mark Vishik,
Tópico(s): Quantum chaos and dynamical systems
1994 - Juliusz Schauder University Center for Nonlinear Studies | Topological Methods in Nonlinear Analysis
... Navier-Stokes equations. Dedicated to the memory ofProfessor Mark I. Vishik.b(x, r) = p(x, r) + 1 4πr ...
Tópico(s): Computational Fluid Dynamics and Aerodynamics
2014 - IOP Publishing | Russian Mathematical Surveys
... survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini- ...
Tópico(s): Nonlinear Dynamics and Pattern Formation
2023 - IOP Publishing | Успехи математических наук
The name of Mark Vishik is one of the very first names that I heardwhen I started research in mathematics in 1964. One year before in1963, Mark published an article that had a very deep influence on thetheory of nonlinear partial differential equations all along the1960s, although ...
Tópico(s): Numerical methods for differential equations
2003 - American Institute of Mathematical Sciences | Discrete and Continuous Dynamical Systems
Andrew Comech, Alexander Komech, M. M. Vishik,
Mark Vishik is a truly original and superb mathematician. He has made many fundamental contributions in the theory ...
Tópico(s): Aquatic and Environmental Studies
2023 - Springer Nature | Trends in mathematics
In early 1960s Mark Vishik wrote a series of articles on quasilinear elliptic and parabolic equations and systems.
Tópico(s): Mathematical and Theoretical Analysis
2023 - Springer Nature | Trends in mathematics
Mark Iosifovich Vishik was my husband Vladimir Chepyzhov's advisor during his years as a student in the Faculty ...
Tópico(s): advanced mathematical theories
2014 - IOP Publishing | Russian Mathematical Surveys
... are based on the stories of my father, Mark Vishik, the way I remember them, as well as ...
2023 - Springer Nature | Trends in mathematics
2023 - Springer Nature | Trends in mathematics
Andrew Comech, Alexander Komech, M. I. Vishik,
РЁ. РўРӌСЇРӌРҝ, РЁРҳСЂР«Рҡ ЫѥрЫРүЫЄРөРҧСӔР„СҜСҖ лѥЫРҳРҝСҘРӌРє РҡРҧСЅ РҡРӌСҺСҺРҳСҐРҳР„СҘРӌРөРҧСӔР„СҜСҖ СҔСҐРөРўР„РҳР„РӌРє СҢРҧРҧРӌлрРӌСҦРҳСҒРҝР«РүР« СЂРӌР»Рө. РҕСҒР». РЁРөСЂРҳРё. Р№РөСҔРҝ 2, 5(21), 194–196 (1947)
Tópico(s): Weber, Simmel, Sociological Theory
2023 - Springer Nature | Trends in mathematics
Andrew Comech, Alexander Komech, M. M. Vishik,
M. Agranovich: Remarks on strongly elliptic systems in Lipschitz domains
Tópico(s): Differential Equations and Numerical Methods
2023 - Springer Nature | Trends in mathematics
We apply the dynamical approach to the study of the second ordersemi-linear elliptic boundary value problem in a cylindrical domainwith a small parameter $\varepsilon$ at the second derivative with respect tothe variable $t$ corresponding to the axis of the cylinder.We prove that, under natural assumptions on the nonlinear interactionfunction $f$ and the external forces $g(t)$, this problem possessesthe uniform attractor $\mathcal A_\varepsilon$ and that these attractors tendas $\varepsilon \to 0$ to the ...
Tópico(s): Advanced Mathematical Physics Problems
2014 - American Institute of Mathematical Sciences | Communications on Pure &Applied Analysis
Tópico(s): Sociopolitical Dynamics in Russia
2002 - American Mathematical Society | Translations - American Mathematical Society/Translations
Vladimir V. Chepyzhov, Mark Vishik,
Tópico(s): Nonlinear Dynamics and Pattern Formation
1992 - Springer Science+Business Media | Rendiconti del Seminario Matematico e Fisico di Milano
Andrew Comech, Alexander Komech, M. M. Vishik,
In this book, friends & pupils of Mark Vishik remember his life & work. Mark Vishik is a prominent figure in the theory of partial differential equations.
Tópico(s): Numerical methods for differential equations
2023 - Springer Nature | Trends in mathematics
In these notes I share my recollections of Mark Vishik that have remained in my memory. I tried ... passed. When it comes to my work with Mark Vishik on the theory of attractors, I remember the ...
Tópico(s): Mathematical Biology Tumor Growth
2023 - Springer Nature | Trends in mathematics
... 1965, when I was a third-year student, Mark Iosifovich Vishik became my scientific advisor. At the time, the ...
Tópico(s): Advanced Mathematical Physics Problems
2023 - Springer Nature | Trends in mathematics
Mark Iosifovich Vishik was a great man. I could just limit myself to listing the names of the mathematicians ...
Tópico(s): advanced mathematical theories
2023 - Springer Nature | Trends in mathematics
One day, Vladimir Arnold asked me to write a review of an article submitted to >. The paper had to do with the classification of stationary two-dimensional flow of an ideal incompressible fluid. In my eyes, the work was uninteresting and vapid, full of nothing but trivialities. That is what I wrote in my review.
Tópico(s): Aquatic and Environmental Studies
2023 - Springer Nature | Trends in mathematics
Tópico(s): Numerical methods for differential equations
2023 - Springer Nature | Trends in mathematics