Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$
Sergey Zelik Jon Pennant
Communications on Pure & Applied Analysis 2013, 12(1): 461-480 doi: 10.3934/cpaa.2013.12.461
We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.
keywords: infinite-energy solutions. unbounded domains Cahn-Hilliard equation
Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces
Sergey Zelik
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 781-810 doi: 10.3934/dcdsb.2015.20.781
We give a comprehensive study of strong uniform attractors of nonautonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space to be verified and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
keywords: Non-autonomous equations energy method. dissipative systems asymptotic compactness uniform attractors
Uniform exponential attractors for a singularly perturbed damped wave equation
Pierre Fabrie Cedric Galusinski A. Miranville Sergey Zelik
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 211-238 doi: 10.3934/dcds.2004.10.211
Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
keywords: reaction-diffusion equations time boundary layer. Singularly perturbed damped wave equations uniform exponential attractors
Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent
Sergey Zelik
Communications on Pure & Applied Analysis 2004, 3(4): 921-934 doi: 10.3934/cpaa.2004.3.921
The paper is devoted to study of the longtime behavior of solutions of a damped semilinear wave equation in a bounded smooth domain of $\mathbb R^3$ with the nonautonomous external forces and with the critical cubic growth rate of the nonlinearity. In contrast to the previous papers, we prove the dissipativity of this equation in higher energy spaces $E^\alpha$, $0<\alpha\le 1$, without the usage of the dissipation integral (which is infinite in our case).
keywords: nonautonomous attractors Damped wave equations critical growth rate regularity of attractors.
Analytical proof of space-time chaos in Ginzburg-Landau equations
Dmitry Turaev Sergey Zelik
Discrete & Continuous Dynamical Systems - A 2010, 28(4): 1713-1751 doi: 10.3934/dcds.2010.28.1713
We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDS's with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDS's. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.
keywords: extended systems attractors of PDE's in unbounded domains soliton interaction normal hyperbolicity center-manifold reduction multi-pulse solutions lattice dynamical systems.
Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities
Varga K. Kalantarov Sergey Zelik
Communications on Pure & Applied Analysis 2012, 11(5): 2037-2054 doi: 10.3934/cpaa.2012.11.2037
We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.
keywords: attractors Brinkmann-Forchheimer equations maximal regularity nonlinear localization.
Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions
Anna Kostianko Sergey Zelik
Communications on Pure & Applied Analysis 2015, 14(5): 2069-2094 doi: 10.3934/cpaa.2015.14.2069
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
keywords: inertial manifold. spatial averaging principle Cahn-Hilliard equation
On the Lyapunov dimension of cascade systems
Sergey Zelik
Communications on Pure & Applied Analysis 2008, 7(4): 971-985 doi: 10.3934/cpaa.2008.7.971
In this paper we obtain sharp upper estimates on the uniform Lyapunov dimension of a cascade system in terms of the corresponding Lyapunov exponents of their components. The obtained result is applied for estimating the Lyapunov and fractal dimensions of the attractors of nonautonomous dissipative systems generated by PDEs of mathematical physics.
keywords: Nonautonomous dynamical systems skew-product flows Lyapunov and fractal dimensions.
Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit
Mark I. Vishik Sergey Zelik
Communications on Pure & Applied Analysis 2014, 13(5): 2059-2093 doi: 10.3934/cpaa.2014.13.2059
We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter $\varepsilon$ at the second derivative with respect to the variable $t$ corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function $f$ and the external forces $g(t)$, this problem possesses the uniform attractor $\mathcal A_\varepsilon$ and that these attractors tend as $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolic equation. Moreover, in case where the limit attractor $\mathcal A_0$ is regular, we give the detailed description of the structure of the uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, and estimate the symmetric distance between the attractors $\mathcal A_\varepsilon$ and $\mathcal A_0$.
keywords: regular attractors. uniform attractors dynamical approach Elliptic boundary value problems
A result on the existence of global attractors for semigroups of closed operators
V. Pata Sergey Zelik
Communications on Pure & Applied Analysis 2007, 6(2): 481-486 doi: 10.3934/cpaa.2007.6.481
In this note, we establish a general result on the existence of global attractors for semigroups $S(t)$ of operators acting on a Banach space $\mathcal X$, where the strong continuity $S(t)\in C(\mathcal X,\mathcal X)$ is replaced by the much weaker requirement that $S(t)$ be a closed map.
keywords: global attractors Semigroups of operators connected attractors. closed operators abstract Cauchy problems

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