Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium

In this paper we give a detailed study of the global attractors for parabolic equations governed by the p-Laplacian in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is presented by showing that their ε-Kolmogorov entropy behaves as a polynomial of the variable 1/ε as ε tends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ε-entropy of infinite-dimensional attractors are also obtained.     

Infinite‐dimensional exponential attractors for fourth‐order nonlinear parabolic equations in unbounded domains

We study in this article the long‐time behavior of solutions of fourth‐order parabolic equations in bfR³. In particular, we prove that under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess infinite‐dimensional exponential attractors whose Kolmogorov's ε‐entropy satisfies an estimate of the same type as that obtained previously for the ε‐entropy of the global attractor. Copyright © 2011 John Wiley & Sons, Ltd.     

On the dimension of global attractors for porous media equations

We provide a short proof on the infinite dimensionality of global attractors for a class of porous media equations. The proof is mainly based on the Z₂ index theory and proper use of energy functions and is completely different from the approaches in the existing literatures (M. Efendiev and S. Zelik, Finite and infinite dimensional attractors for porous media equations, Proc. London Math. Soc. 2008, 96:51–77; M. Efendiev, Infinite dimensional attractors for porous medium equations in heterogeneous medium, Math. Meth. Appl. Sci. 2012, DOI: 10.1002/mma.2619). Copyright © 2013 John Wiley & Sons, Ltd.     

Uniform attractors for non-autonomous motion equations of viscoelastic medium

Sufficient conditions for existence of minimal uniform trajectory attractors and uniform global attractors of non-autonomous evolution equations in Banach spaces are obtained. It is not assumed that the symbol space of an equation is a compact metric space and that the family of trajectory spaces corresponding to this symbol space is translation-coordinated or closed in any sense. Using these results, existence of minimal uniform trajectory attractors and uniform global attractors for weak solutions of the boundary value problem for motion equations of an incompressible viscoelastic medium with the Jeffreys constitutive law is shown.     

Approximation of the long‐term dynamics of the dynamical system generated by the multilayer quasigeostrophic equations of the ocean

In this article, we study the multilayer quasigeostrophic equations of the ocean. More precisely, we discretize these equations in time using the implicit Euler scheme and using the classical and uniform discrete Gronwall lemmas we prove that the approximate solution is uniformly bounded in H^?1, L² and H¹. Using the uniform stability of the scheme and the theory of the multivalued attractors, we then prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time‐step approaches zero. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1041–1065, 2016     

The initial value problem for a multi‐dimensional radiation hydrodynamics model with viscosity

In this paper, we study the existence and time‐asymptotic behavior of solutions to the Cauchy problem for the equations of radiation hydrodynamics with viscosity in ?³. The global existence of the solutions is obtained by using the energy method. With more elaborate energy estimates, we also give some decay rates of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

Convergence to steady state and attractors for doubly nonlinear equations

This paper is concerned with the asymptotic behaviour of a class of doubly nonlinear parabolic-type equations. For degenerate equations, we prove thanks to the Lojasiewicz inequality that the solutions to some nonautonomous equations converge to a steady state. In the nondegenerate case, the existence of exponential attractors is shown by using the ?-trajectories method.     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

The global attractor of g-Navier–Stokes equations with linear dampness on

In this paper, the long time behaviors of g-Navier–Stokes equations with linear dampness on R² were investigated. By using the energy equation method, the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted Sobolev space. Moreover, the estimation of the Hausdorff and Fractal dimensions of such attractors were also obtained.     

Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

In this paper, the 2D Navier‐Stokes‐Voight equations with 3 delays in is considered. By using the Faedo‐Galerkin method, Lions‐Aubin lemma, and Arzelà‐Ascoli theorem, we establish the global well‐posedness of solutions and the existence of pullback attractors in H¹.     

Homogenization of a fully coupled thermoelasticity problem for a highly heterogeneous medium with a priori known phase transformations

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two‐phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an a priori known interface movement because of phase transformations. After transforming the moving geometry to an ? ‐periodic, fixed reference domain, we establish the well‐posedness of the model and derive a number of ? ‐independent a priori estimates. Via a two‐scale convergence argument, we then show that the ? ‐dependent solutions converge to solutions of a corresponding upscaled model with distributed time‐dependent microstructures. Copyright © 2017 John Wiley & Sons, Ltd.     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

Principle of localization of solutions of the Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type

In the case where initial data are generalized functions of the Gevrey-distribution type for which the classical notion of equality of two functions on a set is well defined, we establish the principle of local strengthening of the convergence of a solution of the Cauchy problem to its limit value as t → +0 for one class of degenerate parabolic equations of the Kolmogorov type with 2^?b \overrightarrow {2b} -parabolic part whose coefficients are continuous functions that depend only on t.     

Upper semicontinuity of global attractors for a viscoelastic equations with nonlinear density and memory effects

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations in a three‐dimensional space, for f growing up to the critical exponent and dependent on ρ ∈ [0,4), as ρ→0⁺. This equation models extensional vibrations of thin rods with nonlinear material density ?(?_tu) = |?_tu|^ρ and presence of memory effects. This type of problems has been extensively studied by several authors; the existence of a global attractor with optimal regularity for each ρ ∈ [0,4) were established only recently. The proof involves the optimal regularity of the attractors combined with Hausdorff's measure.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

Homogenization of a non‐linear degenerate parabolic problem in a highly heterogeneous periodic medium

We consider the homogenization of a time‐dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having comparable heat capacities and conductivities, separated by a third material with thickness of the same order ε as the basic periodicity cell but having a much lower conductivity such that the resulting interstitial heat flow is scaled by a factor λ tending to zero with a rate λ=λ(ε). The heat flux vectors a_j, j=1,2,3 are non‐linear, monotone functions of the temperature gradient. The heat capacities c_j(x) are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ=lim_ε→0ε^p/λ and identify the homogenized problem depending on whether δ is zero, strictly positive finite or infinite. Copyright © 2005 John Wiley & Sons, Ltd.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.