Attractor for a conserved phase‐field system with hyperbolic heat conduction

We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.     

An identification problem for a conserved phase‐field model with memory

This paper is devoted to recovering a scalar memory kernel in a conserved phase‐field model. For such a problem local in time existence and uniqueness results are proved. The technique used allows to show also the continuous dependence on the kernel of the solution to the direct problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.     

An adaptive,multilevel scheme for the implicit solution of three‐dimensional phase‐field equations

Phase‐field models, consisting of a set of highly nonlinear coupled parabolic partial differential equations, are widely used for the simulation of a range of solidification phenomena. This article focuses on the numerical solution of one such model, representing anisotropic solidification in three space dimensions. The main contribution of the work is to propose a solution strategy that combines hierarchical mesh adaptivity with implicit time integration and the use of a nonlinear multigrid solver at each step. This strategy is implemented in a general software framework that permits parallel computation in a natural manner. Results are presented that provide both qualitative and quantitative justifications for these choices.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Finite‐dimensional attractors and exponential attractors for degenerate doubly nonlinear equations

We consider the following doubly nonlinear parabolic equation in a bounded domain Ω??³: where the nonlinearity f is allowed to have a degeneracy with respect to ?_tu of the form ?_tu|?_tu|^p at some points x∈Ω. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite‐dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials

Our aim in this article is to study the long time behaviour of a family of singularly perturbed Cahn‐Hilliard equations with singular (and, in particular, logarithmic) potentials. In particular, we are able to construct a continuous family of exponential attractors (as the perturbation parameter goes to 0). Furthermore, using these exponential attractors, we are able to prove the existence of the finite dimensional global attractor which attracts the bounded sets of initial data for all the possible values of the spatial average of the order parameter, hence improving previous results which required strong restrictions on the size of the spatial domain and to work on spaces on which the average of the order parameter is prescribed. Finally, we are able, in one and two space dimensions, to separate the solutions from the singular values of the potential, which allows us to reduce the problem to one with a regular potential. Unfortunately, for the unperturbed problem in three space dimensions, we need additional assumptions on the potential, which prevents us from proving such a result for logarithmic potentials. Copyright © 2004 John Wiley & Sons, Ltd.     

Existence of global attractors for the three‐dimensional Brinkman–Forchheimer equation

In this paper, we investigate the asymptotic behavior of solutions of the three‐dimensional Brinkman–Forchheimer equation. We first prove the existence and uniqueness of solutions of the equation in L², and then show that the equation has a global attractor in H² when the external forcing term belongs to L². Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity

Under what condition, a process which exists a $(E,E)$-pullback exponential attractor implies the existence of $(E,V)$- pullback exponential attractor when $V$ embedded in $E$? We answer this question in this paper. As an application of this result, we prove the existence of pullback exponential attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity in $L^q(\Omega)(\forall q\geq 2)$ and $H_0^1(\Omega)$.     

Robust exponential attractors for a class of non-autonomous semi-linear second-order evolution equation with memory and critical nonlinearity

In this article, we investigate a class of non-autonomous semi-linear second-order evolution with memory terms, expressed by the convolution integrals, which account for the past history of one or more variables. First, the asymptotic regularity of solutions is proved, while the nonlinearity is critical and the time-dependent external forcing term is assumed to be only translation-bounded (instead of translation-compact), and then the existence of compact uniform attractors together with its structure and regularity is established. Finally, the existence of robust family of exponential attractors is constructed.     

Pullback exponential attractors for evolution processes with the difference of 2 solutions lacking smoothing property

In this paper, we construct the pullback exponential attractors for evolution processes in which the difference of 2 solutions lacks the smoothing property. To do this, by the uniform squeezing property of the corresponding discrete process, we add the points to the pullback attractor such that every new set of it has the finite fractal dimension and pullback exponentially attracts every bounded subset of the phase space. As the applications, we establish the existence of pullback exponential attractors for non‐autonomous reaction‐diffusion equation without any restriction on the growing order of nonlinear term and non‐autonomous strongly damped wave equation in with critical nonlinearity.     

Two types of upper semi‐continuity of bi‐spatial attractors for non‐autonomous stochastic p‐Laplacian equations on

We consider the long time behavior of solutions for the non‐autonomous stochastic p‐Laplacian equation with additive noise on an unbounded domain. First, we show the existence of a unique ‐pullback attractor, where q is related to the order of the nonlinearity. The main difficulty existed here is to prove the asymptotic compactness of systems in both spaces, because the Laplacian operator is nonlinear and additive noise is considered. We overcome these obstacles by applying the compactness of solutions inside a ball, a truncation method and some new techniques of estimates involving the Laplacian operator. Next, we establish the upper semi‐continuity of attractors at any intensity of noise under the topology of . Finally, we prove this continuity of attractors from domains in the norm of , which improves an early result by Bates et al.(2001) who studied such continuity when the deterministic lattice equations were approached by finite‐dimensional systems, and also complements Li et al. (2015) who discussed this approximation when the nonlinearity f(·,0) had a compact support. Copyright © 2017 John Wiley & Sons, Ltd.     

Dimension estimate of the exponential attractor for the chemotaxis‐growth system

We consider a chemotaxis‐growth model which takes into account diffusion, chemotaxis, production of chemical substance, and growth. We present estimates from above and below of the fractal dimension dim?? of the exponential attractor ?? in terms of the coefficients of the system. Comparisons are made between the sizes of the global and exponential attractors. Numerical simulations are presented which confirm the analytical results obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Global existence to a three‐dimensional non‐linear thermoelasticity system arising in shape memory materials

This paper is concerned with the unique global solvability of a three‐dimensional (3‐D) non‐linear thermoelasticity system arising from the study of shape memory materials. The system consists of the coupled evolutionary problems of viscoelasticity with non‐convex elastic energy and non‐linear heat conduction with mechanical dissipation. The present paper extends the previous 2‐D existence result of the authors Reference [1] to 3‐D case. This goal is achieved by means of the Leray–Schauder fixed point theorem using technique based on energy arguments and DeGiorgi method. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

This paper considers the two‐dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non‐selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non‐selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite dimensional global attractor for a class of doubly nonlinear parabolic equations

Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.