Finite Dimensional Exponential Attractors Research Articles

Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in ell ^2, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).

Robust family of exponential attractors for isotropic crystal models

Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter β > 0), of two phase‐field equations, namely, the Allen–Cahn and the Cahn–Hilliard equations. In both cases, we construct robust families of exponential attractors, that is, attractors that are continuous with respect to the perturbation parameter. Copyright © 2015 John Wiley & Sons, Ltd.

Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations ut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.

Asymptotic behavior for a semilinear second order evolution equation

This paper is devoted to the qualitative analysis for a second order evolution equation u(tt) -Delta(u)-Delta u(t)-epsilon Delta u(tt)+f(u) = g(x) (epsilon is an element of [0, 1]) with critical nonlinearity. Some uniformly (w.r.t. epsilon is an element of [0,1]) asymptotic regularity about the solutions has been established for both g(x) is an element of L(2)(Omega) and g(x) is an element of H(-1), which shows that the solutions are exponentially approaching a more regular fixed subset uniformly (w.r.t. epsilon is an element of [0,1]). As an application of this regularity result, a family {epsilon(epsilon)}(epsilon is an element of[0,1]) of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with a strongly damped wave equation (epsilon = 0), the upper semicontinuity, at epsilon = 0, of the global attractors has been proved.

Finite-Dimensional Attractors and Exponential Attractors for the Navier--Stokes Equations of Compressible Flow

We prove that the uniform attractor for the Navier--Stokes equations of compressible flow with quasi-periodic external forces has finite fractal dimension. As a byproduct of our analysis, we also obtain the existence of finite-dimensional exponential attractors for the Navier--Stokes system.

Exponential Attractors for the Generalized Ginzburg-Landau Equation

In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation

Finite dimensional exponential attractor for the phase field model

We consider the phase field equations in arbitrary space dimension. We show that the corresponding boundary value problems are well-posed when assuming that the initial data is square integrable and prove the existence of a maximal attractor and of an inertial set.

Finite dimensional exponential attractors for semilinear wave equations with damping

We consider the initial value problem for a class of second order evolution equations that includes, among others, the 3D sine-Gordon equation with damping and the 3D Klein-Gordon type equations with damping. We show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate.