Asymptotic Analysis

The stationary Stokes problem in a n-dimensional domain with a rapidly oscillating ( n − 1 ) dimensional boundary prescribed with Neumann boundary condition and periodicity along the lateral sides is considered. We identify the limit equation in a fixed domain with a one dimensional Laplacian-type problem, satisfied on the region covered by the oscillating part of the boundary, coupled with steady Stokes in the complement. The existence and uniqueness of solution for the coupled limit problem is established and, finally, the weak convergences of velocities are improved to strong convergence by introducing corrector terms.

In this paper we analyze the system of equations that models the behaviour of materials with a double porous structure. We introduce dissipation mechanisms in both structures. We show existence, uniqueness and analyticity for the solutions of the system. As consequences, exponential stability and impossibility of localization for the solutions are obtained.

In this paper we study pullback attractors of multi-valued dynamical systems that are asymptotically convergent. It is shown that, under certain conditions, the components of the pullback attractor of a dynamical system can converge in time to those of the pullback attractor of the limiting dynamical system. Particular examples are asymptotically autonomous and asymptotically periodic pullback attractors. Different criteria theorems requiring different conditions are established and their applicability and advantages are highlighted.

In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick’s function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Γ-convergence theory.

We consider a hyperbolic quasilinear perturbation of the Navier–Stokes equations in three space dimensions. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works. Furthermore, we prove the convergence of solutions to relaxed system towards solutions to the classical Navier–Stokes problem.

This article is about the Z d -periodic Green function  G n ( x , y ) of the multiscale elliptic operator  L u = − div ( A ( n · ) · ∇ u ) , where  A ( x ) is a Z d -periodic, coercive, and Hölder continuous matrix, and n is a large integer. We prove here pointwise estimates on  G n ( x , y ) , ∇ x G n ( x , y ) , ∇ y G n ( x , y ) and  ∇ x ∇ y G n ( x , y ) in dimensions  d ⩾ 2 . Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.