Reiterated Ergodic Algebras and Applications (original) (raw)

Homogenization of degenerate porous medium type equations in ergodic algebras

Advances in Mathematics, 2013

We consider the homogenization problem for general porous medium type equations of the form ut = ∆f (x, x ε , u). The pressure function f (x, y, •) may be of two different types. In the type 1 case, f (x, y, •) is a general strictly increasing function; this is a mildly degenerate case. In the type 2 case, f (x, y, •) has the form h(x, y)F (u) + S(x, y), where F (u) is just a nondecreasing function; this is a strongly degenerate case. We address the initial-boundary value problem for a general, bounded or unbounded, domain Ω, with null (or, more generally, steady) pressure condition on the boundary. The homogenization is carried out in the general context of ergodic algebras. As far as the authors know, homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time. We also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis. Further, we include some new results on algebras with mean value, specially a new criterion establishing the null measure of level sets of elements of the algebra, which is useful in connection with the homogenization of porous medium type equations in the type 2 case.

Homogenization of a generalized Stefan problem in the context of ergodic algebras

Journal of Functional Analysis, 2015

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function u(x, t), ∂ ∂t ∂uΨ(x/ε, x, u) − ∇x • ∇ηψ(x/ε, x, t, u, ∇u) ∋ f (x/ε, x, t, u), on a bounded domain Ω ⊆ R n , t ∈ (0, T), together with initial-boundary conditions, where Ψ(z, x, •) is strictly convex and ψ(z, x, t, u, •) is a C 1 convex function, both with quadratic growth, satisfying some additional technical hypotheses. As functions of the oscillatory variable, Ψ(•, x, u), ψ(•, x, t, u, η) and f (•, x, t, u) belong to the generalized Besicovitch space B 2 associated with an arbitrary ergodic algebra A. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to the usual L 2 convergence in the cartesian product Π × R n , where Π is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with bounded sequences in L 2 .

On the weakly almost periodic homogenization of fully nonlinear elliptic and parabolic equations

2014

A function f ∈ BUC(R d) is said to be weakly* almost periodic, denoted f ∈ W * AP(R d), if there is g ∈ AP(R d), such that, M(|f − g|) = 0, where BUC(R d) and AP(R d) are, respectively, the space of bounded uniformly continuous functions and the space of almost periodic functions, in R d , and M(h) denotes the mean value of h, if it exists. We give a very simple direct proof of the stochastic homogenization property of the Dirichlet problem for fully nonlinear uniformly elliptic equations of the form F (ω, x ε , D 2 u) = 0, x ∈ U , in a bounded domain U ⊆ R d , in the case where for almost all ω ∈ Ω, the realization F (ω, •, M) is a weakly* almost periodic function, for all M ∈ S d , where S d is the space of d×d symmetric matrices. Here (Ω, µ, F) is a probability space with probability measure µ and σ-algebra F of µ-measurable subsets of Ω. For each fixed M ∈ S d , F (ω, y, M) is a stationary process, that is, F (ω, y, M) =F (T (y)ω, M) := F (T (y)ω, 0, M), where T (y) : Ω → Ω is an ergodic group of measure preserving mappings such that the mapping (ω, y) → T (y)ω is measurable. Also, F (ω, y, M), M ∈ S d , is uniformly elliptic, with ellipticity constants 0 < λ < Λ independent of (ω, y) ∈ Ω × R d. The result presented here is a particular instance of the general theorem of Caffarelli, Souganidis and Wang, in CPAM 2005. Our point here is just to show a straightforward proof for this special case, which serves as a motivation for that general theorem, whose proof involves much more intricate arguments. We remark that any continuous stationary process verifies the property that almost all realizations belong to an ergodic algebra, and that W * AP(R d) is, so far, the greatest known ergodic algebra on R d .

Homogenization of Nonlinear PDEs in the Fourier-Stieltjes Algebras

Siam Journal on Mathematical Analysis, 2009

We introduce the Fourier-Stieltjes algebra in R n which we denote by FS(R n ). It is a subalgebra of the algebra of bounded uniformly continuous functions in R n , BUC(R n ), strictly containing the almost periodic functions, whose elements are invariant by translations and possess a mean-value. Thus, it is a so called algebra with mean value, a concept introduced by Zhikov and Krivenko (1986). Namely, FS(R n ) is the closure in BUC(R n ), with the sup norm, of the real valued functions which may be represented by a Fourier-Stieltjes integral of a complex valued measure with finite total variation. We prove that it is an ergodic algebra and that it shares many interesting properties with the almost periodic functions. In particular, we prove its invariance under the flow of Lipschitz Fourier-Stieltjes fields. We analyse the homogenization problem for nonlinear transport equations with oscillatory velocity field in FS(R n ). We also consider the corresponding problem for porous medium type equations on bounded domains with oscillatory external source belonging to FS(R n ). We further address a similar problem for a system of two such equations coupled by a nonlinear zero order term. Motivated by the application to nonlinear transport equations, we also prove basic results on flows generated by Lipschitz vector fields in FS(R n ) which are of interest in their own.

Homogenization in algebras with mean value

2013

In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic al...

Homogenization of Reynolds Equations

2007

This Licentiate thesis is focussed on some new questions in homogenization theory, which have been motivated by some concrete problems in tribology. From the mathematical point of view, these quest ...

Almost periodic homogenization of a generalized Ladyzhenskaya model for incompressible viscous flow

2013

We study the existence and almost periodic homogenization of some model of generalized Navier–Stokes equations. We establish an existence result for nonstationary Ladyzhenskaya equations with a given nonconstant density and an external force depending nonlinearly on the velocity. In the case of a nonconstant density of the fluid, we study the asymptotic behavior of the velocity field by combining some compactness arguments with the sigma-convergence method. Bibliography: 37 titles.

Homogenization of periodic linear degenerate PDEs

Journal of Functional Analysis, 2008

It is well-known under the name of 'periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs.

Periodic homogenization under a hypoellipticity condition

Nonlinear Differential Equations and Applications NoDEA, 2014

In this paper we study a periodic homogenization problem for a quasilinear elliptic equation that present a partial degeneracy of hypoelliptic type. A convergence result is obtained by finding uniform barrier functions and the existence of the invariant measure to the associate diffusion problem that is used to identify the limit equation.