Homogenization of Nonlinear PDEs in the Fourier–Stieltjes Algebras

Abstract

We introduce the Fourier–Stieltjes algebra in $\mathbb{R}^n$ which we denote by $\mathrm{FS}(\mathbb{R}^n)$. It is a subalgebra of the algebra of bounded uniformly continuous functions in $\mathbb{R}^n$, $\mathrm{BUC}(\mathbb{R}^n)$, 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 [Matem. Zametki, 33 (1983), pp. 571–582]. Namely, $\mathrm{FS}(\mathbb{R}^n)$ is the closure in $\mathrm{BUC}(\mathbb{R}^n)$, 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 analyze the homogenization problem for nonlinear transport equations with oscillatory velocity field in $\mathrm{FS}(\mathbb{R}^n)$. We also consider the corresponding problem for porous medium type equations on bounded domains with oscillatory external source belonging to $\mathrm{FS}(\mathbb{R}^n)$. 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 $\mathrm{FS}(\mathbb{R}^n)$ which are of interest on their own.