HOMOGENIZATION OF NONLINEAR PDE’S IN THE FOURIER-STIELTJES ALGEBRAS HERMANO FRID AND JEAN SILVA Abstract. 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. 1. Introduction The purpose of this paper is to introduce a large algebra with mean value (w.m.v.), strictly containing the almost periodic functions and to consider the homogenization problem for some nonlinear partial differential equations with oscillatory behavior governed by functions belonging to that algebra w.m.v.. Namely, denoting by BUC(R n ) the space of bounded uniformly continuous functions, we are going to deal with the algebra FS(R n ) defined as the closure in the sup norm of the functions in BUC(R n ) whose Fourier transform is a complex-valued measure with finite total variation. We show that this algebra shares many important properties with the almost periodic functions. In particular, it is an ergodic algebra, which also contains the perturbations of almost periodic functions by continuous functions vanishing at infinity. We then consider the homogenization problem for certain nonlinear PDE’s. More specifically, we begin by analysing the homogenization of nonlinear transport equations where the associated vector field belongs to the algebra FS(R n ). This discussion extends and improves the one corresponding to the same problem in [2], in the context of almost periodic functions, as well as the pioneering one provided by W. E in [18] in the context of periodic functions. Next, we consider the homogenization problem for a porous medium type equation on a bounded domain with a stiff oscillatory external source in FS(R n ). The latter was addressed in [3] for the Cauchy problem in R n with oscillatory external source belonging to a general ergodic algebra and “well-behaved” initial data, i.e., initial data which are solutions of the associated steady equation in the fast variable. Here we restrict the discussion to FS(R n ) which allows us to consider more general initial data not necessarily “well-behaved”. Finally, we also address the homogenization problem for a system of two such porous medium type equations coupled by a nonlinear zero-order term. 1991 Mathematics Subject Classification. Primary: 35B40, 35B35; Secondary: 35L65, 35K55. Key words and phrases. two-scale Young Measures, homogenization, algebra with mean value, transport equation, porous medium equation. 1