# Preprint 2009-026

# Homogenization of Nonlinear PDE's in the Fourier–Stieltjes Algebras

## Hermano Frid and Jean C. 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.