On the almost periodic homogenization of non-linear scalar conservation laws

The paper deals with the homogenization problem of non-linear scalar conservations laws ∂tuε+∇x·(a(xε)f(uε))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t u_\varepsilon + \nabla _x\cdot (a(\frac{x}{\varepsilon })f(u_\varepsilon ))=0$$\end{document}. The vector field a is assumed to be incompressible and its components are assumed to be almost periodic. We prove that the weak limit of the family {uε}ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u_{\varepsilon }{\}}_{\varepsilon >0}$$\end{document} is the mean value of a function, which we call U(z, x, t), that is the entropy solution of a similar conservation law in the macroscopic variables, but with coefficients depending on the microscopic variables and has the property that the function z↦∫R+n+1U(z,x,t)φ(x,t)dxdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\mapsto \int _{\mathbb {R}_+^{n+1}}U(z,x,t)\varphi (x,t)\,dx\,dt$$\end{document} is also almost periodic for any φ∈Cc(R+n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C_c(\mathbb {R}_+^{n+1})$$\end{document}.