Homogenization and corrector result for a coupled parabolic hyperbolic system

Abstract

The article deals with the homogenization of a coupled system of two equations: a shallow water equation and the equation for the long-term dynamics of sand dunes with small parameter ϵ. The first one is a hyperbolic partial differential equation, while the second one is a parabolic partial differential equation. In previous work, we showed existence and uniqueness results and performed a general homogenization of the coupled system. Here we give a more precise homogenization. For that, we use the asymptotic expansion of the solution and the coefficients of the system. Besides, we obtain corrector results. We also extend the existence theorem of previous work by proving that the solution of the parabolic equation is bounded independently of the small parameter.