Existence Result For a Model Coupling a Quasi-Linear Parabolic Equation and a Linear Hyperbolic System

We prove globally-in-time existence of solution for a problem coupling the linear Lam\'e system and the quasi-linear Stokes equation. A solution of this global coupled problem is viewed as the fixed point of some non-linear operator $T$. We construct, using a regularization procedure, a sequence $(T^\epsilon)_\epsilon$ of auxiliary approximating compact operators. Then we establish, using a combination of Banach and Schaeffer fixed point theorems, the existence of fixed points to every operator $T^\epsilon$. Finally we prove that these fixed points converge to the fixed point of $T$