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

TL;DR

This paper establishes the global-in-time existence of solutions for a coupled system involving a linear hyperbolic Lamé system and a quasi-linear parabolic Stokes equation, using fixed point and regularization techniques.

Contribution

It introduces a novel approach combining regularization and fixed point theorems to prove existence for a coupled hyperbolic-parabolic PDE system.

Findings

Existence of solutions proved for the coupled system.

Construction of auxiliary approximating operators.

Convergence of approximate solutions to the true solution.

Abstract

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 $T$