EXISTENCE RESULT FOR A MODEL COUPLING A QUASI-LINEAR PARABOLIC EQUATION AND A LINEAR HYPERBOLIC SYSTEM

We study a coupled Fluid - Structure system describing the motion of an elastic solid interacting with an incompressible viscous fluid in two dimensions. The behavior of the solid is described by the Lam ' e system of linear elasticity and the fluid obeys the incompressible stokes equations. The quasi-linear nature of the considered Stokes equation is characterized by the nonlinear dependence of the stress tensor on the gradient of the fluid velocity; this encompasses the case of Newtonian as well as non-Newtonian fluids. At the Fluid Solid interface, natural conditions are imposed, continuity of the velocities and of the Cauchy stress forces. The fluid and the solid are coupled through these conditions. By this interaction, the fluid deforms the boundary of the solid which in turn influences the fluid motion. We prove the existence of globally-in-time solution for the problem coupling the linear Lam ' e system and the quasi-linear Stokes equation. To achieve this, we interpret the solution as the fixed point of some non-linear operator T associated to the global problem. Then we construct, using a regularization procedure, a sequence (T-is an element of)(is an element of) of auxiliary compact operators that approximate T. Next we establish, using a combination of Banach and Schaeffer fixed point theorems, the existence of fixed points to every operator T-is an element of, these auxiliary fixed point are actually solution of auxiliary problems. Finally we prove that these fixed points converge to the fixed point of T.