Well-posedness of the motion of a rigid body immersed in a compressible inviscid fluid

TL;DR

This paper proves the local existence and uniqueness of solutions for a rigid body moving in a compressible inviscid fluid, addressing a complex coupled hyperbolic-ODE system with boundary conditions.

Contribution

It introduces a novel approach to handle the coupled fluid-solid system by reformulating in a fixed domain and analyzing an approximate system with non-characteristic boundary.

Findings

Established local well-posedness of the coupled system.

Developed a fixed-point method for the approximate system.

Derived uniform estimates enabling passage to the limit.

Abstract

We consider a rigid body freely moving in a compressible inviscid fluid within a bounded domain $\Omega\subset\mathbb{R}^3$. The fluid is thereby governed by the non necessarily isentropic compressible Euler equations, while the rigid body obeys the conservation of linear and angular momentum. This forms a coupled system comprising an ODE and the initial boundary value problem (IBVP) of a hyperbolic system with characteristic boundary in a moving domain, where the fluid velocity matches the solid velocity along the normal direction of the solid boundary. We establish the existence of a unique local classical solution to this coupled system. To construct the solution, we first perform a change of variables to reformulate the problem in a fixed spatial domain, and then analyze an approximate system with a non-characteristic boundary. For this nonlinear approximate system, we use the better regularity for the trace of the pressure on the boundary to contruct a solution by a fixed-point argument in which the fluid motion and the solid motion are updated in successive steps. We are then able to derive estimates independent of the regularization parameter and to pass to the limit by a strong compactness arguments.