Existence of 3D strong solutions for a system modeling a deformable solid inside a viscous incompressible fluid

TL;DR

This paper proves the existence of strong solutions for a coupled fluid-solid system in three dimensions, accommodating limited deformation regularity, and extends previous results to more realistic physical constraints.

Contribution

It adapts and completes 3D results for a fluid-structure interaction system with limited deformation regularity, introducing new variable transformations and proving local and global existence.

Findings

Global existence for small data and near-identity deformations

Reformulation of the system in fixed domains using new variable changes

Extension of 3D results to less regular deformations

Abstract

In this paper we study a coupled system modeling the movement of a deformable solid immersed in a fluid. For the solid we consider a given deformation that has to obey several physical constraints. The motion of the fluid is modeled by the incompressible Navier-Stokes equations in a time-dependent bounded domain of $\R^3$, and the solid satisfies the Newton's laws. Our contribution consists in adapting and completing some results of ARMA 2008 in dimension 3, in a framework where the regularity of the deformation of the solid is limited. We rewrite the main system in domains which do not depend on time, by using a new means of defining a change of variables, and a suitable change of unknowns. We study the corresponding linearized system before setting a local-in-time existence result. Global existence is obtained for small data, and in particular for deformations of the solid which are arbitrarily close to the identity.