Smoothness of weak solutions to a nonlinear fluid-structure interaction model


The nonlinear fluid-structure interaction coupling the Navier-Stokes equations with a dynamic system of elasticity is considered. The coupling takes place on the boundary (interface) via the continuity of the normal component of the Cauchy stress tensor. Due to a mismatch of parabolic and hyperbolic regularity, previous results in the literature dealt with either a regularized version of the model, or with very smooth initial conditions leading to local existence only. In contrast, in the case of small but rapid oscillations of the interface, in [3] the authors established existence of finite energy weak solutions that are defined globally. This is achieved by exploiting new hyperbolic trace regularity results which provide a way to deal with the mismatch of parabolic and hyperbolic regularity. The goal of this paper is to establish regularity of weak solutions, for initial data satisfying the appropriate regularity and compatibility conditions imposed on the interface. It is shown that weak solutions equipped with smooth initial data become classical.

Publication Title

Indiana University Mathematics Journal