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

Authors: Viorel Barbu, Zoran Grujic, Irena Lasiecka and Amjad Tuffaha

Issue: Volume 57 (2008), Issue 3, 1173-1208

Abstract: The added nonlinear fluid-structure interaction coupling the Navier-Stokes equations with a added 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 added mismatch of parabolic and hyperbolic regularity, previous results in the literature dealt with either a added 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 [V. Barbu, Z. Gruji\'c, I. Lasiecka, and A. Tuffaha, \textit [Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model}, Contemporary Mathematics \textbf{440} (2007), 55-82] the authors established existence of \textit{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 \textit{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.