Partial boundary regularity for the Navier-Stokes equations in time-dependent domains

Abstract

We consider the incompressible Navier-Stokes equations in a moving domain whose boundary is prescribed by a function $\eta=\eta(t,y)$ (with $y\in\mathbb R^2$) of low regularity. This is motivated by problems from fluid-structure interaction. We prove partial boundary regularity for boundary suitable weak solutions assuming that $\eta$ is continuous in time with values in the fractional Sobolev space $W^{2-1/p,p}_y$ for some $p>15/4$ and we have $\partial_t\eta\in L_t^{3}(W^{1,q_0}_y)$ for some $q_0>2$. The existence of boundary suitable weak solutions is a consequence of a new maximal regularity result for the Stokes equations in moving domains which is of independent interest.