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 $y\in R^2$) of low regularity. This is motivated by problems from fluid-structure interaction and our result applies, in particular, for linearised Koiter shells with dissipation. We prove partial boundary regularity for boundary suitable weak solutions assuming that η is continuous in time with values in the fractional Sobolev space $W_y^{2-1/p,p}$ for some $p>15/4$ and we have ${\partial }_t\eta \in L_t^3\left(W_y^{1,q_0}\right)$ 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.