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

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 $W_y^{2-1\mathrm{/}p,p}$ for some $p>15\mathrm{/}4$ and we have ${\mathrm{\partial }}_t\eta \in L_t^3(W_y^{1,q_0})$ 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.