Consider the Stokes equation in a bounded domain $\Omega$ \begin{equation} \begin{split} \partial_{t}u-\Delta u+\nabla p & =0, \\ \text{div}u & =0, \\ u(0,x) & =0, \end{split} \end{equation} with evolutional Dirichlet boundary condition $$u(t,x)=\psi(t,x)\quad\text{on}\ \partial\Omega.$$
I want to know the $H^{1}$ estimate of the solution $u$, i.e. the a priori estimate of $$\sup\limits_{0\leq t\leq T}|\nabla u|_{L^{2}(\Omega)}+\int_{0}^{t}|\nabla u|_{H^{1}(\Omega)},$$ I guess the first thing to do is find an extension $\Psi$ of the boundary data $\psi$, then consider a new unknown $u-\Psi$. But I have no idea how to determine the time derivative $\partial_{t}\Psi$.
Any references or suggestions? Thanks!