My question:
Assume I have a function $ u \in H^2(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega))$. Now I want to bound the gradient of $u$. Can I deduce that $u \in H^1(0,T;H^1(\Omega))$ and under which requirements on $\Omega$ (smoothness of boundary?, dimension?, star shaped?) would this be true? Does anyone have a clue how to approach this question?