A regularity result in Bochner spaces, simmetry of second derivatives (time and space)

Question

For a function $u:(0,T)\to L^2(\Omega)$, assume that we have shown that $\frac{\partial u}{\partial x_i}=\partial_i u \in L^\infty(0,T;L^2(\Omega))$ and also that $\partial_t u \in L^\infty(0,T;L^2(\Omega))$. Can we deduce that $\partial_t(\partial_i u) \in L^\infty(0,T;L^2(\Omega))$ as well? In that case, do the partial derivatives commute, that is $\partial_t(\partial_i u)=\partial_i(\partial_t u)$ a.e. in $(0,T)\times\Omega$?