Suppose I know that $\nabla \cdot (a(t)\nabla u(t))$ is such that $\nabla \cdot (a\nabla u) \in L^2(0,T;L^2(\Omega))$ on some bounded domain $\Omega$. Here $a\colon [0,T] \to \Omega$ is such that $a \in L^\infty(0,T;L^\infty(\Omega))$ and $c_1 < a < c_2$ a.e. where the constants are positive.
I know that $u(t) \in H^2(\Omega)$ for a.e. $t$.
Is there a way I can conclude that $u\colon [0,T] \to H^2(\Omega)$ is measurable? Measurable in the sense of the standard definition eg. here.