I am working with a nonlinear PDE in which one of the nonlinearities have the form $$\left|\nabla \int_0^t udt\right|^2$$
If $\Omega$ is a smooth bounded domain in 2 or 3 dimensions and I have $u \in C([0,T];H^1(\Omega))$ can I write $$\left|\int_0^t \nabla u \right|?$$ What if u is only integrable in time but $H^2$ in space, can I put the gradient inside?
My gut says I can exchange the gradient and integral in this case, but I was unable to find a reference that covers it. Could anyone please help me find a reference for this?