How to get $2\langle \nabla u, \partial_t \nabla u \rangle=\frac{d}{dt}\|\nabla u\|^2_{L^2(U)}$

Question

In the book "Partial Differential Equations"-Lawrence C. Evan, Section 5.2, we get $2\langle u'(t), u(t)\rangle=\frac{d}{dt}\|u(t)\|^2_{L^2(U)}$ when $u\in L^2(0,T; H^1_0(U))$ and $u'\in L^2(0,T; H^{-1}(U)).$

When will we get the formula

$$2\langle \nabla u(.,t), \partial_t \nabla u(.,t) \rangle=\frac{d}{dt}\|\nabla u(.,t)\|^2_{L^2(U)}~~?$$

Answer 1

Denote $\frac{du}{dt}$ by $u'$, if $u,u' \in L^2(0,T;H_0^1(U))$ then $$\left(\|\nabla u\|_{L^2(U)}^{2}\right)'=\langle \nabla u, \nabla u \rangle'=\langle \nabla u', \nabla u \rangle+\langle \nabla u, \nabla u' \rangle=2\langle \nabla u, \nabla u' \rangle,$$ since $\langle f, g \rangle'=\langle f', g \rangle +\langle f, g' \rangle$, where $\langle \cdot, \cdot \rangle$ denotes the inner product in $L^2(U)$.