about the functions of $L^2(0,T;H^{1}_{0} (U))$

Question

I have a function $u \in L^2(0,T;H^{1}_{0} (U))$ with $u' \in L^2(0,T;L^2 (U))$ and $u''\in L^{2}(0,T;H^{-1} (U))$.

My professor wrote :

$$ u' \in W^{1,2}(0,T; H^{-1}(U)).$$

I don't know how to conclude this affirmation.

I was wondering about and if I define the function

$F\colon L^2(0,T;L^2 (U)) \rightarrow L^2(0,T;H^{-1}(U))$ given by $F(u) = \tilde{u}$, where $\langle \tilde{u}(t), f\rangle_{H^{-1}(U)H^{1}_{0}(U)} = \int_{0}^{T} f(x) u(t) \ dx$ . I can show that $F$ is linear and $\lVert F u \rVert\leq\lVert u \rVert$. I don't know if this help . Maybe this function help me how to see a element of $L^2(0,T;L^2 (U))$ in $L^2(0,T;H^{-1}(U))$. Someone can help me ?

Thanks in advance.

Answer 1

Your argument is right @Leandro. You have proved that every element $v\in L^2(0,T:L^2(U))$, is a element of $L^2(0,T:H^{-1}(U))$. By using the definition of $W^{1,2}$, you can conclude that $$u'\in W^{1,2}(0,T;H^{-1}(U))$$