Evans PDE chapter 5.9 theorem 4 (mappings into better space), Evans wrote in the proof:
In addition, $\bar{u}^{'}\in L^2(0,T;L^2(V))$ , with the estimate:
\begin{equation} ||\bar{u}^{'}||_{L^2(0,T;L^2(V))}\leq C||u^{'}||_{L^2(0,T;L^2(U))} \end{equation}
This follows if we consider difference quotients in the t-variable, remember the methods in 5.8.2, and observe also that $E$ is a bounded linear operator from $L^2(U)$ into $L^2(V)$.
Who can give some detail about this, I can't understand. Any suggestions may be helpful.
Thank all of you!
From Theorem 3 (ii) in Section 5.8.2, we only need to show that $$ \Vert D^h\bar{u}\Vert_{L^2(0,T;L^2(V))}\leq C\Vert u^{'}\Vert_{L^2(0,T;L^2(U))}. $$ Since $\bar{u}=Eu$ and $E$ is a bounded linear operator, $$ \Vert D^h\bar{u}\Vert_{L^2(0,T;L^2(V))}\leq C\Vert D^hu\Vert_{L^2(0,T;L^2(U))}. $$ From Theorem 3 (i) in Section 5.8.2, we have $$ \Vert D^hu\Vert_{L^2(0,T;L^2(U))}\leq C\Vert u^{'}\Vert_{L^2(0,T;L^2(U))}. $$ The proof is completed by combining above two inequalities.