Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$, and consider for $T>0$ a function $u \in H^1(0,T;L^2(\Omega))$. Is it possible to speak about $u_t(T^-)$, i.e. the one-sided derivative in $T$ from the left? In general, is the expression $$ \int_\Omega u_t(T^-) u(T) \, dx $$ well defined?
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No, it is not well defined. The derivative $u_t$ is not more regular than any function in $L^2(0,T; L^2(\Omega))$ and you cannot perform point evaluations on those functions. – gerw Aug 09 '17 at 07:39
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Yes, I agree. I was confused about the fact that given $u \in H^1(0,T;L^2(\Omega))$ then $u \in C([0,T];L^2(\Omega))$, but I realized that's not sufficient for having well-defined one-sided derivatives. – Mauro Aug 09 '17 at 08:59