Hi,


let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose 
$$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial\_tu\in L^2(0,T;W^{-1,2}(\Omega)),$$
where $W^{-1,2}(\Omega)=(W^{1,2}\_0(\Omega))^\*$. Is there a result stating that from those regularities one can deduce $u\in C([0,T];L^2(\Omega))$? 


Thanks a lot,
Richard



Continuity with values in L^2
In the case $\Omega=\mathbb{R}^n$ we have
\begin{equation}
 L^2([0,T];W^{1,2}(\Omega))\cap W^{1,2}([0,T];W^{-1,2}(\Omega))\hookrightarrow BUC([0,T];X),
\end{equation}
where $X$ is given via real interpolation:
\begin{equation}
 X=\big(W^{1,2}(\Omega),W^{-1,2}(\Omega)\big)\_{1/2, 2}=B^0\_{2,2}(\Omega).
\end{equation}
This is basically contained in *Linear and quasilinear parabolic problems I* by H. Amann (Theorem III.4.10.2).
Since $B^0\_{2,2}(\Omega)=L^2(\Omega)$ the desired result follows. Now the case of smooth bounded $\Omega$ should follow via extension and restriction.


