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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

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up vote 2 down vote accepted

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.

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Thanks a lot, that was the hint I was looking for, Richard – Richard Gustier Aug 28 '12 at 7:45

Part of your hypotheses are that $u \in L^2(I, W^{1,2}) \cap H^1(I, W^{-1,2})$, where $I = (0,T)$, $H^1 = W^{1,2}$ in $t$ (first Sobolev space), so interpolation gives that $u \in H^{1/2}(I, L^2)$ which just misses continuity. Adding in that $u \in L^\infty$ and using interpolation again probably does the trick, but you should check that.

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Set $V:=L^2$ and $H:=W^{-1,2}$ and define the maximal regularity space $$MR_2(0,T;H,D):=W^{1,2}(0,T;H)\cap L^2(0,T;D),$$ where $D$ is the domain of the unbounded operator on $H$ associated with the quadratic form $Q(u):=\|u\|_V^2$. Then it is well-known (reference in some book by J. Lions, but now I cannot find an exact one) that $MR_2(0,T;H,D)$ is continuously embedded in $C([0,T];V)$. This is almost what you want. Not quite (I have not checked, but in this case it looks like $D=W^{1,2}_0$, unlike in your assumptions; and, worse, you are not assuming $u$ to be in $W^{1,2}(0,T;H)$), but also not far away, if you are able to use your assumption that $u\in L^\infty$ to define an equivalent norm on the set of those $$\{u\in L^2(0,T;W^{1,2}) \hbox{ s.t. } u_t\in L^2(0,T;W^{−1,2})\}.$$

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