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edit approved Jun 13, 2014 at 9:32

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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}\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}\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.

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.

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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created Aug 27, 2012 at 17:13

user8707

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.