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I need to prove (if true) that the space $W^{1,2}(0,T;\mathbb{R}^d)$ is compactly embedded in $C(0,T;\mathbb{R}^d)$. The proof for the continuous embedding part is straightforward and is given in PDE book by Evans. However could someone give me some ideas on how to prove the compact part. This question pops up in proving certain bounds for existence of solutions to a simple GENERIC problem. Is there some extension of Azrela-Ascoli theorem to say Bochner spaces which can be used here. |
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I guess this is the solution. Would be grateful if someone could verify the arguments. The proof of the continuous embedding $W^{1,2}(0,T;\mathbb{R}^d)\hookrightarrow \mathbb{L}^2(0,T;\mathbb{R}^d)$ follows from Evans 5.9.2. So let's begin with the assumption that we have a bounded sequence $(x_n)$ in $X:=W^{1,2}(0,T;\mathbb{R}^d)$, $\sup\limits_{n}\|x_n\|_X<c$ (i.e. we have uniform bounds). (This assumption may be incorrect (is it??). But it actually holds in the original problem.) For proving the compact embedding we make use of Azrela-Ascoli theorem which requires (1) Uniform bounds on $(x_n)$ in the space $X$ which we have. (2) Equicontinuity: Let $0\leq s\leq t\leq T$. \begin{equation} \|x_n(t)-x_n(s)\|_{\mathbb{R}^d}\leq\int\limits_{s}^t\|\dot{x}_n(\sigma)\|_{\mathbb{R}^d}d\sigma\leq \|\dot{x}_n\|_{\mathbb{L}^2(0,T;\mathbb{R}^d)}(t-s)^{\frac{1}{2}}\leq C(t-s)^{\frac{1}{2}}, \end{equation} The first inequality is due to fundamental theorem of calculus (and existence of weak derivative). By Azrela-Ascoli we have managed to show the compactness of $(x_n)$ in the Holder space $C^{\frac{1}{2}}(0,T;\mathbb{R}^d)$. By similar arguments we can show that Holder space $C^\alpha$ is compactly embedded in $C^\beta$ where $\beta>\alpha$. Hence we are done. Note: There is a subtlety involved in the usage of Azrela-Ascoli theorem. We require that the space $\overline{\{x_n(t): t\in[0,T]\}}$ is compact in $\mathbb{R}^d$. Therefore this embedding would have not been possible for any range space $X$ instead of $\mathbb{R}^d$. |
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