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If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".

Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\Omega))$, then $u\in C([0,T];L^2(\Omega))$.

This is a very special case, dose anybody know some similar result(more general)?

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    $\begingroup$ What do you mean by "similar" or "more general"? $\endgroup$
    – Deane Yang
    Commented Mar 20, 2014 at 3:33
  • $\begingroup$ I want a result in this way: $u\in X$ and $u_t\in Y$ implies $u\in C^\alpha([0,T];Z)$. Here $X, Y, Z$ are some functional spaces. "similar"--get some continuity in time by interpolation. "general"--$X,Y,Z$ have more choices. $\endgroup$
    – user44565
    Commented Mar 20, 2014 at 12:12
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    $\begingroup$ See Wloka's book "PDE", approx. p. 390. $\endgroup$
    – TaQ
    Commented Mar 20, 2014 at 18:35

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