I have a function which lives in $f(x,t)∈L^2(0,T;H^{1/2})∩L^\infty(0,T;L^2)$ for a certain time interval. I also know that $\partial_{t} \ f(x,t)∈L^2(0,T;H^{−1})$. Can I assure that the function lives in $f(x,t)∈C(0,T;L^2)$, i.e., is continuous in time with values in $L^2$?
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Because $f\in H^1(0,T;H^{-1})$, hence $f\in C(0,T;H^{-1})$, you may infer that $t\mapsto f(t)$ is continuous into $L^2$ equipped with its weak topology. To prove that it is continous into $L^2$ equipped with its strong (normed) topology, you need that $t\mapsto\|f(t)\|$ be continous ; this is not guaranted by your assumptions, essentially because the spaces $H^{1/2}$ and $H^{-1}$ are not in duality.
Edits. Because we already know that $t\mapsto f(t)$ is continuous from $(0,T)$ into $L^2_w$, we see that the continuity into $L^2$ for the strong topology is equivalent to the continuity of $t\mapsto\|f(t)\|_L^2$. This is why $f\in L^\infty(0,T;H^{-1/2})$ doesn't help you.
answered Jul 15, 2016 at 15:43
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$\begingroup$ Thanks! I got it, you said that I can only infer that $f \in C_{w}(0,T;L^{2})$! What kind of assumptions should I had to try to prove this result in the strong topology? I also have that $f\in L^{\infty}(0,T;H^{-1/2})$ . There is no way no get some interpolation result, which after I can deduce the strong continuity? Thanks in advanced! $\endgroup$ Commented Jul 22, 2016 at 10:42
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$\begingroup$ See my edits, and don't forget to vote the answer if you find it useful. $\endgroup$ Commented Jul 22, 2016 at 14:44