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Let us consider $\theta,s\in\mathbb{R}$, and the standard Sobolev spaces $H^s(\mathbb{R})$ and $H^\theta(\mathbb{R})$, with no apriori relation between $s$ and $\theta$. Consider a function $$ u\in C([0,T],H^\theta(\mathbb{R}))\cap L^\infty([0,T],H^s(\mathbb{R})). $$ Finally, consider a sequence of times $t_n\to0$. Does the continuity of $u$ with values in $H^\theta$ implies $$ u(t_n)\rightharpoonup u(0) \quad\hbox{in} \quad H^s(\mathbb{R})? $$ Of course, this is trivial if $\theta\geq s$ . I am wondering what about the case $\theta<s$.

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Yes, it is true also in the case $\theta < s$.

Note that (see Lemma II.5.9 in "Mathematical Tools for the Navier-Stokes Equations and Related Models") $$u \in C([0,T];H^\theta) \cap L^\infty(0,T;H^s) = C([0,T];H^s_{weak}).$$

That means $t \mapsto \langle \psi, u(t) \rangle$ is continuous for all $\psi \in (H^s)'$. That means for $t_n \to 0$ we have

$$\langle \psi, u(t_n) \rangle \to \langle \psi, u(0) \rangle,$$

for all $\psi \in (H^s)'$. In other words,

$$u(t_n) \rightharpoonup u(0)$$

in $H^s$.

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  • $\begingroup$ Thank you very much for you answer. I have one small question, in the case $\theta<s$ does these hypothesis also implies that $u\in C([0,T]; H^{s'})$ for any $\theta\leq s'<s$? In other words, if we allow a little loss in space-regularity, is it posible to improve the the weak continuity in $H^s$ to strong continuity in $H^{s'}$ for $s'<s$? $\endgroup$
    – W2S
    Sep 1, 2020 at 17:27
  • $\begingroup$ Maybe if I add, for example, an extra hypothesis like $\partial_tu\in L^\infty(0,T;H^\alpha)$ with $\alpha\leq \theta$? Might that work? $\endgroup$
    – W2S
    Sep 1, 2020 at 21:47

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