I'm dealing with KdV equation with the fourth order regularizer $\epsilon \partial_{xxxx}u^\epsilon$. If we know the parametrized solution $u^\epsilon$ is bounded in $C(I; H2(\Omega))$, can we extract a convergent subsequence in $C(I; H1(\Omega))$?
In this case, from the PDE, we additionally know $\partial_t u \in C(I; H^{-10}(\Omega))$, for example. But Aubin-Lions's lemma is invalid for the space $L^\infty$.
And obviously, Azela-Ascoli and Frechet-Kolmogrov type results can not straightforwardly be applied here.
Here is a counter-example that $C(I;H^2)$ does not embed compactly into $C^1(I;H^1)$. Take $\phi \in H^2(\Omega)$ with $\phi\ne0$. Define $u_n = \sin(nt)\phi(x)$. This is a bounded sequence in $C(I;H^2)$, which does not converge strongly in $C(I;H^1)$, as $u_n(t)$ does not converge strongly for all $t$.