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If a sequence is $\{u_n\}$ bounded $L^\infty(0,T;H^2(\Omega))$, can I find a subsequence $\{u_{n_k}\}$ that converges in $L^p(0,T;H^1(\Omega))$ for some $p$? I don't think this is true, but I can't come up with a counterexample. Any hints? Also, what further assumptions can one add to make this s...

In mathematics, Bochner spaces are a generalization of the concept of L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not necessarily the space R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } of real or complex numbers. The space L p...