On page 109 of book Vorticity and Incompressible flow by Majda and Bertozzi,there are some arguments which, at least to me, are not so clear.
The argument that there exists a subsequence is a classic one,and i have the proof.Then i understand that since we have a uniform bound from a previous step we can find a subsequence by the previous argument converges to some u in $H^m$.
First question:Who ensures that the limit u is the same limit of sequence? I think that the answer is:
Since we have prove (in previous steps) that $u_ε \to u_v$ in $C([0,T],H^s)$ with s<m,any subsequence of $u_ε$ converges to $u_v$ in that space and since $H^s \subset H^m$ and the fact that limit is unique we have that $u_v=u$.
Second question: how we reach to the fact that u is in $L^\infty ([0,T],H^m)$ ?
Third question:How the fact that the derivative of a function with respect to time is bounded is related to the fact that the function is Lipschitz continuous? I think that those two facts are equivalent,but I can't see a proof.
The solution to the first question is correct. The second is similar but there is a typo in the text; the important equation is (3.64). This shows we are bounded in $L^\infty H^m$ which is a dual space so Banach Alaoglu gives weak* convergence. Now the earlier argument tells us that the limit is the same function we found earlier.
For the third question this is the equivalence of (bounded) functions with bounded derivatives and the Sobolev space $W^{1,\infty}$: I believe it goes by the name Radamacher’s theorem and if memory serves me well, depending on how much prereqs you have you may find a proof in Evans’ PDE book or the book on fine properties of functions