I'm reading this paper. In it there the following argument (see page 240).

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Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $\frac{\partial b(\overline{v}_n)}{\partial t}$ is only in $H^1(0,T;H^{-1})\cap L^\infty(0,T;L^s)$. He just integrates it in time.. I don't follow.

Secondly, I am lost with the spaces when he says that $\int_0^T \overline{v}_n(s)\;ds$ is a Cauchy sequence in the space $L^\infty(0,T;W^{1,r}_0)$. And then he uses the monotone convergence theorem to say that $\overline{v}_n$ convreges strongly in $L^1(\Omega \times (0,T))$ to some $\overline{v}$. Again a different space. Any explanation appreciated.



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