A question about PDE argument involving monotone convergence theorem and Sobolev space

Question

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

Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $b(\overline{v}_n)$ is only in $H^1(0,T;H^{-1})\cap L^\infty(0,T;L^s)$ and $\overline{v}_n \in L^2(0,T;H^1)$. 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 $\overline{v}$. Any explanation appreciated.

I posed this here on MSE but did not receive an answer.

Answer 1

Hy

First: take $\varphi_n(t) w(x)$ where $\varphi(t)\in C_0([0,T))$ and $w\in H^1_0(\Omega)$. If $\varphi_n$ is well chosen and converges to the characteristic function of the set $[0,t]$ you will understand the formulation.

Second: to be short, you have to mix the fact that $\int_0^t f_n(s,x)$ is a Cauchy sequence in $L^1(\Omega)$, that ($n$ fixed) $\int_0^t f_n(s,x)$ is continuous in $L^1(0,T; L^1(\Omega))$ and the De Giorgi regularity result. Since $\bar{v}_n$ is monotonically inscreasing in $n$ you have almost convergence to some $\bar{v}$ and then proving that $\bar{v}_n$ converges strongly to $v$ in $L^1(0,T;L^1(\Omega))$ is equivalent to prove that $\int_0^T\int_\Omega \bar{v}_n dxdt$ goes to $\int_0^T\int_\Omega \bar{v} dxdt$. From the previous result (the Cauchy sequence in $L^\infty(0,T;W^{1,q}_0(\Omega))$) you obtain the result.