I am currently studying a paper on weak solutions to the Navier-Stokes equations.
In this paper, the author sets $T>0$, and uses the time interval $[0,T)$ for all of his statements early on. However, we get to a point where, having constructed approximate solutions $u_m$, we wish to show uniform (with respect to time) weak convergence to some weak solution $u$, using the Arzela-Ascoli theorem.
At this point, the author suddenly switches to the closed time interval $[0,T]$. It seems that we must do this, as the statement of the Arzela-Ascoli theorem clearly stipulates that the interval our series of functions acts on be closed and bounded.
For $T<\infty$, this should be no problem. Our approximate solutions are continuous in $t$, so we can extend our functions to include the point $t=T$. The problem is that the paper never mentions any requirement that $T < \infty$.
Up until now, I have assumed that the reason why we use the half-closed interval $[0,T)$ for these studies of Navier-Stokes is because we are including the possibility that $T = \infty$. Is this mistaken?
My main question is, in order to use Arzela-Ascoli, are we forced to assume $T<\infty$ so that we can use the closed interval $[0,T]$? Or is there a way to apply Arzela-Ascoli on the interval $[0,\infty)$ so that we can still include the $T=\infty$ case?
No, unfortunately Arzela-Ascoli requires the domain of the functions to be compact. For example take $f_n: [0, \infty) \to \mathbb R$ defined by $$ f_n(x) = \begin{cases} 0 & x \leq n \\ (x - n) & n \leq x \leq n + 1 \\ 1 & x \geq n + 1. \end{cases}$$ Then $\{f_n\}$ is bounded and equicontinuous however has no convergent subsequence since $\lVert f_n - f_m \rVert_{\infty} = 1$ for all $n \neq m$. Thus there is no convergent subsequence.
I haven't read the paper so there may be additional assumptions in this specific context that might help but in general Arzela-Ascoli requires a compact domain.