a question about a Sobolev space notation

Question

I am reading a paper [ref. 1] on singularity conditions for Euler equations in $\mathbb{R}^3$. It mentioned that "For smooth solutions $u \in C([0, T);W^{2,q}(\mathbb{R}^3)), q > 3$...".

Does "$C$" mean "smooth" here? Why not $C^\infty$

Thanks for the help!

ref.1 D. Chae, P. Constantin, On a Type I singularity condition in terms of the pressure for the Euler equations in R3, International Mathematics Research Notices, rnab014 (2021)

https://arxiv.org/abs/2012.11948

Answer 1

$C([0,T);X)$ (in your case $X=W^{2,q}(\mathbb{R}^3)$ is the set of continuous functions $u:[0,T)\longrightarrow X$ with $\|u\|_{C([0,T);X)}=\displaystyle\sup_{0\leq t<T}\|u(t)\|_X<\infty$.

This is the definition given by Evan's PDE book. This is a pretty standard definition.

I didn't read the whole paper. Skimming through it, I think the author means functions smooth in spacetime $[0,T)\times\mathbb{R}^3$ that is also living in the above mentioned space.