What does the space $C^2([0,T]; H^2(\Omega))$ mean?

Question

Does the space $C^2([0,T]; H^2(\Omega))$ mean: $C^2$ in the time direction and $H^2$ in the space direction?

Thank you very much!

Answer 1

Informally, that's a reasonable way to think about it.

Formally, this is the space of all paths $\gamma : [0,T] \to H^2(\Omega)$ having two continuous derivatives (in time). The time derivative is defined in the usual way $\gamma'(t) = \lim_{\epsilon \to 0} \frac{1}{\epsilon}(\gamma(t+\epsilon)-\gamma(t))$, where the limit should be taken in the $H^2(\Omega)$ norm topology. The second derivative is defined analogously, and $\gamma'' : [0,T] \to H^2(\Omega)$ should be continuous, again with respect to the $H^2(\Omega)$ norm topology.

It's a Banach space under a norm such as

$$\|\gamma\| = \sup_{t \in [0,T]} \|\gamma(t)\|_{H^2(\Omega)} + \sup_{t \in [0,T]}\|\gamma'(t)\|_{H^2(\Omega)} + \sup_{t \in [0,T]} \|\gamma''(t)\|_{H^2(\Omega)}.$$