About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

Question

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions.

I notice that in the energy method of hyperbolic equations, we always put $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$, more generally, in $\bigcap^s_{ℓ=0}W^{ℓ,\infty}([0, T ), H^{s−ℓ}(Ω))$. And I have two questions about these spaces.

1. I am really curious about why we always use this kind of spaces, I mean, this space looks like very strange such as we let the sum of the power $ℓ$ of $W$ (or $C$) and the power $s−ℓ$ of $H$ be just equal to $s$.

2. Is the following equality correct? and why? $$ C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1}) $$

Waiting for your answers, thanks!

Answer 1

I may answer my question by myself because I find a nice answer in Ringstrom's book "The Cauchy problem in general relativity". In Chapter 4, he shows me a energy: $$ E_k=\frac{1}{2}\sum_{|\alpha\le k|}\int_{\mathbb{R}^n}[(\partial^\alpha\partial_t u)^2+|\nabla \partial^\alpha u|^2]dx $$

For the linear case, of course, this energy is conserved, so according to this energy we should put $u(t) \in H^{k+1}$ and $u'(t) \in H^{k}$.

This is just basic ideas.