The norm of a distribution (generalized function) in a Sobolev Spaces involving time.

Question

Being a distribution a linear continuous functional in the test function space $\mathcal{D}(0,t)$ (for the purpose of my question), and $$||u ||_{L^p(0,T;V)}:= \left(\int^T_0||u||^p_V \ dt\right)^{1/p}$$.

In the picture below, we see defined the norm for the Sobolev space of order 1(involving time).

What is $$||\dot u||_{L^2(0,T;V)}?$$ more specifically, how do I interpret $$||\dot u||_{V}?$$

Also, how can $\left< \dot u, \phi\right>$ be an element of $W$?

Answer 1

Q: What is $||\dot u||_{L^2(0,T;V)}?$

A: $$\|\dot u\|_{L^2(0,T;V)}=\left(\int_0^T \|\dot u(t)\|_V^2 \text{ d}t \right)^{1/2}$$

Q: More specifically, how do I interpret $||\dot u||_{V}?$

A: If $\dot u \in L^2(0,T;V)$, then $\dot u(t) \in V$ for a.e. $t \in (0,T)$, hence taking the integral of $\|\dot u(t)\|_V^2$ makes sense.

Q: Also, how can $\left< \dot u, \phi\right>$ be an element of $W$?

A: Note that these are Bochner (or Banach-valued) integrals, i.e. $$\langle \dot u, \phi \rangle = \int_0^T \underbrace{\phi(t)}_{\in \mathbb{R}} \underbrace{\dot u(t)}_{\in W} \text{ d}t \in W.$$