Notion for weak derivatives of $L^p(0,T,X)$-functions

Question

A definition in Evan's PDE-book from chapter 5.9.2 says (let $X$ be a Banach space):

Let $u\in L^1(0,T,X)$. We say $v\in L^1(0,T,X)$ is the weak derivative of $u$ provided $$\int_0^T \phi'(t)u(t)dt = -\int_0^T\phi(t)v(t)dt$$ for all scalar test functions $\phi\in C_c^\infty(0,T)$.

(Alright, and then the space $W^{1,p}(0,T,X)$ consists of all functions $u\in L^p(0,T,X)$ such that a weak derivative exists and belongs to $L^p(0,T,X)$).

A following theorem in the same chapter starts like this:

Suppose $u\in L^2(0,T,H_0^1(\Omega))$ with $u'\in L^2(0,T,H^{-1}(\Omega))$ ...

I don't get what is meant by that notion and I didn't find any information in the book. So it should be intuitively clear from the above definition. Maybe there is someone who can tell me.

Answer 1

It means that the weak derivative of $u$, $u'$ is an element of $L^2(0,T;H^{-1}(\Omega))$, and by definition it satisfies for all scalar test functions $\phi$ the identity $$\int_0^T \phi'(t) u(t) = -\int_0^T u'(t)\phi(t)$$ where the right hand side is an element of $H^{-1}(\Omega)$ (we have integrated out the time). The equality makes sense since the left hand side is an element of $H^1(\Omega)$, and $H^1 \subset H^{-1}$.

I guess Evans should have defined a weaker notion of a weak time derivative.