I am studying PDE using Evans' book and I have two main confusions (probably stupid questions to experts) regarding weak derivatives in Banach spaces.
First confusion: $\def\u{\mathbf u}$ $\def\v{\mathbf v}$
DEFINITION. Let $\u \in L^1(0, T; X)$. We say $\v \in L^1(0, T; X)$ is the weak derivative of $\u$, written $$\u'=\v,$$ provided $$\int_0^T \phi'(t) \u(t) \, dt = -\int_0^T \phi(t) \v(t) \, dt$$ for all scalar test functions $\phi \in C_c^\infty (0, T)$.
However, a subsequent theorem begins with an assumption of
THEOREM $\mathbf 3$ (More calculus). Suppose $\u \in L^2(0, T; H_0^1(U))$, with $\u' \in L^2(0, T; H^{-1}(U))$.
Now, the main confusion here is that $H^{-1}$ is only the dual space of $H^1_0$, not a subset nor a superset of $H^1_0$, whereas in the definition, both the function and its weak derivative take values in the same Banach space $X$. Is there some kind of hidden identification of spaces?
Second confusion:
THEOREM $\mathbf 2$ (Calculus in an abstract space). Let $\u \in W^{1,p} (0, T; X)$ for some $1 \leq p \leq \infty$. Then
(i) $\u \in C([0,T]; X)$ (after possibly being redefined on a set of measure zero), and
(ii) $\u(t) = \u(s) +\displaystyle\int_s^t \u'(\tau) \, d\tau$ for all $0 \leq s \leq t \leq T$.
(iii) Furthermore, we have the estimate $$\max_{0 \leq t \leq T} \|\u(t)\| \leq C\|\u\|_{W^{1,p} (0,T;X)}, \tag{7}$$ the constant $C$ depending only on $T$.Proof. $1$. Extend $\u$ to be $\mathbf 0$ on $(-\infty, 0)$ and $(T, \infty)$, and then set $\u^\varepsilon = \eta_\varepsilon * \u$, $\eta_\varepsilon$ denoting the usual mollifier on $\mathbb R^1$. We check as in the proof of Theorem $1$ in $\S5.3.1$ that $\u^{\varepsilon'} = \eta_\varepsilon * \u'$ on $(\varepsilon, T-\varepsilon)$.
Then as $\varepsilon \to 0$, $$\begin{cases} \u^\varepsilon \to \u & \text{in } L^p(0, T; X), \\ (\u^\varepsilon)' \to \u' & \text{in } L^p (0, T; X). \end{cases} \tag{8}$$ Fixing $0<s<t<T$, we compute $$\boxed{\u^\varepsilon (t) = \u^\varepsilon (s) +\int_s^t \u^\varepsilon {}' (\tau) \, d\tau.}$$
Some kind of "fundamental theorem of calculus for Bochner integrals" seems to be used here. Am I correct that the functions are $C^1$ after mollification, so some version of the "fundamental theorem of calculus for Bochner integrals" can be applied? Here, at least the weak derivative and the function belong to the same space, as in the definition. But this is not the case in the proof of the subsequent theorem. Dual functions suddenly appear as derivatives:
Proof. $1$. Extend $\u$ to the larger interval $[-\sigma, T+\sigma]$ for $\sigma>0$, and define the regularizations $\u^\varepsilon = \eta_\varepsilon * \u$, as in the earlier proof. Then for $\varepsilon$, $\delta>0$, $$\frac{d}{dt} \|\u^\varepsilon (t) -\u^\delta (t)\|_{L^2(U)}^2 = 2 \bigl(\u^{\varepsilon'} (t) -\u^{\delta'} (t), \u^\varepsilon (t) -\u^\delta (t)\bigr)_{L^2(U)}.$$ Thus \begin{align} \|\u^\varepsilon (t) -\u^\delta (t)\|_{L^2(U)}^2 &= \|\u^\varepsilon (s) -\u^\delta (s)\|_{L^2(U)}^2 \\ &{}+2\int_s^t \langle \u^{\varepsilon'} (\tau) -\u^{\delta'} (\tau), \u^\varepsilon (\tau) -\u^\delta (\tau)\rangle \, d\tau \tag{11} \end{align}
None of these two "fundamental theorems of calculus" are derived explicitly and I am really confused!
Best Answer
For the first question, we have $$H_0^1(\Omega) \hookrightarrow L^2(\Omega) \hookrightarrow H^{-1}(\Omega),$$ so $H_0^1(\Omega)$ is indeed isomorphic to a subspace of $H^{-1}(\Omega)$. Evans notes this at the beginning of section 5.9.
Yes, there is a FTOC for Bochner integrals and that would imply the given (boxed) equation (for $\mathbf{u}^\varepsilon$). For more info on that, you can Google "Bochner integral fundamental theorem of calculus" and you should be able to readily find something of interest. For example, you may find the following helpful.
https://tinyurl.com/y2sxnvdg
I think looking up the FTOC for Bochner integrals (along with the above embeddings) should clear up the last question, but if you need more clarification/info I'm happy to help.