Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$

Question

Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution equations in PDE, we may concern ourselves with spaces of functions $[0,T] \to X$ where $X$ is a Banach space of functions $\Omega \to \mathbb{R}$ ($X = L^p(\Omega), W^{k,p}(\Omega)$, etc.). For functions of this form, we can introduce notions of measurability and weak (time) derivatives in connection with the Bochner integral (e.g. as in Evans PDE).

But also to any function $\textbf{f} : [0,T] \to X$ we can associate a function $f : [0,T] \times \Omega \to \mathbb{R}$ in the obvious way: $f(t,x) = \textbf{f}(t)(x)$. Then $f$ is a function defined on a subset of $\mathbb{R}^{N+1}$ and so has natural notions of measurability and weak time partial derivatives. This suggests the fairly natural questions: is $f$ measurable iff $\textbf{f}$ is (note that if $X$ isn't separable, there are a few different notions of measurability to consider for $\textbf{f}$)? And is $\textbf{f}$ weakly differentiable in time iff $\partial_t f$ exists in the weak sense?

More generically, the question is how can the highly related notions defined separately for function-space-valued functions and for functions on $\mathbb{R}^{N+1}$ be connected? It seems like we'd really want them to agree, at least in nice cases, but I'm not quite convinced they will.

The book I've looked at (Evans, Wloka) haven't touched on these seemingly natural questions, so any pointers to resources that cover this sort of thing would also be useful.

Answer 1

First note that the vectors of the Lebesgue spaces $L^p(\Omega)$ are not functions $\Omega\to\mathbb R$ or ($\mathbb C$, depending on whether real or complex valued functions are considered) but a.e. equivalence classes of them. In usual PDE considerations of evolution equations, there are made lots of implicit "identifications" that make understanding the correct state of matters difficult to the beginner. Obviously from this the OP question arises. In the classical book

Dunford, Nelson; Schwartz, Jacob T., Linear operators. I. General theory. Pure and Applied Mathematics. Vol. 7. New York and London: Interscience Publishers. (1958)

there is an explicit theorem (probably Theorem 17 on Page 198, although I have not had the possibility to check this) that for $1\le p<+\infty$ justifies the identification that $\iota:E=L^p(I,L^p(\Omega))\simeq F=L^p(I\times\Omega)$ holds in the sense that $\iota$ is a topological linear homeomorphism $E\to F$ when defined as follows. Given a vector $\vec e$ in $E$ that is an a.e. class of functions $\bar e:I\to L^p(\Omega)$, let $\bar e\in\vec e$ be arbitrary. Then $\bar e$ is a function $I\to L^p(\Omega)$ and there is a measurable function $f:I\times\Omega\to\mathbb K$ ($=\mathbb R$ or $\mathbb C$) such that the function $\bar f:I\to L^p(\Omega)$ defined by $t\mapsto[\,f(t,\cdot\,)\,]$ is a.e. equivalent to $\bar e$ and hence is in $\vec e$. If $g$ is a.e. equivalent to $f$, also $\bar g\in\vec e$ holds, and conversely any representative $f$ of a vector $\vec f$ in $F$ determines in this manner a unique vector in $E$. Then we put $\iota:\vec e\mapsto\vec f$. Note, however, that there are also nonmeasurable functions $f:I\times\Omega\to\mathbb K$ such that the function $t\mapsto[\,f(t,\cdot\,)\,]$ defines a vector in $E$, provided that ZF is suitably complemented. Put in more detail, ZFC complemented with the continuum hypothesis allowes one to prove that every representative $\bar c$ of every vector of $E$ has uncountably many choice functions $c$ such that the function $\hat c$ defined by $(t,s)\mapsto c\,t\,s$ is nonmeasurable.

The above result is the basis for the implicit "exponential" identifications made in PDE theory.