A possible interpretations of real-valued function $u(x,t)$ as a Banach space-valued function $u(t)(x)$?

Question

I work on a PDE problem in the area of conservation laws. In order to apply the technique I want, I need to consider the shock wave. That is one of the solutions of a problem and also a real-valued function that maps $S \times [0,T]$ into $\mathbb{R}$.

In my problem I have to think of it as a function that has values in some Banach space-valued space. This question concernes the possibilities for that Banach space. Details are given below.

One of the solutions of the conservation law problem

$$\partial_t u+\partial_x f(u)=0,$$

with Riemann initial conditions

$$ u(x,0)= \begin{cases} u_l, x < 0, \\[2ex] u_r, x > 0, \end{cases} $$

is a discontinuous real-valued function called the shock wave that is given with:

$$(1) \hspace{0.2cm} u(x,t)= \begin{cases} u_l, x < c \cdot t \\[2ex] u_r, x > c \cdot t, \end{cases} $$

where $c$, $u_l$, $u_r$ are constants, $t \in [0,T]$, $x \in S \subset \mathbb{R}$.

In the book: Weak and measure-valued solutions to evolutionary partial differentional equations - Malek, Necas, Rokyta, Ruzicka, 1996, the authors write about viewing the real-valued function as a Banach space-valued function (Section 1.2.6).

In short: Let's say we have some real valued function $v(x,t):S \times [0,T] \rightarrow \mathbb{R}$ where $S \subset \mathbb{R}$. We could think of $v$ in a different way. For $v(x,t)$ the map: $$ v(t):x \mapsto v(x,t) $$

is an element of some function space (for example Sobolev, Lebesgue,...). Then the function:

$$ t \mapsto v(t) $$

maps the interval $[0,T]$ into that function space.

The idea is to regard a function of time and space as a collection of functions of space that is parametrized by time. And instead of a real valued function $v(x,t)$ we would have the Banach space-valued function $v(t)(x)$.

My question is:

If we interpret the real valued function $u(x,t)$ given in $(1)$ as a Banach space-valued function $u(t)(x)$, in which spaces would $u(t)(x)$ belong?

My BIG list of possibilities of the space-valued Banach spaces in which I think that is possible to consider the shock waves:

1. $ L^{p}(0,T;L^{\infty}(S)), 1 \leq p \leq \infty$
2. $ L^{p}(0,T;BV(S)), 1 \leq p \leq \infty$
3. $ L^{p}(0,T;L^{1}(S)), 1 \leq p \leq \infty$
4. $ L^{p}(0,T;\mathcal{M}(S)), 1 \leq p \leq \infty$
5. $ L^{p}(0,T;\mathcal{D}^{'}(S)), 1 \leq p \leq \infty$
6. $ C([0,T];L^{\infty}(S))$
7. $ C([0,T];BV(S))$
8. $ C([0,T];L^{1}(S))$
9. $ C([0,T];\mathcal{M}(S))$
10. $ C([0,T];\mathcal{D}^{'}(S))$
11. $ C_w([0,T];L^{\infty}(S))$
12. $ C_w([0,T];BV(S))$
13. $ C_w([0,T];L^{1}(S))$
14. $ C_w([0,T];\mathcal{M}(S))$
15. $ C_w([0,T];\mathcal{D}^{'}(S))$

Above $C_w(...)$ are the spaces of weakly continuous Banach space-valued functions, $\mathcal{M}$ is some measure space such as space of Radon measures, $\mathcal{D}^{'}$ is the space of distributions and $BV$ is the space of functions with the bounded variation.

I am particulary interested in the possibilities of the measure-valued and distribution-valued functions spaces. Also the spaces of cadlag functions would go well here definitely but I can't use them in this problem.

I know that the list above should be shorter but I want to consider all the possibilities. So adding some spaces to this list would be great. And if some of the spaces shouldn't be here, write it please and I will correct it.