Meaning of combined $L^p$ and Sobolev spaces in fluid dynamics

Question

I have a problem here with the definition on the page 372 of $$L^2((0,+\infty),L^2).$$ What functions form this space? I do not also understand this spaces:

$$L^p L^q$$ and

$$L^2 \dot{H}^1.$$

Answer 1

This notation is slightly compacted to make it easier to read, but it can be confusing for the uninitiated. A spatial domain $\Omega\subset\mathbb{R}^n$ is fixed and we can construct the standard function spaces on $\Omega$; e.g., $L^q(\Omega)$, $H^1(\Omega)$, $\dot{H}_1(\Omega)$, etc. You should be familiar with these. Notice that elements of these sets can be considered functions of $x$, the spatial variable and the different spaces impose different amounts of regularity on these functions and their derivatives.

However, if we wish to construct weak solutions to the Navier-Stokes equations, we need some measure of the regularity with respect to time as well. A simple case looks something like $u\in C^1([0,T],X)$, where $X$ is some Banach space. This statements means that the mapping $t\mapsto u(t)$, $t\in[0,T]$ is $C^1$ with respect to the topology on $X$. In the same way we typically extend spaces of continuous functions to $L^p$ spaces, we can extend this definition to include measureable maps from $[0,T]$ (or any other interval of time) from $[0,T]$ to $X$ with some degree of regularity.

For a concrete example, consider the space in the first bullet point: $u\in L^\infty((0,T),L^2)\cap L^2((0,T),H^1).$ This can be read as follows:

$u$ is a map from $(0,T)\to X\subset L^2(\Omega)\cap H^1(\Omega)$ such that the map $t\mapsto u(t)$ is essentially bounded with respect to the topology on $L^2(\Omega)$ and is square-integrable on $(0,T)$ with respect to the topology on $H_1(\Omega)$.

In concrete terms, these last two conditions read

$$\left\|~\|u(t)\|_{L^2(\Omega)}~\right\|_{L^\infty(0,T)} = \sup_{s\in(0,T)}\|u(s)\|_{L^2(\Omega)}\text{ is finite.}$$

and

$$\left\|~\|u(t)\|_{H^1(\Omega)}~\right\|_{L^2(0,T)}=\left(\int_0^T\|u(s)\|_{H^1}^2~\mathrm{d}s\right)^{\frac{1}{2}}\text{ is finite.}$$

With this in mind, we can now read off the shorthand notation. The time interval is hard to determine from the context, but it appears to be the case that $$L^pL^q=L^p([0,T],L^q(\Omega)),\quad\text{and}\quad L^2H^1 = L^2([0,T],H^1(\Omega)).$$ This condensed notation $u\in XY$ for two functions spaces $X$ and $Y$ tends to mean that $u$ is a $X$-regular mapping from the real line to $Y$, representing the time dependence, and $Y$ represents the spatial regularity of the elements $u(t)$.

Note: I have abused notation a bit in my definition of spaces on $\Omega$, since the functions here should be vector-valued. If one wished to be very clear, one should write, for example, $L^2(\Omega\to\mathbb{R}^n)$ and one must use the appropriate norms.