Notation Involving Sobolev Spaces I Cannot Find the Meaning Of

Question

I've been reading some research papers in PDEs and I've run into a notation a couple of times that googling has not helped me define. It looks something like $W^{1,p}(0,T; L^2(\Omega))$. I know what each of these things mean individually. $$L^p(\Omega)=\{u: \int_{\Omega}|u|^p<\infty\}$$ $$W^{1,p}=\{u:u\in L^p, Du\in L^p\}$$ and I understand that the $(0,T)$ mean we are talking about over the time (since it is usually talking about the heat equation or something similar).

I would understand if it were $W^{1,p}(0,T; \Omega)$ as I would guess that would be $$\{u:\int_{[0,T]}\int_{\Omega}|u|^p<\infty, \int_{[0,T]}\int_{\Omega}|Du|^p<\infty\}$$ (though I could be wrong). I just don't know what to do with the $L^2$ inside the $W^{1,p}.$ Is it a way to say that we're assuming $u\in L^2$? Is it saying we're integrating over $L^2(\Omega)$ somehow?

Any help with understanding what this notation means would be helpful.

Answer 1

The solution of the PDE you are trying to solve is a function

$$u : [0,T] \times \Omega \to \mathbb{R}.$$

In other words, given a time and a spatial coordinate, you get a value. By currying, this is equivalent to a function $U$ from $[0,T]$ (the range) to functions from $\Omega$ to $\mathbb{R}$. From this point of view, for any time $t$, you have a function $U(t)(\cdot) = u(t, \cdot)$ on $\Omega$.

The notation $W^{1,p} (0,T; \mathbb{L}^2 (\Omega))$ is coherent with this second point of view: the indices $0$, $T$ correspond to the domain of $U$, while $\mathbb{L}^2 (\Omega)$ is its range. In other words, $U$ is seen as a function from $[0,T]$ taking its values in $\mathbb{L}^2 (\Omega)$. The derivative of $U$ is then the partial derivative of $u$ with respect to time.

To expand things a little in your example (a reference would be nice to check that I did not make a mistake here):

• for (almost?) any $t \in [0,T]$, the function $u(t, \cdot)$ belongs to $\mathbb{L}^2 (\Omega)$:

$$\left( \int_\Omega |u(t,x)|^2 \ dx \right)^{\frac{1}{2}} < +\infty.$$

• the function $U: t \mapsto u(t, \cdot)$ belongs to $\mathbb{L}^1 (0,T; \mathbb{L}^2 (\Omega))$:

$$\int_0^T\left( \int_\Omega |u(t,x)|^2 \ dx \right)^{\frac{1}{2}} \ dt< +\infty.$$

• the function $U: t \mapsto u(t, \cdot)$ belongs to $W^{1,p} (0,T; \mathbb{L}^2 (\Omega))$:

$$\left( \int_0^T\left( \int_\Omega |u(t,x)|^2 \ dx \right)^{\frac{p}{2}} \ dt \right)^{\frac{1}{p}} < +\infty \\ \text{ and } \\ \left( \int_0^T\left( \int_\Omega |\partial_t u(t,x)|^2 \ dx \right)^{\frac{p}{2}} \ dt \right)^{\frac{1}{p}}< +\infty.$$