I'm reading Lemarie-Rieusset's book Recent developments in the Navier-Stokes problem and have the following issue: he defines a weak solution to the Navier-Stokes equations on $(0,T)\times\mathbb R^d$ as a distribution vector field $u(t,x)$ in $(\mathcal D'((0,T)\times\mathbb R^d)^d$ such that
- $u$ is locally square integrable on $(0,T)\times\mathbb R^d$
- There exists a pressure $p$ in $\mathcal D'((0,T)\times\mathbb R^d)$ such that $u,p$ satisfy the equations in the sense of distributions.
The second point is quite clear. But what on earth does it mean for a distribution (or vector of them) to be square integrable? He proceeds throughout the chapter (and probably the rest of the book too) to treat things that he defined as distributions as though they take arguments. Does this have a precise mathematical meaning or is he dropping formalities out of convenience? To me it seems like this defeats the purpose of considering weak solutions.
My guess is that the author makes a hypothesis that the solution is a distribution which can be represented by a $L^2_{loc}$ function in the sense that there exists a function $g\in L^2_{loc}$ such that for any test function $\phi$ $$\langle u,\phi\rangle=\int_{(0,T)\times \Bbb R^d}g(t,x)\phi(t,x)\,\mathrm dt\,\mathrm dx$$