Weak-star approximation of smooth functions in weak $L^p$-space

Question

It is well known that the weak space $L^{p,\infty}$ has less density property contrary to standard $L^p$ space. Related to this one, I'm struggling to prove the following statement which is given in the paper of Baker-Seregin-Sverak:

Proposition. Let $u_0 \in L^{3,\infty}$ be divergence-free in the sense of distributions. Then there exists a sequence $u_0^{(k)} \in C_{0,0}^\infty(\mathbb{R}^3)$ such that $$ u_0^{(k)}\rightarrow u_0\quad \text{weakly star in } L^{3,\infty}. $$

Here $C_{0,0}^\infty(\mathbb{R}^3)$ is the space of all smooth vector fields with compact support whose divergence is free.

I have no idea to prove the above statement. Approximation by smooth functions is easy by using mollification, but I have no idea to obtain a suitable sequence as stated in the proposition. In the case of $L^p$ with $1<p<\infty$, by using the Hahn-Banach theorem and De Rham's theorem, we can show that $$ L^p_\sigma = \left\{ u \in L^p : \int_{\mathbb{R}^3} u\cdot \nabla \phi \,dx=0\quad \text{for all } \phi \in D^{1,p'}\right\}. $$ Here $L^p_\sigma$ is the closure of $C_{0,0}^\infty$ under $L^p$-norm and $D^{1,p'}$ is the space of all functions $u$ such that $\nabla u \in L^{p'}$.

Thanks for your time.

Answer 1

The statement as written above is incorrect. It should be weak-$\ast$ convergence instead of strong convergence. This is one of the key features of $L^{p,\infty}$: Test functions are not dense. As a general comment, in the Navier-Stokes context, local well-posedness holds for initial data in the strong closure of divergence-free test functions in $L^{3,\infty}$ but probably not for general large divergence-free fields in $L^{3,\infty}$. For example, non-uniqueness of large self-similar solutions is conjectured in the papers of Jia-Sverak, with numerical evidence later given by Guillod-Sverak.

To do the weak-$\ast$ approximation, my suggestion would be to mollify, apply a smooth cut-off, and then use Bogovskii's operator to add a correction which makes it divergence free (see Chapter 3 of Galdi's book for an extensive discussion, and there is also a statement in Tsai's book). You can use real interpolation to bound the Bogovskii operator on Lorentz spaces. This procedure is very useful for localizing divergence-free functions.