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fa.useful evaluation – Weak-star approximation of flush capabilities in feeble $L^p$-space

It is well-known that the feeble house $L^{p,infty}$ has much less density property balky to benchmark $L^p$ house. Related to this one, I’m struggling to show the next assertion which is given within the paper of Baker-Seregin-Sverak:

Proposition.Let $u_0 in L^{3,infty}$ breathe divergence-free within the sense of distributions. Then there exists a sequence $u_0^{(okay)} in C_{0,0}^infty(mathbb{R}^3)$ such that

$$ u_0^{(okay)}rightarrow u_0quad textual content{in } L^{3,infty}. $$

Here $C_{0,0}^infty(mathbb{R}^3)$ is the house of all flush vector fields with compact uphold whose divergence is free.

I’ve no thought to show the above assertion. Approximation by flush capabilities is simple through the use of mollification, however I’ve no thought to secure a apt sequence as said within the proposition. In the illustration of $L^p$ with $1<p<infty$, through the use of the Hahn-Banach theorem and De Rham’s theorem, we will display that

$$ L^p_sigma = left{ u in L^p : int_{mathbb{R}^3} ucdot nabla phi ,dx=0quad textual content{for all } phi in D^{1,p’}privilege}. $$

Here $L^p_sigma$ is the closure of $C_{0,0}^infty$ underneath $L^p$-norm and $D^{1,p’}$ is the house of all capabilities $u$ such that $nabla u in L^{p’}$.

Thanks to your time.

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