On the regularity set and angular integrability for the Navier–Stokes equation

We investigate the size of the regular set for suitable weak solutions of the Navier--Stokes equation, in the sense of Caffarelli--Kohn--Nirenberg cite{CKN}. We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space ${t>0}$ in an appropriate limit. In particular, we obtain that if the $L^{2}$ norm with weight $|x|^{-rac12}$ of the data tends to 0, the regular set invades ${t>0}$; this result improves Theorem D of cite{CKN}.