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 (Commun Pure Appl Math 35:771-831, 1982). 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 in an appropriate limit. In particular, we obtain that if the norm with weight of the data tends to 0, the regular set invades ; this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771-831, 1982).