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

Abstract

We investigate the size of the regular set for suitable weak solutions of the Navier--Stokes equation, in the sense of Caffarelli--Kohn--Nirenberg. 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 $|x{|}^{-\frac{1}{2}}$ of the data tends to 0, the regular set invades $\{t>0\}$.

References in corpus (3)

• Angular Regularity and Strichartz Estimates for the Wave Equation
• Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with angular integrability
• Singular integrals and maximal functions: the disk multiplier revisited

Cited by in corpus (3)

• An $\epsilon$-regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data
• Stability properties of the regular set for the Navier--Stokes equation
• Singular integrals with angular integrability