The regularity of weak solutions of the 3D Navier-Stokes equations in \(B^{-1}_{\infty,\infty}\). (English) Zbl 1186.35137

Cheskidov, A.; Shvydkoy, R.

Arch. Ration. Mech. Anal. 195, No. 1, 159-169 (2010).

Summary: We show that if a Leray-Hopf solution \(u\) of the three-dimensional Navier-Stokes equation belongs to \(C((0,T]; B^{-1}_{\infty,\infty})\) or its jumps in the \(B^{-1}_{\infty,\infty}\)-norm do not exceed a constant multiple of viscosity, then \(u\) is regular for \((0, T]\). Our method uses frequency local estimates of the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion.

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05	Navier-Stokes equations for incompressible viscous fluids
35B65	Smoothness and regularity of solutions to PDEs
35D30	Weak solutions to PDEs

Keywords:

Navier-Stokes equation; weak solutions; Leray-Hopf solution; Ladyzhenskaya-Prodi-Serrin criteria

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