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