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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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