Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier–Stokes equations in dimension three. We show that it is locally L 4 3 , q for any q > 4 3 , which improves from the current result of L 4 3 , ∞ . Similar improvements in Lorentz space are also obtained...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2021-08, Vol.241 (2), p.683-727
Main Authors: Vasseur, Alexis, Yang, Jincheng
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
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Fluid dynamics
Fluid flow
Fluid- and Aerodynamics
Inequality
Mathematical analysis
Mathematical and Computational Physics
Navier-Stokes equations
Physics
Physics and Astronomy
Spacetime
Theoretical
Transport equations
Velocity
Vorticity equations
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Summary:We study the second spatial derivatives of suitable weak solutions to the incompressible Navier–Stokes equations in dimension three. We show that it is locally L 4 3 , q for any q > 4 3 , which improves from the current result of L 4 3 , ∞ . Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-021-01661-4