Nonuniqueness of weak solutions to the Navier-Stokes equation

Pages 101-144 from Volume 189 (2019), Issue 1 by Tristan Buckmaster, Vlad Vicol

Abstract

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

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Keywords
, , , , , ,
Mathematical Subject Classification
Primary: , , ,
DOI
https://doi.org/10.4007/annals.2019.189.1.3
MR
3898708
zbMATH
07003146
Milestones
Received: 5 October 2017
Revised: 9 October 2018
Accepted: 10 October 2018
Published online: 11 January 2019

Authors

Tristan Buckmaster

Princeton University, Princeton, NJ

Vlad Vicol

Princeton University, Princeton, NJ

Current address:

Courant Institute of Mathematical Sciences, New York University, New York, NY