Nonuniqueness of weak solutions to the Navier-Stokes equation

TL;DR

This paper demonstrates the nonuniqueness of weak solutions to the 3D Navier-Stokes equations with finite energy and shows that certain Euler solutions can be approximated as limits of Navier-Stokes solutions.

Contribution

It proves the nonuniqueness of finite energy weak solutions to the 3D Navier-Stokes equations and links dissipative Euler solutions to Navier-Stokes via vanishing viscosity limits.

Findings

Weak solutions of 3D Navier-Stokes are not unique.

Dissipative Euler solutions can be obtained as limits of Navier-Stokes solutions.

Nonuniqueness impacts understanding of fluid dynamics models.

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 Holder 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.