Vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

TL;DR

This paper demonstrates that the vanishing viscosity limit of Navier-Stokes solutions can select unique weak solutions of the Euler equations, highlighting a potential criterion for solution uniqueness in fluid dynamics.

Contribution

It provides a specific example where vanishing viscosity selects a unique Euler solution among non-unique weak solutions, suggesting a broader applicability of this principle.

Findings

Vanishing viscosity limit yields a unique shear flow solution.

Existence of non-unique weak solutions to Euler equations.

Vanishing viscosity can serve as a selection principle.

Abstract

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray-Hopf weak solutions for the Navier-Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier-Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.