Some remark on the existence of infinitely many nonphysical solutions to the incompressible Navier-Stokes equations

TL;DR

This paper demonstrates the existence of infinitely many nonphysical solutions with infinite energy for the incompressible Navier-Stokes equations in two dimensions, highlighting potential issues in solution uniqueness.

Contribution

It establishes the existence of infinitely many distributional solutions with infinite energy for both Navier-Stokes and Burgers equations in unbounded domains.

Findings

Infinitely many solutions with infinite energy exist for 2D Navier-Stokes.

Infinitely many solutions exist for Burgers equation in  .

Solutions are distributional and nonphysical.

Abstract

We prove that there exist infinitely many distributional solutions with infinite kinetic energy to the incompressible Navier-Stokes equations in $ \mathbb{R}^2 $. We prove as well the existence of infinitely many distributional solutions for Burgers equation in $ \mathbb{R} $.