Vanishing viscosity non-unique solutions to the forced 2D Euler Equations

2025/07/25 by Dallas Albritton, Albritton, Dallas, Maria Colombo +3
Mathematics · Earth and Planetary Sciences · Engineering · #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing #Fluid dynamics and aerodynamics studies

paper · pdf · doi:10.48550/arxiv.2507.19257

Abstract

The forced 2D Euler equations exhibit non-unique solutions with vorticity in L^p, p > 1, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit ν→ 0⁺ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity ν and consider O(ε) size perturbations of the initial datum. We discover a uniqueness threshold ε ∼ ν^κ_{\rm c}, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions.

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