On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

TL;DR

This paper demonstrates the failure of unconditional uniqueness for mild solutions of Navier-Stokes equations in certain Besov spaces, and constructs stationary singular solutions using convex integration, with implications for fractional variants.

Contribution

It introduces non-uniqueness results for mild solutions in Besov spaces and constructs stationary singular solutions via convex integration, extending to fractional Navier-Stokes equations.

Findings

Unconditional uniqueness fails in all Besov spaces with negative regularity.

Constructed non-trivial stationary singular solutions using convex integration.

Proved similar results for fractional Navier-Stokes equations.

Abstract

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.