Non-uniqueness of Weak Solutions to Hyperviscous Navier-Stokes Equations -- On Sharpness of J.-L. Lions Exponent

TL;DR

This paper demonstrates that for the 3D Navier-Stokes equations with fractional hyperviscosity below a critical exponent, weak solutions are not unique, highlighting the sharpness of Lions' exponent using convex integration.

Contribution

It establishes non-uniqueness of weak solutions for fractional hyperviscous Navier-Stokes equations when the exponent is less than 5/4, confirming the sharpness of Lions' exponent.

Findings

Non-uniqueness of weak solutions for $ heta < 5/4$

Application of convex integration to fractional Navier-Stokes

Sharpness of J.-L. Lions' exponent $5/4$

Abstract

Using the convex integration technique for the three-dimensional Navier-Stokes equations introduced by T. Buckmaster and V. Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier-Stokes equations with fractional hyperviscosity $(-\Delta)^{\theta}$, whenever the exponent $\theta$ is less than J.-L. Lions' exponent $5/4$, i.e., when $\theta < 5/4$.