Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations

TL;DR

This paper demonstrates that Leray solutions to the fractional dissipative Navier-Stokes equations are ill-posed when the dissipation exponent is less than 1/5, using convex integration methods originally developed for Euler equations.

Contribution

It establishes ill-posedness for a range of fractional dissipation exponents in the Navier-Stokes equations, extending the understanding of solution behavior in these regimes.

Findings

Ill-posedness for $eta<1/5$ in fractional Navier-Stokes

Convex integration methods applicable to this problem

Results extend to exponents up to just below 1/2

Abstract

We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-\Delta)^\alpha$ with exponent $\alpha < \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and L\'aszl\'o Sz\'ekelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.