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

created by delellis on 04 Sep 2017
modified on 01 Dec 2020

[BibTeX]

Published Paper

Inserted: 4 sep 2017
Last Updated: 1 dec 2020

Journal: Comm. Math. Phys.
Volume: 362
Number: 2
Pages: 659-688
Year: 2018

ArXiv: 1708.05666 PDF

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 $[\frac{1}{5}, \frac{1}{2}[$.