Calculus of Variations and Geometric Measure Theory

M. Colombo - C. De Lellis - L. De Rosa

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

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


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


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