Calculus of Variations and Geometric Measure Theory
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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



Inserted: 4 sep 2017
Last Updated: 4 sep 2017

Year: 2017

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


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