*Published Paper*

**Inserted:** 4 sep 2017

**Last Updated:** 1 dec 2020

**Journal:** Comm. Math. Phys.

**Volume:** 362

**Number:** 2

**Pages:** 659-688

**Year:** 2018

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

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