Ill-posedness of leray solutions for the hypodissipative navier–stokes equations

Abstract

We prove the ill-posedness of Leray solutions to the Cauchy problem for the hypodissipative Navier–Stokes equations, when the dissipative term is a fractional Laplacian $(−Δ)^α$ with exponent $α<\frac{1}{5}$. The proof follows the “convex integration methods” introduced by the second author and László Székelyhidi Jr. for the incompressible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5},\frac{1}{2}[$.