Illposedness of incompressible fluids in supercritical Sobolev spaces

Abstract

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0<s<\frac{5}{2}$, we construct a $C_c^{\infty }$ initial velocity field with arbitrarily small $H^s$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large ${\dot{H}}^s$ norm inflation almost instantaneously. In the viscous case, the same ${\dot{H}}^s$ norm inflation occurs in the 3D Navier-Stokes equation for $0<s<\frac{1}{2}$, where $s=\frac{1}{2}$ is scaling critical for this equation.

v2: 24 pages, results unchanged, fixed mistakes and typos