Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces

Abstract

We prove that the incompressible Navier-Stokes equations exhibit norm inflation in ${\dot{B}}_{p,q}^s({\mathbb{R}}^3)$ with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical ${\dot{B}}_{p,q}^s$ near the scaling-critical line $s=-1+\frac{3}{p}$ except at $s=0$. The growth mechanism differs depending on the sign of the regularity index $s$: forward energy cascade driven by mixing for $s>0$ and backward energy cascade caused by un-mixing for $s<0$. The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations.

31 pages