Positive-regularity norm inflation for the 3D hypodissipative Navier--Stokes equations

Abstract

We prove same-space norm inflation for the three-dimensional hypodissipative Navier--Stokes equations with dissipation $(-\Delta {)}^{\alpha }$, $0<\alpha <1$. Let $2\le p<\infty$ and \[ 0<s<1-2α+\frac3p. \] In the Besov case, let also $1\le q\le \infty$. There exist divergence-free $C_c^{\infty }({\mathbb{R}}^3)$ initial data that are arbitrarily small in $W^{s,p}$, respectively in $B_{p,q}^s$, while the corresponding unique local smooth solution becomes arbitrarily large in the same space in arbitrarily short time. The proof adapts the anisotropic vortex-ring mixing mechanism to fractional dissipation; the strict scaling-supercritical gap makes both the curvature error and the nonlocal dissipative error perturbative.