Norm Inflation for Generalized Navier-Stokes Equations

TL;DR

This paper demonstrates that for a generalized Navier-Stokes equation with fractional Laplacian, solutions can exhibit rapid norm inflation in certain Besov spaces, even when global regularity is established.

Contribution

It extends previous results on norm inflation to a broader class of fractional Navier-Stokes equations, including supercritical cases.

Findings

Existence of smooth solutions with small initial data in certain Besov spaces

Rapid norm inflation in negative smoothness Besov spaces

Norm inflation occurs even for bove the known regularity threshold

Abstract

We consider the incompressible Navier-Stokes equation with a fractional power $\alpha\in[1,\infty)$ of the Laplacian in the three dimensional case. We prove the existence of a smooth solution with arbitrarily small in $\dot{B}_{\infty,p}^{-\alpha}$ ($2<p \leq \infty$) initial data that becomes arbitrarily large in $\dot{B}_{\infty,\infty}^{-s}$ for all $s> 0$ in arbitrarily small time. This extends the result of Bourgain and Pavlovi\'{c} for the classical Navier-Stokes equation which utilizes the fact that the energy transfer to low modes increases norms with negative smoothness indexes. It is remarkable that the space $\dot{B}_{\infty,\infty}^{-\alpha}$ is supercritical for $\alpha >1$. Moreover, the norm inflation occurs even in the case $\alpha \geq 5/4$ where the global regularity is known.