Norm Inflation for Generalized Navier-Stokes Equations

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 ${\dot{B}}_{\infty ,p}^{-\alpha }$ ($2<p\le \infty$) 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 Pavlović 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 $\alpha \ge 5/4$ where the global regularity is known.

12 pages