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 $\dot{B}{\infty,p}^{-\alpha}$ ($2<p \leq \infty$) 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.