Norm Inflation for Generalized Navier-Stokes Equations

We consider the incompressible Navier-Stokes equation with a fractional power $α\in[1,\infty)$ of the Laplacian in the three dimensional case. We prove the existence of a smooth solution with arbitrarily small in $\dot{B}_{\infty,p}^{-α}$ ($2<p \leq \infty$) 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 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 $\dot{B}_{\infty,\infty}^{-α}$ is supercritical for $α>1$. Moreover, the norm inflation occurs even in the case $α\geq 5/4$ where the global regularity is known.

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