article

Title: Norm inflation for generalized Navier-Stokes equations

Authors: Mimi Dai and Alexey Cheskidov

Issue: Volume 63 (2014), Issue 3, 869-884

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 initial data in $\dot{B}_{\infty,p}^{-\alpha}$ ($2<p\leq\infty$) 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 [J.\ Bourgain and N.\ Pavlovi\'c, \textit{Ill-posedness of the Navier-Stokes equations in a critical space in 3D}, J.\ Funct.\ Anal.\ \textbf{255} (2008), no. 9, 2233--2247] for the classical Navier-Stokes equation, a result which uses 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 $\alpha\geq\frac{5}{4}$ where the global regularity is known.$