Ill-posedness for the Navier-Stokes equations in critical Besov spaces ${\dot{B}}_{\infty ,q}^{-1}$

Abstract

We study the Cauchy problem for the incompressible Navier-Stokes equation \begin{align} u_t -Δu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= δu_0. \label{NS} \end{align} For arbitrarily small $\delta >0$, we show that the solution map $\delta u_0\to u$ in critical Besov spaces ${\dot{B}}_{\infty ,q}^{-1}$ ($\forall q\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BM{O}^{-1}$. Taking notice of the embedding ${\dot{B}}_{\infty ,q}^{-1}\subset BM{O}^{-1}$ ($q\le 2$), we see that for sufficiently small $\delta >0$, $u_0\in {\dot{B}}_{\infty ,q}^{-1}$ ($q\le 2$) can guarantee that the Navier-Stokes equation has a unique global solution in $BM{O}^{-1}$, however, this solution is instable in ${\dot{B}}_{\infty ,q}^{-1}$ and the solution can have an inflation in ${\dot{B}}_{\infty ,q}^{-1}$ for certain initial data. So, our result indicates that two different topological structures in the same space may determine the well and ill posedness, respectively.

25 Pages, in this new version, we add a reference of Iwabuchi T. and Nakamura M

References in corpus (3)

• An alternative approach to regularity for the Navier-Stokes equations in critical spaces
• Analyticity for the (generalized) Navier-Stokes equations with rough initial data
• Ill-posedness of the incompressible Navier-Stokes equations in ${\dot{F}}_{\infty }^{-1,q}(R^3)$