Ill-posedness of the Navier-Stokes equations in a critical space in 3D

Abstract

We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in ${\dot{B}}_{\infty }^{-1,\infty }$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in the Schwartz class $\mathcal{S}$ that are arbitrarily small in ${\dot{B}}_{\infty }^{-1,\infty }$ can produce solutions arbitrarily large in ${\dot{B}}_{\infty }^{-1,\infty }$ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in ${\dot{B}}_{\infty }^{-1,\infty }$ at the origin.

16 pages, no figures

References in corpus (2)

• Ill-posedness for nonlinear Schrodinger and wave equations
• Instability of the periodic nonlinear Schrodinger equation