Ill-posedness of basic equations of fluid dynamics in Besov spaces

Abstract

We give a construction of a divergence-free vector field $u_0\in H^s\cap B_{\infty ,\infty }^{-1}$, for all $s<1/2$, such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B_{\infty ,\infty }^{-1}$. For the Euler equation a similar result is proved in all Besov spaces $B_{r,\infty }^s$ where $s>0$ if $r>2$, and $s>n(2/r-1)$ if $1\le r\le 2$.

9 pages