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

TL;DR

This paper demonstrates ill-posedness of fundamental fluid dynamics equations in certain Besov spaces by constructing initial data leading to discontinuous solutions at initial time.

Contribution

It constructs specific divergence-free initial data in Besov spaces causing solutions to be discontinuous at t=0 for Navier-Stokes and Euler equations.

Findings

Solutions are discontinuous at t=0 in specified Besov spaces.

Ill-posedness is shown for Navier-Stokes in B^{-1}_{\infty,\infty}.

Similar ill-posedness results are established for Euler equations in various Besov spaces.

Abstract

We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, 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 $B^{-1}_{\infty,\infty}$. For the Euler equation a similar result is proved in all Besov spaces $B^s_{r,\infty}$ where $s>0$ if $r>2$, and $s>n(2/r-1)$ if $1 \leq r \leq 2$.