Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces

Jinlu Li, Yanghai Yu, Weipeng Zhu

Published 2022-04-18Version 1

This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in $\R^d$ is well-posed from $\dot{B}_{p, q}^{-2}$ to $\mathbb{P} \dot{B}_{p, q}^{0}$ with $p=d$ and $1 \leq q < 2$. In this paper, we demonstrate that for the case $1\leq q<2$ the 4D stationary Navier-Stokes equations is ill-posed from $\dot{B}_{4, q}^{-2}(\R^4)$ to $\mathbb{P} \dot{B}_{4, q}^{0}(\R^4)$ by showing that a sequence of external forces is constructed to show discontinuity of the solution map at zero. Indeed in such case of $q$, there exists a sequence of external forces which converges to zero in $\dot{B}_{4, q}^{-2}$ and yields a sequence of solutions which does not converge to zero in $\dot{B}_{4, q}^{0}$.