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

TL;DR

This paper investigates the well-posedness of stationary Navier-Stokes equations in critical Besov spaces, proving ill-posedness for certain parameters and well-posedness for others, thereby resolving an open question for dimensions four and higher.

Contribution

It establishes the ill-posedness in specific Besov spaces for dimensions d≥4 and confirms well-posedness in the case d=3,4, solving an open problem in the field.

Findings

Ill-posedness for 1≤q<d/2 in dimensions d≥4.

Well-posedness for q=2 in dimensions d=3,4.

Complete resolution of the open question for d≥4.

Abstract

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 $1 \leq q \leq 2$. In this paper, we prove that for the case $1\leq q<\frac d2$ with $d\geq4$ the stationary Navier-Stokes equations is ill-posed from $\dot{B}_{d, q}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d, q}^{0}(\R^d)$ 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 $\dot{B}_{d, q}^{-2}$ and yields a sequence of solutions which does not converge to zero in $\dot{B}_{d, q}^{0}$. In particular, we also prove that the stationary Navier-Stokes equations is well-posed from $\dot{B}_{d, 2}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d, 2}^{0}(\R^d)$ with $d=3,4$. Based on these two cases, we demonstrate that the above open question for the dimension $d\geq4$ has been solved completely.