论文标题
固定的Navier-Stokes方程的不良适合位置
Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces
论文作者
论文摘要
本文提出了一个开放问题的一些进展，该问题提出了Tsurumi（Arch。Mech。Mech。Anal。234：2，2019）：无论是否固定的Navier-Stokes方程$ \ r^d $都很好地源自$ \ dot {b} _ {b} _ { \ dot {b} _ {p，q}^{0} $，$ p = d $和$ 1 \ leq q \ leq 2 $。在本文中，我们证明，对于$ 1 \ leq q <\ frac d2 $带有$ d \ geq4 $的固定navier-stokes方程不属于$ \ dot {b} _ {d，q}^，q}^{ - 2}^{ - 2}（ - q}^{0}（\ r^d）$，通过显示一系列外部力的序列以显示零以零的不连续性。实际上，在这种情况下，存在一系列外部力的序列，这些序列在$ \ dot {b} _ {d，q}^{ - 2} $中收敛至零，并产生一系列解决方案，这些序列不会在$ \ dot {b} _ {b} _ {d，q} _} _ {d，q}^0} $中收敛到零。特别是，我们还证明了固定的navier-stokes方程是从$ \ dot {b} _ {d，2}^{ - 2}（\ r^d）$ to $ \ dot {b} _ {\ r^d）$ to $ \ mathbb {p} \ dot {p} \ dot {b} _ {b} _ {d} _ {d} _ {d}^0} $ d = 3.4基于这两种情况，我们证明了尺寸$ d \ geq4 $的上述打开问题已完全解决。
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.
