The present paper is concerned with stability of the stationary solution of the Burgers equation in exterior domains in ${\mathbb{R}}^n$. In the previous papers [5, 6, 7] the asymptotic behavior of radially symmetric solutions for the multi‐dimensional Burgers equation in exterior domains in ${\mathbb{R}}^n,n\ge 3$, has been considered. The results [5, 6, 7] are restricted to stability of radially solutions within the class of spherically one dimensional flow. However, from a viewpoint of fluid dynamics, it is the rare case that such a radially symmetric stationary wave remains to be a radial flow under the initial disturbance. Hence it seems to be natural to handle the non‐radially symmetric perturbed fluid motion even from the radially symmetric one. On the other hand, Kozono and Ogawa [8] showed the asymptotic stability of stationary solutions for the incompressible Navier–Stokes equation on multi‐dimensional spaces. In this paper we apply their method [8] to the multidimensional Burgers equation, and show the asymptotic stability for stationary wave on ${\mathbb{R}}^n$.

中文翻译：

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具有外部多维初始扰动的Burgers方程的径向对称平稳波的稳定性。

本文讨论了Burgers方程的平稳解在外部域中的稳定性。 ${\mathbb{[R}}^{ñ}$。在先前的论文[5，6，7]中，多维Burgers方程在外部区域中的径向对称解的渐近行为${\mathbb{[R}}^{ñ}，ñ\ge 3$，已被考虑。结果[5、6、7]限于球形一维流类别内的径向解的稳定性。但是，从流体动力学的观点来看，这种径向对称的驻波在初始扰动下仍然是径向流的情况很少见。因此，即使是径向对称的流体，也要处理非径向对称的扰动流体运动是很自然的。另一方面，Kozono和Ogawa [8]显示了多维空间上不可压缩Navier-Stokes方程的平稳解的渐近稳定性。在本文中，我们将他们的方法[8]应用于多维Burgers方程，并证明了平稳波在上的渐近稳定性。${\mathbb{[R}}^{ñ}$。