Stability of the radially symmetric stationary wave of the Burgers equation with multi‐dimensional initial perturbations in exterior domain

• Published in: Mathematische Nachrichten Vol. 293; no. 12; pp. 2348 - 2362
• Main Author: Hashimoto, Itsuko
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.12.2020
• Subjects: asymptotic behavior; Asymptotic methods; Asymptotic properties; Burgers equation; Dimensional stability; Domains; Flow stability; Fluid dynamics; Fluid flow; Incompressible flow; Linear equations; Nonlinear equations; One dimensional flow; Radial flow; radially symmetric solution
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.201900233

The present paper is concerned with stability of the stationary solution of the Burgers equation in exterior domains in Rn. In the previous papers [5, 6, 7] the asymptotic behavior of radially symmetric solutions for the multi‐dimensional Burgers equation in exterior domains in Rn,n≥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 Rn.