高维Burgers方程外区域问题球对称解的渐近行为  

Asymptotics of radially symmetric solutions for the exterior problem of multidimensional Burgers equation

作　　者：杨彤[1] 赵会江[2,3] 赵青松 Tong Yang;Huijiang Zhao;Qingsong Zhao

机构地区：[1]香港城市大学数学系,中国香港 [2]武汉大学数学与统计学院,武汉430072 [3]武汉大学计算科学湖北省重点实验室,武汉430072

出　　处：《中国科学：数学》2021年第6期1057-1072,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11731008和11671309)资助项目。

摘　　要：本文考虑高维Burgers方程外区域问题球对称解的大时间渐近行为,主要关注在球对称初始扰动下球对称稳态波的非线性稳定性.对这一问题,Hashimoto和Matsumura(2019)给出了保证其球对称稳态波存在性的一个充分条件,但是由于这一稳态波不再是单调的,他们只能在更强的假设下证明其非线性稳定性.本文的主要目的是在Hashimoto和Matsumura给出的保证这一稳态波存在的条件下证明其非线性稳定性.此外,还得到了该外区域问题的整体球对称解收敛到上述稳态波的关于时间变元的代数和指数衰减率估计.本文的稳定性分析是基于空间加权的能量方法,问题的关键在于构造适当的权函数来控制由于稳态波的非单调性及边界条件的出现所导致的困难.至于关于时间变元的衰减估计,除了这一空间加权的能量方法之外,还利用了由Kawashima和Matsumura在1985年引入的空间-时间加权的能量方法.We are concerned with the large-time behavior of radially symmetric solutions to the exterior problem of multidimensional Burgers equation and focus on the nonlinear stability of its radially symmetric stationary waves under radially symmetric initial perturbation.For such a problem,a sufficient condition to guarantee the existence of such a stationary wave is obtained by Hashimoto and Matsumura in 2019,but since the stationary wave is no longer monotonic,its nonlinear stability is justified only for the case where an additional assumption is imposed.The main purpose of this paper is to verify the time asymptotically nonlinear stability of such a stationary wave under the condition imposed by Hashimoto and Matsumura to guarantee its existence.Moreover,we also derive the temporal convergence rates,both algebraically and exponentially,of solutions of the above exterior problem to the stationary wave.Our stability analysis is based on a space weighted energy method with a suitable chosen weight function,while for the temporal decay rates,in addition to such a space weighted energy method,we also use the space-time weighted energy method introduced by Kawashima and Matsumura in 1985.

关 键 词：高维Burgers方程 外区域问题 球对称稳态波 非线性稳定性 空间-时间加权的能量方法 

分 类 号：O175[理学—数学]