粘性冲击与标量守恒律的长时间行为

IF 1 3区数学 Q1 MATHEMATICS

Communications on Pure and Applied Analysis Pub Date : 2023-01-01 DOI:10.3934/cpaa.2023119

Thierry Gallay, Arnd Scheel

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引用次数: 0

摘要

我们通过极限集的结构研究了标量粘性守恒律的长时间行为。通过建立对任意单调初始数据的收敛性，证明了$ \ ω $-极限集总是包含常数或激波。在Burgers方程的特殊情况下，我们回顾并完善了根据概率度量参数化整个解决方案的结果，并且我们构建了初始数据，其中$ \omega $-极限集没有被简化为单个冲击的转换。最后，我们提出了几个与长时间动力学描述有关的开放性问题。

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Viscous shocks and long-time behavior of scalar conservation laws

We study the long-time behavior of scalar viscous conservation laws via the structure of $ \omega $-limit sets. We show that $ \omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ \omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.

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来源期刊

Communications on Pure and Applied Analysis 数学-数学

CiteScore

1.90

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.