On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-08-30 DOI:10.1137/24m1640628

Jeanne Lin, Dmitry E. Pelinovsky, Björn de Rijk

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引用次数: 0

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2323-2363, September 2024.
Abstract.We are interested in the dynamics of interfaces, or zeros, of shock waves in general scalar viscous conservation laws with a locally Lipschitz continuous flux function, such as the modular Burgers equation. We prove that all interfaces coalesce within finite time, leaving behind either a single interface or no interface at all. Our proof relies on mass and energy estimates, regularization of the flux function, and an application of the Sturm theorems on the number of zeros of solutions of parabolic problems. Our analysis yields an explicit upper bound on the time of extinction in terms of the initial condition and the flux function. Moreover, in the case of a smooth flux function, we characterize the generic bifurcations arising at a coalescence event with and without the presence of odd symmetry. We identify associated scaling laws describing the local interface dynamics near collision. Finally, we present an extension of these results to the case of antishock waves converging to asymptotic limits of opposite signs. Our analysis is corroborated by numerical simulations of the modular Burgers equation.

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论标量粘性守恒定律中多重冲击的消亡

SIAM 应用动力系统期刊》第 23 卷第 3 期第 2323-2363 页，2024 年 9 月。摘要.我们对一般标量粘性守恒定律中冲击波的界面（或零点）动力学感兴趣，该守恒定律具有局部 Lipschitz 连续通量函数，例如模态伯格斯方程。我们证明，所有界面都会在有限时间内凝聚，只留下一个界面或根本没有界面。我们的证明依赖于质量和能量估计、通量函数的正则化，以及关于抛物线问题解的零点数的斯特姆定理的应用。根据初始条件和通量函数，我们的分析得出了消亡时间的明确上限。此外，在光滑通量函数的情况下，我们还描述了在存在和不存在奇数对称性的凝聚事件中出现的一般分岔。我们确定了描述碰撞附近局部界面动力学的相关缩放定律。最后，我们将这些结果扩展到反冲波收敛到相反符号渐近极限的情况。我们的分析得到了模态伯格斯方程数值模拟的证实。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.