Majda and ZND Models for Detonation: Nonlinear Stability vs. Formation of Singularities

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-09-04 DOI:10.1137/23m1544945

Paul Blochas, Aric Wheeler

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引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6137-6191, October 2024.
Abstract. In this paper we explore the boundaries of damping estimates by comparing and contrasting two closely related models of combustion, the Majda and ZND models. We are especially concerned with studying the behavior of perturbations of discontinuous waves. On the one hand, we show that singularities form in the unweighted Lipschitz norm on both sides of the shock for both models, extending classical results of John in [Comm. Pure Appl. Math., 27 (1974), pp. 377–405] and Liu in [J. Differential Equations, 33 (1979), pp. 92–111] to suitable variable coefficient systems This involves adapting John’s argument to perturbations of nonconstant waves instead of perturbations of constants. On the other hand, we show instability in exponentially weighted Sobolev spaces for ZND and stability for the Majda model in similarly weighted spaces. This involves proving high order energy estimates, using convective effects and the partial decay coming from a damping term, while being careful with the boundary terms. We note that the convective effects are the origin of the instability in the weighted norm in the ZND model.

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爆炸的 Majda 和 ZND 模型：非线性稳定性与奇点的形成

SIAM 数学分析期刊》，第 56 卷第 5 期，第 6137-6191 页，2024 年 10 月。摘要在本文中，我们通过比较和对比两个密切相关的燃烧模型--Majda 模型和 ZND 模型--来探索阻尼估计的边界。我们特别关注研究不连续波的扰动行为。一方面，我们证明了这两个模型在冲击两侧的非加权 Lipschitz norm 中形成奇点，从而将 John 在 [Comm. Pure Appl.另一方面，我们证明了 ZND 在指数加权 Sobolev 空间中的不稳定性，以及 Majda 模型在类似加权空间中的稳定性。这涉及到利用对流效应和阻尼项产生的部分衰减来证明高阶能量估计，同时注意边界项。我们注意到，对流效应是 ZND 模型中加权规范不稳定性的根源。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.