{"title":"Time asymptotic behavior of non-equilibrium flows in one space dimension","authors":"Yanni Zeng","doi":"10.1016/j.physd.2025.134647","DOIUrl":null,"url":null,"abstract":"<div><div>We study long time behavior of polyatomic gas flows in both translational and vibrational non-equilibrium. The author previously established global existence of solution and obtained optimal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> time-decay rates for the solution towards an equilibrium state for the Cauchy problem. The current paper is a continuation in studying the solution behavior. An asymptotic solution is constructed explicitly using a heat kernel along the particle path and two Burgers kernels along the equilibrium acoustic directions. Convergence of the exact solution to the asymptotic solution is studied in a pointwise sense in both space and time to give a complete picture of wave propagation. The study lays a foundation for a future work on solution behavior around a shock wave, a mechanism that induces Richtmyer–Meshkov instability in mixing problems .</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134647"},"PeriodicalIF":2.7000,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925001265","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We study long time behavior of polyatomic gas flows in both translational and vibrational non-equilibrium. The author previously established global existence of solution and obtained optimal time-decay rates for the solution towards an equilibrium state for the Cauchy problem. The current paper is a continuation in studying the solution behavior. An asymptotic solution is constructed explicitly using a heat kernel along the particle path and two Burgers kernels along the equilibrium acoustic directions. Convergence of the exact solution to the asymptotic solution is studied in a pointwise sense in both space and time to give a complete picture of wave propagation. The study lays a foundation for a future work on solution behavior around a shock wave, a mechanism that induces Richtmyer–Meshkov instability in mixing problems .
期刊介绍:
Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.