{"title":"N-wave-like properties for entropy solutions to scalar parabolic–hyperbolic conservation laws","authors":"Hiroshi Watanabe","doi":"10.1016/j.nonrwa.2024.104265","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we consider qualitative properties for entropy solutions to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equations have both properties of hyperbolic equations and those of parabolic equations, it is difficult to investigate the behavior of solutions to (CP). In our previous works, we focused on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with multiple discontinuity. Moreover, we constructed rarefaction wave type sub-, super-solutions to (CP) and investigated their properties.</div><div>In the present paper, we investigate “<span><math><mi>N</mi></math></span>-wave-like properties” for entropy solutions to (CP) while we are not able to construct an analogue of <span><math><mi>N</mi></math></span>-waves. In particular, we derive generalized one-sided Lipschitz estimates (Oleinik type entropy estimates) and decay estimates for entropy solutions to (CP). Based on the decay estimates, we discuss the asymptotic profiles of entropy solutions to (CP) under some specific setting.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104265"},"PeriodicalIF":1.8000,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824002049","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
In this paper, we consider qualitative properties for entropy solutions to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equations have both properties of hyperbolic equations and those of parabolic equations, it is difficult to investigate the behavior of solutions to (CP). In our previous works, we focused on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with multiple discontinuity. Moreover, we constructed rarefaction wave type sub-, super-solutions to (CP) and investigated their properties.
In the present paper, we investigate “-wave-like properties” for entropy solutions to (CP) while we are not able to construct an analogue of -waves. In particular, we derive generalized one-sided Lipschitz estimates (Oleinik type entropy estimates) and decay estimates for entropy solutions to (CP). Based on the decay estimates, we discuss the asymptotic profiles of entropy solutions to (CP) under some specific setting.
期刊介绍:
Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems.
The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.