Singular limits for a parabolic-elliptic regularization of scalar conservation laws

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• 作者：Andrea Corli^a ; ^{andrea.corli@unife.it} ; Christian Rohde^b
• 关键词：Scalar conservation laws ; Nonconvex flux ; Undercompressive shock waves ; Parabolic&ndash ; elliptic regularization ; Traveling waves
• 刊名：Journal of Differential Equations
• 出版年：2012
• 期刊代码：74_00220396
• 类别：mt
• 出版时间：1 September, 2012
• 卷：253
• 期：5
• 页码：1399-1421
• 文件大小：257 K

摘要

We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic-elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive-dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic-elliptic system can be understood as a low-order approximation of the third-order diffusive-dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic-elliptic system completes the paper.