Scalar conservation laws with discontinuous flux function. I: The viscous profile condition
type: Item
instance of: scholarly article
title: Scalar conservation laws with discontinuous flux function. I: The viscous profile condition
zbMATH Open document ID: 0845.35067
DOI: 10.1007/BF02099361
published in: Communications in Mathematical Physics
publication date: 1996-09-11T00:00:00Z
review text: The equation \({\partial u\over \partial t} {\partial\over \partial x} (H(x) f(u)(1- H(x)) g(u))= 0\), where \(H\) is Heviside's step function, arises on continuous sedimentation of solid particles in a liquid, in two-phase flow in porous media etc. The discontinuity of the flux function causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular \(2\times 2\) nonstrictly hyperbolic system. This augmentation is nonunique and a natural definition is given by means a viscous profiles. A viscous profile is a stationary solution of \(u_t(F^\delta)_x= \varepsilon u_{xx}\), where \(F^\delta\) is a smooth approximation of the discontinuous flux, i.e. \(H\) is smoothed. In terms of the \(2\times 2\), the discontinuity at \(x= 0\) is either a regular Lax, an under- or over-compressive or a degenerate shock wave. In some cases, depending on \(f\) and \(g\), there is unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). It is proved the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution at \(x= 0\) and the viscous profile condition.
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 76T99
zbMATH DE Number: 844127
zbMATH Keywords: discontinuous flux
zbMATH Keywords: viscous profiles
author: Stefan Diehl
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02099361
OpenAlex ID: W1994609182
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description: scientific article; zbMATH DE number 844127