Advanced

Scalar conservation laws with discontinuous flux function: I. The viscous profile condition.

Diehl, Stefan LU (1996) In Communications in Mathematical Physics 176. p.23-44
Abstract
The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution ofu t +(F ) x =u xx , whereF is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. In terms of the 2×2 system, the discontinuity atx=0 is either a regular Lax, an... (More)
The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution ofu t +(F ) x =u xx , whereF is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. In terms of the 2×2 system, the discontinuity atx=0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending onf andg, there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx=0 and the viscous profile condition. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Communications in Mathematical Physics
volume
176
pages
23 - 44
publisher
Springer
external identifiers
  • scopus:0030529879
ISSN
1432-0916
DOI
10.1007/BF02099361
language
English
LU publication?
yes
id
8a26a983-9a8e-4c9a-aa4b-08ae6e744ad5 (old id 792790)
alternative location
http://www.springerlink.com/content/r031176365vv3552/fulltext.pdf
http://projecteuclid.org/euclid.cmp/1104285902
date added to LUP
2009-02-13 14:16:25
date last changed
2017-08-06 04:43:43
@article{8a26a983-9a8e-4c9a-aa4b-08ae6e744ad5,
  abstract     = {The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution ofu t +(F ) x =u xx , whereF is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. In terms of the 2×2 system, the discontinuity atx=0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending onf andg, there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx=0 and the viscous profile condition.},
  author       = {Diehl, Stefan},
  issn         = {1432-0916},
  language     = {eng},
  pages        = {23--44},
  publisher    = {Springer},
  series       = {Communications in Mathematical Physics},
  title        = {Scalar conservation laws with discontinuous flux function: I. The viscous profile condition.},
  url          = {http://dx.doi.org/10.1007/BF02099361},
  volume       = {176},
  year         = {1996},
}