Scalar conservation laws with discontinuous flux function: I. The viscous profile condition.
(1996) In Communications in Mathematical Physics 176. p.2344 Abstract
 The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in twophase flow, in trafficflow 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 nonstrictly hyperbolic system. This augmentation is nonunique 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 twophase flow, in trafficflow 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 nonstrictly hyperbolic system. This augmentation is nonunique 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 underor 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:
https://lup.lub.lu.se/record/792790
 author
 Diehl, Stefan ^{LU}
 organization
 publishing date
 1996
 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
 14320916
 DOI
 10.1007/BF02099361
 language
 English
 LU publication?
 yes
 id
 8a26a9839a8e4c9aaa4b08ae6e744ad5 (old id 792790)
 alternative location
 http://www.springerlink.com/content/r031176365vv3552/fulltext.pdf
 http://projecteuclid.org/euclid.cmp/1104285902
 date added to LUP
 20160404 07:07:42
 date last changed
 20220129 01:44:13
@article{8a26a9839a8e4c9aaa4b08ae6e744ad5, abstract = {{The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in twophase flow, in trafficflow 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 nonstrictly hyperbolic system. This augmentation is nonunique 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 underor 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 = {{14320916}}, language = {{eng}}, pages = {{2344}}, 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}}, doi = {{10.1007/BF02099361}}, volume = {{176}}, year = {{1996}}, }