Existence and uniqueness of lax-type solutions to the Riemann problem of scalar balance law with singular source term

Yuan Chang, Shih Wei Chou, John M. Hong, Ying Chieh Lin

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Abstract

We give a new approach of constructing the generalized entropy solutions to the Riemann problem of scalar nonlinear balance laws with singular source terms. The source term is singular in the sense that it is a product of delta function and a discontinuous function, which is undefined in distribution. By re-formulating the source term, we study the corresponding perturbed Riemann problem. The existence and stability of perturbed Riemann solutions is established under some entropy condition so that the generalized entropy solutions of Riemann problem can be interpreted as the limit of corresponding perturbed Riemann solutions. The self-similarity of generalized entropy solutions is also obtained, which means that Lax's method in [13] can be extended to scalar nonlinear balance laws with singular source terms.

Original languageEnglish
Pages (from-to)431-464
Number of pages34
JournalTaiwanese Journal of Mathematics
Volume17
Issue number2
DOIs
StatePublished - 2013

Keywords

  • Characteristic method
  • Conservation laws
  • Lax's method
  • Nonlinear balance laws
  • Perturbed Riemann problems
  • Riemann problems

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