On the piecewise smoothness of entropy solutions to scalar conservation laws for a larger class of initial data

Tao TANG*, Jinghua Wang, Yinchuan Zhao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

We prove that if the initial data do not belong to a certain subset of Ck, then the solutions of scalar conservation laws are piecewise Ck smooth. In particular, our initial data allow centered compression waves, which was the case not covered by Dafermos (1974) and Schaeffer (1973). More precisely, we are concerned with the structure of the solutions in some neighborhood of the point at which only a Ck+1 shock is generated. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except for a certain subset of ℒ(ℝ) of the first category. It should be pointed out that this subset is smaller than those used in previous works. We point out that Thom's theory of catastrophes, which plays a key role in Schaeffer (1973), cannot be used to analyze the larger class of initial data considered in this paper.

Original languageEnglish
Pages (from-to)369-389
Number of pages21
JournalJournal of Hyperbolic Differential Equations
Volume4
Issue number3
DOIs
Publication statusPublished - Sep 2007

Scopus Subject Areas

  • Analysis
  • Mathematics(all)

User-Defined Keywords

  • A set of first category
  • Conservation laws
  • Piecewise smooth solutions

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