Unique Conservative Solutions to a Variational Wave Equation

Alberto Bressan, Geng Chen, Qingtian Zhang

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $${u_{tt} - c (u) (c(u)u_{x}) x = 0}$$utt-c(u)(c(u)ux)x=0. Given a solution u(t, x), even if the wave speed c(u) is only Hör continuous in the t – x plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables X, Y, constant along characteristics, we prove that t, x, u, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $${u(0, \cdot) \in H^{1}(I\!R), u_{t} (0, \cdot) \in L^{2}(I\!R).}$$u(0,·)∈H1(IR),ut(0,·)∈L2(IR).

Original languageEnglish (US)
Pages (from-to)1069-1101
Number of pages33
JournalArchive for Rational Mechanics and Analysis
Volume217
Issue number3
DOIs
StatePublished - Sep 17 2015

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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