Uniqueness of conservative solutions for nonlinear wave equations via characteristics

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Abstract

For some classes of one-dimensional nonlinear wave equations, solutions are Hölder continuous and the ODEs for characteristics admit multiple solutions. Introducing an additional conservation equation and a suitable set of transformed variables, one obtains a new ODE whose right hand side is either Lipschitz continuous or has directionally bounded variation. In this way, a unique characteristic can be singled out through each initial point. This approach yields the uniqueness of conservative solutions to various equations, including the Camassa-Holm and the variational wave equation utt − c(u)(c(u)ux)x = 0, for general initial data in H1(R).

Original languageEnglish (US)
Pages (from-to)157-169
Number of pages13
JournalBulletin of the Brazilian Mathematical Society
Volume47
Issue number1
DOIs
StatePublished - Mar 1 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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