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Unique Conservative Solutions to a Variational Wave Equation

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Abstract

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation \({u_{tt} - c (u) (c(u)u_{x}) x = 0}\). 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 \({u(0, \cdot) \in H^{1}(I\!R), u_{t} (0, \cdot) \in L^{2}(I\!R).}\)

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Alberto Bressan & Qingtian Zhang

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Geng Chen

Authors
  1. Alberto Bressan
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  2. Geng Chen
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  3. Qingtian Zhang
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Correspondence to Qingtian Zhang.

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Bressan, A., Chen, G. & Zhang, Q. Unique Conservative Solutions to a Variational Wave Equation. Arch Rational Mech Anal 217, 1069–1101 (2015). https://doi.org/10.1007/s00205-015-0849-y

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  • DOI: https://doi.org/10.1007/s00205-015-0849-y

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