JournalsaihpcVol. 34, No. 2pp. 335–354

Generic regularity of conservative solutions to a nonlinear wave equation

  • Alberto Bressan

    Department of Mathematics, Penn State University, University Park, PA 16802, USA
  • Geng Chen

    School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
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Abstract

The paper is concerned with conservative solutions to the nonlinear wave equation uttc(u)(c(u)ux)x=0u_{tt}−c(u)\left(c(u)u_{x}\right)_{x}\text{} = \text{}0. For an open dense set of C3\mathscr{C}^{3} initial data, we prove that the solution is piecewise smooth in the tx plane, while the gradient uxu_{x} can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.

Cite this article

Alberto Bressan, Geng Chen, Generic regularity of conservative solutions to a nonlinear wave equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 2, pp. 335–354

DOI 10.1016/J.ANIHPC.2015.12.004