# 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

## Abstract

The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt}−c(u)\left(c(u)u_{x}\right)_{x}\text{} = \text{}0$. For an open dense set of $\mathscr{C}^{3}$ initial data, we prove that the solution is piecewise smooth in the *t*–*x* plane, while the gradient $u_{x}$ 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