Preprint 2015-004
Generic regularity of conservative solutions to a nonlinear wave equation
Alberto Bressan and Geng Chen
Abstract: The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt}−c(u)\big(c(u)u_x\big)_x=0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient ux 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.