Generic regularity of conservative solutions to a nonlinear wave equation

Abstract

The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt}-c(u){\left(c(u)u_x\right)}_x=0$. For an open dense set of ${\mathcal{C}}^3$ 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.