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)(c(u)u_{x})_{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.