Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation

TL;DR

This paper proves that conservative solutions to a nonlinear wave equation are generally piecewise smooth with finitely many characteristic curves where the gradient may blow up, using a variable transformation and transversality theory.

Contribution

It establishes the generic regularity and singularity structure of solutions to a nonlinear wave equation for an open dense set of initial data.

Findings

Solutions are piecewise smooth in the t-x plane.

Gradient blow-up occurs along finitely many characteristic curves.

The analysis employs a variable transformation and Thom's transversality theorem.

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 $u_x$ 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.