Representation of Dissipative Solutions to a Nonlinear Variational Wave Equation

TL;DR

This paper develops a novel approach to construct dissipative solutions for a nonlinear variational wave equation, utilizing variable transformations and approximation techniques to handle discontinuities and energy dissipation.

Contribution

It introduces a new method for constructing dissipative solutions to a nonlinear wave equation via variable transformation and approximation, addressing discontinuities in source terms.

Findings

Solutions exhibit monotone decreasing energy over time

Method effectively handles discontinuous source terms

Provides a framework for dissipative solutions in nonlinear wave equations

Abstract

The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov's compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy is a monotone decreasing function of time.