Communications in Mathematical Sciences

Volume 14 (2016)

Number 1

Representation of dissipative solutions to a nonlinear variational wave equation

Pages: 31 – 53

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n1.a2

Authors

Alberto Bressan (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Tao Huang (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

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.

Keywords

nonlinear variational wave equation, global existence, dissipative solutions

2010 Mathematics Subject Classification

35Q35

Full Text (PDF format)

Published 16 September 2015