Archive for Rational Mechanics and Analysis

, Volume 183, Issue 1, pp 163–185

Asymptotic Variational Wave Equations

Article

DOI: 10.1007/s00205-006-0014-8

Cite this article as:
Bressan, A., Zhang, P. & Zheng, Y. Arch Rational Mech Anal (2007) 183: 163. doi:10.1007/s00205-006-0014-8

Abstract

We investigate the equation (ut+(f(u))x)x=f′ ′(u) (ux)2/2 where f(u) is a given smooth function. Typically f(u)=u2/2 or u3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation uttc(u) (c(u)ux)x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.

We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUPUSA
  2. 2.Academy of Mathematics and System SciencesCASBeijingChina
  3. 3.Department of MathematicsPenn State UniversityUPUSA