Asymptotic Variational Wave Equations Article First Online: 04 May 2006 Received: 07 May 2004 Accepted: 03 March 2005 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
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Abstract We investigate the equation (u _{t} +(f (u ))_{x} )_{x} =f ^{′ ′} (u ) (u _{x} )^{2} /2 where f (u ) is a given smooth function. Typically f (u )=u ^{2} /2 or u ^{3} /3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _{tt} − c (u ) (c (u )u _{x} )_{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.

Communicated by the Editors

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Authors and Affiliations 1. Department of Mathematics Penn State University UP USA 2. Academy of Mathematics and System Sciences CAS Beijing China 3. Department of Mathematics Penn State University UP USA