Abstract:We investigate the equation (u_{t}+(f(iu))_{x})x=f''(u)(u_{x})^{2}/2 wheref(u) is a given smooth function. Typicallyf(u)=u^{2}/2 oru^{3}/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equationu_{tt}-c(u)(c(u)u_{x})_{x}=0 which models some liquid crystals with a natural sinusoidalc. 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

fhas Lipschitz continuous second-order derivative. In the case wherefis convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, whenfis not convex, we show that the dissipative solutions do not depend continuously on the initial data.

**Paper:**- Available as PDF (464 Kbytes), Postscript (912 kbytes) or gzipped PostScript (272 Kbytes; uncompress using gunzip).
**Author(s):**- Alberto Bressan, <bressan@math.psu.edu>
- Ping Zhang, <zp@mail.amss.ac.cn>
- Yuxi Zheng, <yzheng@math.psu.edu>
**Publishing information:****Comments:****Submitted by:**- <yzheng@math.psu.edu> May 6 2004.

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Conservation Laws Preprint Server <conservation@math.ntnu.no> 2004-05-07 17:31:21 UTC