01700nas a2200121 4500008004300000245004200043210004200085520136200127100002101489700001601510700001601526856003601542 2007 en_Ud 00aAsymptotic variational wave equations0 aAsymptotic variational wave equations3 aWe 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.\\nWe 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 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.1 aBressan, Alberto1 aPing, Zhang1 aYuxi, Zheng uhttp://hdl.handle.net/1963/218200638nas a2200109 4500008004300000245006800043210006800111520027600179100002100455700001600476856003600492 2006 en_Ud 00aConservative Solutions to a Nonlinear Variational Wave Equation0 aConservative Solutions to a Nonlinear Variational Wave Equation3 aWe establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values.1 aBressan, Alberto1 aYuxi, Zheng uhttp://hdl.handle.net/1963/2184