On Asymptotic Variational Wave Equations
Alberto Bressan, Ping Zhang, Yuxi Zheng
TL;DR
This paper analyzes the nonlinear variational wave equation $(u_t + (f(u))_x)_x = \tfrac12 f''(u) (u_x)^2$, which models unidirectional, weakly nonlinear waves and relates to the variational wave equation $u_{tt} - c(u)(c(u)u_x)_x =0$. It distinguishes two solution notions—conservative solutions that preserve energy and dissipative weak solutions that lose energy at singularities—and develops a rigorous framework based on an energy measure $\mu$ carried by the spatial derivative. The authors prove global well-posedness for conservative solutions (existence, forward/backward flow, and uniqueness under a non-degeneracy condition); for convex flux $f$ they also obtain a well-posed semigroup of dissipative solutions, whereas for non-convex $f$ dissipative solutions can fail to depend continuously on initial data. The paper constructs a semigroup on the extended state space $(u, \mu)$ with an energy transport equation and proves regularity results (Hölder continuity, Lipschitz temporal dependence, continuous dependence on initial data); it discusses vanishing viscosity/dispersion as selection mechanisms and shows non-coercivity of the action functional, with ill-posedness in the non-convex case illustrated by $f(u)=u^3$. This work clarifies when conservative versus dissipative prolongations are well-posed and how energy concentration shapes solution behavior in variational wave models.
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 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.