Global Conservative Solutions to a Nonlinear Variational Wave Equation

TL;DR

The paper proves the global existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ with finite-energy data and smooth, uniformly positive $c()$. It develops a transform-and-fixed-point framework by introducing $R=u_t+c(u)u_x$, $S=u_t-c(u)u_x$, then $w=2\arctan(R)$, $z=2\arctan(S)$ and new coordinates $X,Y$ along forward/backward characteristics, leading to a semilinear hyperbolic system for $(u,w,z,p,q)$. A weighted Banach-space fixed-point argument on truncated domains yields a local solution in the transformed variables, which extends globally and maps back to a globally defined weak solution that is locally Hölder continuous with exponent $1/2$. The energy is encoded by a time-dependent Radon measure $\mu_t$ with density $\tfrac12(u_t^2+c(u)^2u_x^2)$ plus a possible singular part supported where $c'(u)=0$, ensuring $\mu_t(\mathbb{R})=E_0$ for all $t$, and energy conservation is established in a measure form via local conservation laws and a wave-interaction potential $\Lambda(t)$. This framework yields a robust global well-posedness result that captures energy concentration and provides stability with respect to initial data for nonlinear wave models arising in inertia-dominated director dynamics and related systems.

Abstract

We 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.

Figures (6)

• Figure 1: Characteristic curves
• Figure 2: The closed curve $\Sigma$
• Figure 3: Paths of integration
• Figure 4:
• Figure 5: Proving Lipschitz continuity