Weak solutions to a nonlinear variational wave equation with general data

Abstract

We establish the existence of global weak solutions to the initial value problem for a nonlinear variational wave equation u tt − c(u)(c(u)u x) x = 0 with general initial data (u(0), u t (0)) = (u 0 , u 1) ∈ W 1,2 × L 2 under the assumptions that the wave speed c(u) satisfies c (•) 0 and c (u 0 (•)) > 0. Moreover, we obtain high regularity for the spatial derivative ∂ x u of the wave amplitude u away from where c (u) = 0. This equation arises from studies in nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use Young measure method in the setting of L p spaces and method of renormalization to overcome the difficulty that oscillations in a sequence of approximations get amplified by the quadratic growth term of the equation. We use a high space-time estimate for ∂ x u to handle possible concentrations. This result improves our earlier existence result for initial data in the space W 1,∞ × L ∞ to the natural space W 1,2 × L 2 .

Singularity and existence for a wave system of nematic liquid crystals

“…The global weak existence and singularity formation for the Cauchy problem of the extreme case of (1.1) with µ = 0 in one space dimension (1-d) has also been extensively studied [3,8,16,17,18,19,4]. This hyperbolic system describes the model with viscous effects neglected, for which an example with smooth initial data and singularity formation (gradient blowup) in finite time has been provided in [8].…”

A convergent finite-difference method for a nonlinear variational wave equation

“…Either one can use a viscous regularization of the system (1.5) by adding the terms R xx and S xx to the first and the second equation, respectively, and subsequently analyze in detail the behavior of the solution as → 0, see [12]. Alternatively [11,13,14], one can replace the quadratic growth on the right-hand side of equation (1.5) by a linear growth for large values of R 2 and S 2 . More specifically, introduce the function…”

“…In a series of papers, Zhang and Zheng [11,12,13,14,15] have analyzed (1.1) carefully, using the method of Young measures. From their many results we quote the recent one [14, Thm.…”

Generic regularity and Lipschitz metric for the Hunter–Saxton type equations

“…When ρ ≡ 0, the above system becomes the one-component Hunter-Saxton equation 2) which is an asymptotic equation of the variational wave equation used to model nematic liquid crystal [1,11]. The Hunter-Saxton equation (1.2) was first derived in [11] as an asymptotic equation of the variational wave equation, which was considered in [1,2,3,4,8,21,22], for the nematic liquid crystals. The global existences of weak conservative and dissipative solutions of (1.2) were first proved by Hunter and Zheng in [12,13] on the initial value problem, by studying the self-similar solutions, then were treated by several other methods including the Young measure method by Zhang and Zheng in [20], and the characteristic method by Bressan and Constantin [5] and Bressan, Zhang and Zheng [7] on the initial value or initial boundary value problem.…”