Conservative Solutions to a Nonlinear Variational Wave Equation

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(·) is any smooth function with uniformly positive bounded values. Mathematics Subject Classification (2000): 35Q35

Singularity and existence for a wave system of nematic liquid crystals

“…This method was first applied to equation (1.6) with µ = 0 for an existence proof by [3]. When n is arbitrary in S 2 and µ = 0, under the a priori assumption that n(x, t) is uniformly away from (1, 0, 0), the existences of 1-d solutions for (1.3) and (2.6) with β < α have been provided by [18] and [19], respectively.…”

“…Relying on the local bounds (5.10)(5.11), the local solution to (4.20)(5.6)(1.5)(1.4) can be extended to Ω + One may consult paper [3] for details. This completes the sketch of the proof.…”

“…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].…”

“…The lack of regularity makes us only be able to consider the existence of weak solutions instead of classical solutions. The existing global existence results in [3,18,19,4] proved by the method of energy-dependent coordinates will be introduced in Section 2.…”

“…In order to prove these theorems, by the method of energy-dependent coordinates, first used in papers [3] and related Camassa-Holm equation, we dilate the singularity then find the energy dissipated solution. In fact, the energy is dissipated when µ < 0.…”