Rarefactive solutions to a nonlinear variational wave equation of liquid crystals

Ping Zhang, Yuxi Zheng

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

Original languageEnglish (US)
Pages (from-to)381-419
Number of pages39
JournalCommunications in Partial Differential Equations
Volume26
Issue number3-4
DOIs
StatePublished - Jan 1 2001

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Variational Equation
Wave equations
Liquid Crystal
Liquid crystals
Wave equation
Wave Speed
Smooth Solution
Invariant Region
Young Measures
Measure Theory
Nonlinear Wave Equation
Global Existence
Rough
Weak Solution
Smoothing
Phase Space
Cauchy Problem
Regularization
Monotone
Approximation

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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Rarefactive solutions to a nonlinear variational wave equation of liquid crystals. / Zhang, Ping; Zheng, Yuxi.

In: Communications in Partial Differential Equations, Vol. 26, No. 3-4, 01.01.2001, p. 381-419.

Research output: Contribution to journalArticle

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