On the second-order asymptotic equation of a variational wave equation

Ping Zhang, Yuxi Zheng

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We have been interested in studying a nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude, arising naturally from liquid crystals. High-frequency waves of small amplitudes, the so-called weakly nonlinear waves, near a constant state a are governed by two asymptotic equations: the first-order asymptotic equation if a is not a critical point of the sinusoidal function, or the second-order asymptotic equation if a is either a maximal or a minimal point of the sinusoidal function. Our earlier work on the first-order asymptotic equation has greatly helped the study of the nonlinear variational wave equation with monotone wave speed functions. It is apparent in our research that investigation of the second-order asymptotic equation is both crucial and equally illuminating for the study of the nonlinear variational wave equation with sinusoidal wave speed functions. We succeed in this paper in handling what may be appropriately called the 'concentration-annihilation' phenomena in the historical spirit of compensated-compactness (Tartar et al.), concentration-compactness (Lions), and concentration-cancellation or concentration-evanesces (DiPerna and Majda). More precisely, the second-order asymptotic equation has a product term uv2 for which v2 may have concentration on a set where u vanishes in a sequence of approximate solutions, while the product retains no concentration. Although absent in the first-order asymptotic equation, this concentration-annihilation phenomenon has been demonstrated through an explicit example for the nonlinear variational wave equation with sinusoidal wave speed functions in an earlier work. We use this concentration-annihilation to establish the global existence of weak solutions to the second-order asymptotic equation with initial data of bounded total variations.

Original languageEnglish (US)
Pages (from-to)483-509
Number of pages27
JournalRoyal Society of Edinburgh - Proceedings A
Volume132
Issue number2
StatePublished - Dec 1 2002

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Second-order Asymptotics
Variational Equation
Wave equation
Wave Speed
Annihilation
First-order
Concentration-compactness
Compensated Compactness
Existence of Weak Solutions
Nonlinear Waves
Total Variation
Cancellation
Liquid Crystal
Global Existence
Vanish
Critical point
Monotone
Approximate Solution

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "We have been interested in studying a nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude, arising naturally from liquid crystals. High-frequency waves of small amplitudes, the so-called weakly nonlinear waves, near a constant state a are governed by two asymptotic equations: the first-order asymptotic equation if a is not a critical point of the sinusoidal function, or the second-order asymptotic equation if a is either a maximal or a minimal point of the sinusoidal function. Our earlier work on the first-order asymptotic equation has greatly helped the study of the nonlinear variational wave equation with monotone wave speed functions. It is apparent in our research that investigation of the second-order asymptotic equation is both crucial and equally illuminating for the study of the nonlinear variational wave equation with sinusoidal wave speed functions. We succeed in this paper in handling what may be appropriately called the 'concentration-annihilation' phenomena in the historical spirit of compensated-compactness (Tartar et al.), concentration-compactness (Lions), and concentration-cancellation or concentration-evanesces (DiPerna and Majda). More precisely, the second-order asymptotic equation has a product term uv2 for which v2 may have concentration on a set where u vanishes in a sequence of approximate solutions, while the product retains no concentration. Although absent in the first-order asymptotic equation, this concentration-annihilation phenomenon has been demonstrated through an explicit example for the nonlinear variational wave equation with sinusoidal wave speed functions in an earlier work. We use this concentration-annihilation to establish the global existence of weak solutions to the second-order asymptotic equation with initial data of bounded total variations.",
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On the second-order asymptotic equation of a variational wave equation. / Zhang, Ping; Zheng, Yuxi.

In: Royal Society of Edinburgh - Proceedings A, Vol. 132, No. 2, 01.12.2002, p. 483-509.

Research output: Contribution to journalArticle

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