Hydrodynamic limits of the nonlinear Klein–Gordon equation

We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein–Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we...

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Bibliographic Details
Published in:Journal de mathématiques pures et appliquées 2012-09, Vol.98 (3), p.328-345
Main Authors: Lin, Chi-Kun, Wu, Kung-Chien
Format: Article
Language:English
Subjects:
Euler equations
Exact sciences and technology
General mathematics
General, history and biography
Hydrodynamic limits
Klein–Gordon equation
Mathematical analysis
Mathematics
Partial differential equations
Sciences and techniques of general use
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Summary:We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein–Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. The proof involves the modulated energy introduced by Brenier (2000) [1]. On obtient une dérivation mathématique des équations dʼEuler compressibles et incompressibles à partir de lʼéquation de Klein–Gordon non linéaire modulée. Avant la formation de singularités pour le système limite, on démontre dans la limite non relativiste et semi-classique la convergence vers les équations dʼEuler compressibles. Au moyen dʼun changement dʼéchelle supplémentaire en temps, on démontre la limite semi-classique la vitesse de la lumière restant fixée, la limite vers les équations dʼEuler incompressibles. La démonstration utilise lʼénergie modulée introduite par Brenier (2000) [1].
ISSN:0021-7824
DOI:10.1016/j.matpur.2012.02.002