Hydrodynamic limits of the nonlinear Klein-Gordon equation

Chi Kun Lin; Kung Chien Wu

doi:10.1016/j.matpur.2012.02.002

Hydrodynamic limits of the nonlinear Klein-Gordon equation

Chi Kun Lin^*, Kung Chien Wu

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

14 Citations (Scopus)

Abstract

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].

Original language	English
Pages (from-to)	328-345
Number of pages	18
Journal	Journal des Mathematiques Pures et Appliquees
Volume	98
Issue number	3
DOIs	https://doi.org/10.1016/j.matpur.2012.02.002
Publication status	Published - Sept 2012
Externally published	Yes

Keywords

Euler equations
Hydrodynamic limits
Klein-Gordon equation

Access to Document

10.1016/j.matpur.2012.02.002

Cite this

@article{3bf049426ca64318bc18d28ddf53b0fa,

title = "Hydrodynamic limits of the nonlinear Klein-Gordon equation",

abstract = "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].",

keywords = "Euler equations, Hydrodynamic limits, Klein-Gordon equation",

author = "Lin, {Chi Kun} and Wu, {Kung Chien}",

note = "Funding Information: C.K. Lin is supported by National Science Council of Taiwan under the grant NSC98-2115-M-009-004-MY3. K.C. Wu is supported by the Tsz-Tza Foundation in Institute of Mathematics, Academia Sinica, Taipei, Taiwan. K.C. Wu would like to thank Dr. Cl{\'e}ment Mouhot for his kind invitation to visit Cambridge during 2011–2012 academic year.",

year = "2012",

month = sep,

doi = "10.1016/j.matpur.2012.02.002",

language = "English",

volume = "98",

pages = "328--345",

journal = "Journal des Mathematiques Pures et Appliquees",

issn = "0021-7824",

number = "3",

}

TY - JOUR

T1 - Hydrodynamic limits of the nonlinear Klein-Gordon equation

AU - Lin, Chi Kun

AU - Wu, Kung Chien

N1 - Funding Information: C.K. Lin is supported by National Science Council of Taiwan under the grant NSC98-2115-M-009-004-MY3. K.C. Wu is supported by the Tsz-Tza Foundation in Institute of Mathematics, Academia Sinica, Taipei, Taiwan. K.C. Wu would like to thank Dr. Clément Mouhot for his kind invitation to visit Cambridge during 2011–2012 academic year.

PY - 2012/9

Y1 - 2012/9

N2 - 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].

AB - 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].

KW - Euler equations

KW - Hydrodynamic limits

KW - Klein-Gordon equation

UR - http://www.scopus.com/inward/record.url?scp=84865349883&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2012.02.002

DO - 10.1016/j.matpur.2012.02.002

M3 - Article

AN - SCOPUS:84865349883

SN - 0021-7824

VL - 98

SP - 328

EP - 345

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 3

ER -

Hydrodynamic limits of the nonlinear Klein-Gordon equation

Abstract

Keywords

Access to Document

Other files and links

Cite this