No Access

THE NON-RELATIVISTIC LIMIT OF THE EULER–NORDSTRÖM SYSTEM WITH COSMOLOGICAL CONSTANT

JARED SPECK

Department of Mathematics, Rutgers University, Hill Center, 110, Frelinghuysen Road, Piscataway, NJ 08854, USA

https://doi.org/10.1142/S0129055X09003748Cited by:9 (Source: Crossref)

Abstract

In this paper, we study the singular limit c → ∞ of the family of Euler–Nordström systems indexed by the parameters κ² and c, where κ² > 0 is the cosmological constant and c is the speed of light. Using Christodoulou's techniques to generate energy currents, we develop Sobolev estimates that show initial data belonging to an appropriate Sobolev space launch unique solutions to the system that converge to corresponding unique solutions of the Euler–Poisson system with the cosmological constant κ² as c tends to infinity.

This article was finalized while the author was a postdoctoral researcher in the Princeton University Math. Department.

Keywords:

AMSC: 35L80, 35M99, 83C55, 83D05

References

N. Andersson and G. L. Comer, Living Rev. Relativity 10, 83 (2007), http://relativity.livingreviews.org/Articles/lrr-2007-1/. Google Scholar
S. Bauer, Comm. Partial Differential Equations 30, 957 (2005), DOI: 10.1081/PDE-200064434. Crossref, Web of Science, Google Scholar
S. Baueret al., Comm. Math. Phys. 266, 267 (2006), DOI: 10.1007/s00220-006-0015-3. Crossref, Web of Science, ADS, Google Scholar
S. Calogero, Classical Quantum Gravity 20, 1729 (2003), DOI: 10.1088/0264-9381/20/9/310. Crossref, Web of Science, ADS, Google Scholar
S. Calogero, Comm. Math. Phys. 266, 343 (2006), DOI: 10.1007/s00220-006-0029-x. Crossref, Web of Science, ADS, Google Scholar
S. Calogero and H. Lee, Commun. Math. Sci. 2, 19 (2004). Crossref, Google Scholar
D. Christodoulou, Arch. Ration. Mech. Anal. 130, 343 (1995), DOI: 10.1007/BF00375144. Crossref, Web of Science, Google Scholar
D. Christodoulou , The Action Principle and Partial Differential Equations ( Princeton University Press , Princeton, NJ , 2000 ) . Crossref, Google Scholar
D. Christodoulou , The Formation of Shocks in 3-Dimensional Fluids , European Mathematical Society ( Zürich, Switzerland , 2007 ) . Crossref, Google Scholar
Y. Guo and S. Tahvildar-Zadeh, Contemp. Math. 238, 151 (1999). Crossref, Google Scholar
L. Hörmander , Lectures on Nonlinear Hyperbolic Differential Equations ( Springer-Verlag , Berlin, Heidelberg, New York , 1997 ) . Google Scholar
M. Kiessling, Adv. Appl. Math. 31, 132 (2003), DOI: 10.1016/S0196-8858(02)00556-0. Crossref, Web of Science, Google Scholar
S. Klainerman and A. Majda, Comm. Pure Appl. Math. 34, 481 (1981), DOI: 10.1002/cpa.3160340405. Crossref, Web of Science, ADS, Google Scholar
T. Makino, Stud. Math. Appl. 18, 459 (1986), DOI: 10.1016/S0168-2024(08)70142-5. Crossref, Web of Science, Google Scholar
L. Nirenberg, Ann. Scuola Norm. Sup. Pisa (3) 13, 115 (1959). Google Scholar
G. Nordström, Ann. Phys. 42, 533 (1913). Google Scholar
T. A. Oliynyk, Comm. Math. Phys. 276, 131 (2007), DOI: 10.1007/s00220-007-0334-z. Crossref, Web of Science, ADS, Google Scholar
T. A. Oliynyk, Comm. Math. Phys. (2008), arXiv:0810.3752v1. Google Scholar
A. D. Rendall, J. Math. Phys. 33, 1047 (1992), DOI: 10.1063/1.529766. Crossref, Web of Science, Google Scholar
A. D. Rendall, Comm. Math. Phys. 163, 89 (1994), DOI: 10.1007/BF02101736. Crossref, Web of Science, ADS, Google Scholar
C. D. Sogge , Lectures on Nonlinear Wave Equations , Monographs in Analysis, II ( International Press , Boston, MA , 1995 ) . Google Scholar
J. Speck, J. Hyperbolic Differ. Equ. 6(2), 313 (2009), DOI: 10.1142/S0219891609001885. Link, Web of Science, Google Scholar
N. Straumann , General Relativity and Relativistic Astrophysics , Texts and Monographs in Physics ( Springer-Verlag , Berlin , 1984 ) . Crossref, Google Scholar