THE NON-RELATIVISTIC LIMIT OF THE EULER–NORDSTRÖM SYSTEM WITH COSMOLOGICAL CONSTANT
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.
References
- Living Rev. Relativity 10, 83 (2007), http://relativity.livingreviews.org/Articles/lrr-2007-1/. Google Scholar
- Comm. Partial Differential Equations 30, 957 (2005), DOI: 10.1081/PDE-200064434. Crossref, Web of Science, Google Scholar
- Comm. Math. Phys. 266, 267 (2006), DOI: 10.1007/s00220-006-0015-3. Crossref, Web of Science, ADS, Google Scholar
- Classical Quantum Gravity 20, 1729 (2003), DOI: 10.1088/0264-9381/20/9/310. Crossref, Web of Science, ADS, Google Scholar
- Comm. Math. Phys. 266, 343 (2006), DOI: 10.1007/s00220-006-0029-x. Crossref, Web of Science, ADS, Google Scholar
- Commun. Math. Sci. 2, 19 (2004). Crossref, Google Scholar
- 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 - 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 - Adv. Appl. Math. 31, 132 (2003), DOI: 10.1016/S0196-8858(02)00556-0. Crossref, Web of Science, Google Scholar
- Comm. Pure Appl. Math. 34, 481 (1981), DOI: 10.1002/cpa.3160340405. Crossref, Web of Science, ADS, Google Scholar
- Stud. Math. Appl. 18, 459 (1986), DOI: 10.1016/S0168-2024(08)70142-5. Crossref, Web of Science, Google Scholar
- Ann. Scuola Norm. Sup. Pisa (3) 13, 115 (1959). Google Scholar
- Ann. Phys. 42, 533 (1913). Google Scholar
- Comm. Math. Phys. 276, 131 (2007), DOI: 10.1007/s00220-007-0334-z. Crossref, Web of Science, ADS, Google Scholar
- Comm. Math. Phys. (2008), arXiv:0810.3752v1. Google Scholar
- J. Math. Phys. 33, 1047 (1992), DOI: 10.1063/1.529766. Crossref, Web of Science, Google Scholar
- 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. 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