×

Global solutions of the Einstein-Maxwell equations. (Solutions globales des équations d’Einstein-Maxwell.) (French) Zbl 1200.35303

This paper gives a global existence result for the Einstein-Maxwell system with small, asymptotically flat data, in \(n\) space dimensions with \(n\geq 3\), using harmonic coordinates and the Lorentz gauge. The initial vector potential is \({\mathcal O}(r^{(1-n)/2-\alpha})\) for some \(\alpha>0\), where \(r\) is a radial coordinate. The methods rely heavily on those in a preprint by H. Lindblad and I. Rodnianski [arXiv:math/0411109] and in their article [Commun. Math. Phys. 256, No. 1, 43–110 (2005; Zbl 1081.83003)].

MSC:

35Q76 Einstein equations
83C22 Einstein-Maxwell equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence

Citations:

Zbl 1081.83003
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Bizoń (P.), Chmaj (T.), and Schmidt (B.G.).— Critical behavior in vacuum gravitational collapse in 4+1 dimensions, Phys. Rev. Lett. 95, 071102, gr-qc/0506074 (2005).
[2] Choquet-Bruhat (Y.).— Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, p. 144-225 (1952). · Zbl 0049.19201
[3] Christodoulou (D.).— Global solutions of nonlinear hyperbolic equations for small initial data, Commun. Pure Appl. Math. 39, p. 267-282 (1986). MR MR820070 (87c :35111) · Zbl 0612.35090
[4] Christodoulou (D.) and Klainerman (S.).— The global nonlinear stability of the Minkowski space, Princeton UP, (1993). · Zbl 0827.53055
[5] Emparan (R.) and Reall (H.S.).— Black rings, hep-th/0608012, (2006). · Zbl 1108.83001
[6] Hollands (S.) and Ishibashi (A.).— Asymptotic flatness and Bondi energy in higher dimensional gravity, Jour. Math. Phys. 46, 022503, 31, gr-qc/0304054 (2005). MR MR2121709 (2005m :83039) · Zbl 1076.83010
[7] Hollands (S.) and Wald (R.M.).— Conformal null infinity does not exist for radiating solutions in odd spacetime dimensions, Class. Quantum Grav. 21, p. 5139-5145, gr-qc/0407014, (2004). MR MR2103245 (2005k :83039) · Zbl 1081.83030
[8] Hörmander (L.).— Lectures on Nonlinear Hyperbolic Differential Equations,Springer, (1986). · Zbl 0881.35001
[9] Hörmander (L.).— On the fully nonlinear Cauchy problem with small data. II, Microlocal analysis and nonlinear waves (Minneapolis, MN, 1988-1989), IMA Vol. Math. Appl., vol. 30, Springer, New York, 1991, pp. 51-81. MR MR1120284 (94c :35127) · Zbl 0783.35036
[10] Klainerman (S.).— Global Existence for Nonlinear Wave Equations. Communications on Pure and Applied Mathematics, Vol. XXXIII, p. 43-100 (1980). · Zbl 0405.35056
[11] Klainerman (S.).— Uniform Decay Estimates and the Lorentz Invariance of the Classical wave Equation.Communications on Pure and Applied Mathematics, Vol. XXXVIII, p. 321-332 (1985). · Zbl 0635.35059
[12] Klainerman (S.).— The null condition and global existence to nonlinear wave equations. Lectures in Applied Mathematics 23, p. 293-326 (1986). · Zbl 0599.35105
[13] Lindblad (H.) and Rodnianski (I.).— Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256, p. 43-110 (2005). · Zbl 1081.83003
[14] Lindblad (H.) and Rodnianski (I.).— The global stability of Minkowski space-time in harmonic gauge. ArXiv :math.AP/0411109.
[15] Ta-Tsien Li and Yun Mei Chen.— Global classical solutions for nonlinear evolution equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 45, Longman Scientific & Technical, Harlow (1992). MR MR1172318 (93g :35002) · Zbl 0787.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.