En adaptant une méthode de Lindblad et Rodnianski, on prouve l’existence de solutions globales pour les équations d’Einstein-Maxwell en dimension d’espace . Les données initiales considérées sont lisses, asymptotiquement euclidiennes et suffisamment petites. On utilise la jauge harmonique et la jauge de Lorenz.
Adapting a method of Lindblad and Rodnianski, we prove existence of global solutions for the Einstein-Maxwell equations in space dimension . We consider small enough smooth and asymptotically flat initial data. We use harmonic gauge and Lorenz gauge.
@article{AFST_2009_6_18_3_495_0, author = {Loizelet, Julien}, title = {Solutions globales des \'equations {d{\textquoteright}Einstein-Maxwell}}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {495--540}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {6e s{\'e}rie, 18}, number = {3}, year = {2009}, doi = {10.5802/afst.1212}, zbl = {1200.35303}, mrnumber = {2582443}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/afst.1212/} }
TY - JOUR AU - Loizelet, Julien TI - Solutions globales des équations d’Einstein-Maxwell JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2009 SP - 495 EP - 540 VL - 18 IS - 3 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1212/ DO - 10.5802/afst.1212 LA - fr ID - AFST_2009_6_18_3_495_0 ER -
%0 Journal Article %A Loizelet, Julien %T Solutions globales des équations d’Einstein-Maxwell %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2009 %P 495-540 %V 18 %N 3 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1212/ %R 10.5802/afst.1212 %G fr %F AFST_2009_6_18_3_495_0
Loizelet, Julien. Solutions globales des équations d’Einstein-Maxwell. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 3, pp. 495-540. doi : 10.5802/afst.1212. http://www.numdam.org/articles/10.5802/afst.1212/
[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). | MR
[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
[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) | MR | Zbl
[4] Christodoulou (D.) and Klainerman (S.).— The global nonlinear stability of the Minkowski space, Princeton UP, (1993). | MR | Zbl
[5] Emparan (R.) and Reall (H.S.).— Black rings, hep-th/0608012, (2006). | MR | Zbl
[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) | MR | Zbl
[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) | MR | Zbl
[8] Hörmander (L.).— Lectures on Nonlinear Hyperbolic Differential Equations,Springer, (1986). | MR | Zbl
[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) | MR | Zbl
[10] Klainerman (S.).— Global Existence for Nonlinear Wave Equations. Communications on Pure and Applied Mathematics, Vol. XXXIII, p. 43-100 (1980). | MR | Zbl
[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). | MR | Zbl
[12] Klainerman (S.).— The null condition and global existence to nonlinear wave equations. Lectures in Applied Mathematics 23, p. 293-326 (1986). | MR | Zbl
[13] Lindblad (H.) and Rodnianski (I.).— Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256, p. 43-110 (2005). | MR | Zbl
[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) | MR | Zbl
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