In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the vacuum Einstein equations reduce to the Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in of free solutions to the wave equation in dimensions, which is weaker than in dimensions. As in [21], we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully choose an approximate solution with a non trivial behaviour at space-like infinity to enforce convergence to Minkowski space-time at time-like infinity.
@article{SLSEDP_2014-2015____A19_0, author = {C\'ecile Huneau}, title = {Stability in exponential time of {Minkowski} space-time with a space-like translation symmetry}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:19}, pages = {1--14}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2014-2015}, doi = {10.5802/slsedp.77}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.77/} }
TY - JOUR AU - Cécile Huneau TI - Stability in exponential time of Minkowski space-time with a space-like translation symmetry JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:19 PY - 2014-2015 SP - 1 EP - 14 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.77/ DO - 10.5802/slsedp.77 LA - en ID - SLSEDP_2014-2015____A19_0 ER -
%0 Journal Article %A Cécile Huneau %T Stability in exponential time of Minkowski space-time with a space-like translation symmetry %J Séminaire Laurent Schwartz — EDP et applications %Z talk:19 %D 2014-2015 %P 1-14 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.77/ %R 10.5802/slsedp.77 %G en %F SLSEDP_2014-2015____A19_0
Cécile Huneau. Stability in exponential time of Minkowski space-time with a space-like translation symmetry. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 19, 14 p. doi : 10.5802/slsedp.77. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.77/
[1] S. Alinhac – « The null condition for quasilinear wave equations in two space dimensions I », Invent. Math. 145 (2001), no. 3, p. 597–618. | MR | Zbl
[2] S. Alinhac – « An example of blowup at infinity for a quasilinear wave equation », Astérisque (2003), no. 284, p. 1–91, Autour de l’analyse microlocale. | MR | Zbl
[3] A. Ashtekar, J. Bičák & B. G. Schmidt – « Asymptotic structure of symmetry-reduced general relativity », Phys. Rev. D (3) 55 (1997), no. 2, p. 669–686. | MR
[4] R. Bartnik & J. Isenberg – « The constraint equations », in The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, p. 1–38. | MR | Zbl
[5] G. Beck – « Zur Theorie binärer Gravitationsfelder », Zeitschrift für Physik 33 (1925), no. 14, p. 713–728.
[6] B. K. Berger, P. T. Chruściel & V. Moncrief – « On “asymptotically flat” space-times with -invariant Cauchy surfaces », Ann. Physics 237 (1995), no. 2, p. 322–354. | MR | Zbl
[7] Y. Choquet-Bruhat & R. Geroch – « Global aspects of the Cauchy problem in general relativity », Comm. Math. Phys. 14 (1969), p. 329–335. | MR | Zbl
[8] Y. Choquet-Bruhat & V. Moncrief – « Nonlinear stability of an expanding universe with the isometry group », in Partial differential equations and mathematical physics (Tokyo, 2001), Progr. Nonlinear Differential Equations Appl., vol. 52, Birkhäuser Boston, Boston, MA, 2003, p. 57–71. | MR | Zbl
[9] D. Christodoulou & S. Klainerman – The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. | MR | Zbl
[10] P. Godin – « Lifespan of solutions of semilinear wave equations in two space dimensions », Comm. Partial Differential Equations 18 (1993), no. 5-6, p. 895–916. | MR | Zbl
[11] A. Hoshiga – « The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space », Funkcial. Ekvac. 49 (2006), no. 3, p. 357–384. | MR | Zbl
[12] C. Huneau – « Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II », . | arXiv
[13] —, « Stability in exponential time of Minkowski Space-time with a translation space-like Killing field », . | arXiv
[14] F. John – « Blow-up for quasilinear wave equations in three space dimensions », Comm. Pure Appl. Math. 34 (1981), no. 1, p. 29–51. | MR | Zbl
[15] S. Klainerman – « The null condition and global existence to nonlinear wave equations », in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, p. 293–326. | MR | Zbl
[16] S. Klainerman – « Uniform decay estimates and the Lorentz invariance of the classical wave equation », Comm. Pure Appl. Math. 38 (1985), no. 3, p. 321–332. | MR | Zbl
[17] H. Kubo & K. Kubota – « Scattering for systems of semilinear wave equations with different speeds of propagation », Adv. Differential Equations 7 (2002), no. 4, p. 441–468. | MR | Zbl
[18] H. Lindblad – « Global solutions of nonlinear wave equations », Comm. Pure Appl. Math. 45 (1992), no. 9, p. 1063–1096. | MR | Zbl
[19] —, « Global solutions of quasilinear wave equations », Amer. J. Math. 130 (2008), no. 1, p. 115–157. | MR
[20] H. Lindblad & I. Rodnianski – « The weak null condition for Einstein’s equations », C. R. Math. Acad. Sci. Paris 336 (2003), no. 11, p. 901–906. | MR | Zbl
[21] —, « The global stability of Minkowski space-time in harmonic gauge », Ann. of Math. (2) 171 (2010), no. 3, p. 1401–1477. | MR
[22] R. Wald – General Relativity, The University of Chicago press, 1984. | MR | Zbl
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