no. 4
Global analysis of quasilinear wave equations on asymptotically de Sitter spaces
[Analyse globale des équations d’ondes quasilinéaires sur les espaces asymptotiquement de Sitter]
Hintz, Peter1, 2
1 Department of Mathematics Stanford University CA 94305-2125 (USA)
2 Department of Mathematics University of California Berkeley, CA 94720-3840 (USA)
Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1285-1408.

On établit la resolubilité, sur une classe géométrique d’espaces-temps incluant les espaces asymptotiquement de Sitter, de certaines equations d’ondes et de Klein-Gordon dans des espaces de grande régularité. On obtient ces résultats en prouvant l’inversibilité globale d’opérateurs linéaires à coefficients dans des espaces de grande régularité de type L 2 , et en utilisant des processus itératifs pour les problèmes non-linéaires. L’analyse linéaire est fait en deux étapes : premièrement, on développe une théorie de la régularité au moyen d’un calcul pour les opérateurs pseudo-différentiels à coefficients peu réguliers sur variétés à bord, similaire à celui développé par Beals et Reed. Deuxièmement, on étudie le comportement asymptotique des solutions des équations linéaires en utilisant des développements en états résonnants, introduite dans ce contexte par Vasy, dans le cadre de la b-analyse de Melrose.

We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity L 2 -based function spaces and using iterative arguments for the nonlinear problems. The linear analysis is accomplished in two parts: Firstly, a regularity theory is developed by means of a calculus for pseudodifferential operators with non-smooth coefficients, similar to the one developed by Beals and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of solutions to linear equations is studied using resonance expansions, introduced in this context by Vasy using the framework of Melrose’s b-analysis.

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DOI : 10.5802/aif.3039
Classification : 35L70, 35B40, 35S05, 58J47
Keywords: quasilinear wave equations, asymptotically de Sitter spaces, microlocal analysis, resonance expansions
Mot clés : équations d’ondes quasilinéaires, espaces asymptotiquement de Sitter, analyse microlocale, développements en états résonnants

Hintz, Peter 1, 2

1 Department of Mathematics Stanford University CA 94305-2125 (USA)
2 Department of Mathematics University of California Berkeley, CA 94720-3840 (USA) 
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Hintz, Peter. Global analysis of quasilinear wave equations on asymptotically de Sitter spaces. Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1285-1408. doi : 10.5802/aif.3039. https://aif.centre-mersenne.org/articles/10.5802/aif.3039/

[1] Alinhac, S. Geometric analysis of hyperbolic differential equations: an introduction, London Mathematical Society Lecture Note Series, 374, Cambridge University Press, Cambridge, 2010, x+118 pages | DOI

[2] Anderson, Michael T. Existence and stability of even-dimensional asymptotically de Sitter spaces, Ann. Henri Poincaré, Volume 6 (2005) no. 5, pp. 801-820 | DOI

[3] Baskin, Dean A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces, J. Funct. Anal., Volume 259 (2010) no. 7, pp. 1673-1719 | DOI

[4] Baskin, Dean A Strichartz estimate for de Sitter space, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis (Proc. Centre Math. Appl. Austral. Nat. Univ.), Volume 44, Austral. Nat. Univ., Canberra, 2010, pp. 97-104

[5] Baskin, Dean Strichartz estimates on asymptotically de Sitter spaces, Ann. Henri Poincaré, Volume 14 (2013) no. 2, pp. 221-252 | DOI

[6] Baskin, Dean; Vasy, András; Wunsch, Jared Asymptotics of radiation fields in asymptotically Minkowski space (2012) (http://arxiv.org/abs/1212.5141v1)

[7] Beals, Michael; Reed, Michael Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc., Volume 285 (1984) no. 1, pp. 159-184 | DOI

[8] Bony, Jean-Michel Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), Volume 14 (1981) no. 2, pp. 209-246

[9] Christodoulou, Demetrios; Klainerman, Sergiu The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, 41, Princeton University Press, Princeton, NJ, 1993, x+514 pages

[10] Dafermos, Mihalis; Holzegel, Gustav; Rodnianski, Igor A scattering theory construction of dynamical vacuum black holes (2013) (http://arxiv.org/abs/1306.5364)

[11] Dafermos, Mihalis; Rodnianski, Igor Lectures on black holes and linear waves, Evolution equations (Clay Math. Proc.), Volume 17, Amer. Math. Soc., Providence, RI, 2013, pp. 97-205

[12] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999, xii+227 pages | DOI

[13] Duistermaat, J. J.; Hörmander, L. Fourier integral operators. II, Acta Math., Volume 128 (1972) no. 3-4, pp. 183-269

[14] The conformal structure of space-time (Frauendiener, J.; Friedrich, H., eds.), Lecture Notes in Physics, 604, Springer-Verlag, Berlin, 2002, xiv+373 pages (Geometry, analysis, numerics) | DOI

[15] Friedrich, Helmut Existence and structure of past asymptotically simple solutions of Einstein’s field equations with positive cosmological constant, J. Geom. Phys., Volume 3 (1986) no. 1, pp. 101-117 | DOI

[16] Friedrich, Helmut On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Comm. Math. Phys., Volume 107 (1986) no. 4, pp. 587-609 http://projecteuclid.org/euclid.cmp/1104116232

[17] Hassell, Andrew; Melrose, Richard; Vasy, András Microlocal propagation near radial points and scattering for symbolic potentials of order zero, Anal. PDE, Volume 1 (2008) no. 2, pp. 127-196 | DOI

[18] Hintz, Peter; Vasy, András Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces (http://arxiv.org/abs/1404.1348, preprint 2014. To appear in Int. Math. Res. Notices)

[19] Hintz, Peter; Vasy, András Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes (http://arxiv.org/abs/1306.4705, preprint 2013. To appear in Anal. PDE)

[20] Hintz, Peter; Vasy, András Non-trapping estimates near normally hyperbolic trapping, Math. Res. Lett., Volume 21 (2014) no. 6, pp. 1277-1304 | DOI

[21] de Hoop, Maarten; Uhlmann, Gunther; Vasy, András Diffraction from conormal singularities, Ann. Sci. Éc. Norm. Supér. (4), Volume 48 (2015) no. 2, pp. 351-408

[22] Hörmander, Lars Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26, Springer-Verlag, Berlin, 1997, viii+289 pages

[23] Hörmander, Lars The analysis of linear partial differential operators. I-IV, Classics in Mathematics, Springer-Verlag, Berlin, 2007 | DOI

[24] Klainerman, S. The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) (Lectures in Appl. Math.), Volume 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293-326

[25] Kumano-go, Hitoshi; Nagase, Michihiro Pseudo-differential operators with non-regular symbols and applications, Funkcial. Ekvac., Volume 21 (1978) no. 2, pp. 151-192 http://www.math.kobe-u.ac.jp/~fe/xml/mr0518297.xml

[26] Lindblad, Hans; Rodnianski, Igor The global stability of Minkowski space-time in harmonic gauge, Ann. of Math. (2), Volume 171 (2010) no. 3, pp. 1401-1477 | DOI

[27] Marschall, Jürgen Pseudodifferential operators with coefficients in Sobolev spaces, Trans. Amer. Math. Soc., Volume 307 (1988) no. 1, pp. 335-361 | DOI

[28] Mazzeo, Rafe R.; Melrose, Richard B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310 | DOI

[29] Melrose, R. B.; Sjöstrand, J. Singularities of boundary value problems. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 5, pp. 593-617

[30] Melrose, R. B.; Sjöstrand, J. Singularities of boundary value problems. II, Comm. Pure Appl. Math., Volume 35 (1982) no. 2, pp. 129-168 | DOI

[31] Melrose, Richard B. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, 4, A K Peters, Ltd., Wellesley, MA, 1993, xiv+377 pages

[32] Melrose, Richard B. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) (Lecture Notes in Pure and Appl. Math.), Volume 161, Dekker, New York, 1994, pp. 85-130

[33] Meyer, Yves Régularité des solutions des équations aux dérivées partielles non linéaires (d’après J.-M. Bony), Bourbaki Seminar, Vol. 1979/80 (Lecture Notes in Math.), Volume 842, Springer, Berlin-New York, 1981, pp. 293-302

[34] Rodnianski, Igor; Speck, Jared The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant, J. Eur. Math. Soc. (JEMS), Volume 15 (2013) no. 6, pp. 2369-2462 | DOI

[35] Saint-Raymond, Xavier A simple Nash-Moser implicit function theorem, Enseign. Math. (2), Volume 35 (1989) no. 3-4, pp. 217-226

[36] Taylor, Michael E. Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, 100, Birkhäuser Boston, Inc., Boston, MA, 1991, 213 pages | DOI

[37] Taylor, Michael E. Partial differential equations. III, Applied Mathematical Sciences, Springer-Verlag, New York, 1996

[38] Vasy, András The wave equation on asymptotically de Sitter-like spaces, Adv. Math., Volume 223 (2010) no. 1, pp. 49-97 | DOI

[39] Vasy, András The wave equation on asymptotically anti de Sitter spaces, Anal. PDE, Volume 5 (2012) no. 1, pp. 81-144 | DOI

[40] Vasy, András Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., Volume 194 (2013) no. 2, pp. 381-513 | DOI

[41] Witt, Ingo A calculus for classical pseudo-differential operators with non-smooth symbols, Math. Nachr., Volume 194 (1998), pp. 239-284 | DOI

[42] Yagdjian, Karen The semilinear Klein-Gordon equation in de Sitter spacetime, Discrete Contin. Dyn. Syst. Ser. S, Volume 2 (2009) no. 3, pp. 679-696 | DOI

[43] Yagdjian, Karen Global existence of the scalar field in de Sitter spacetime, J. Math. Anal. Appl., Volume 396 (2012) no. 1, pp. 323-344 | DOI

[44] Yagdjian, Karen; Galstian, Anahit Fundamental solutions for the Klein-Gordon equation in de Sitter spacetime, Comm. Math. Phys., Volume 285 (2009) no. 1, pp. 293-344 | DOI

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