The stability of the Minkowski space for the Einstein–Vlasov system

Smulevici, Jacques

Séminaire Laurent Schwartz — EDP et applications (2017-2018), Exposé no. 15, 15 p.

This text serves as an introduction to the article [23] written in collaboration with David Fajman and Jérémie Joudioux, and presented at the Laurent Schwarz Seminar in March 2018. In [23], we establish the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [21, 22]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst non-linear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields.

[1] L. Andersson, P. Blue, and J. Joudioux, Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime, . | arXiv | DOI | MR

[2] H. Andréasson, The Einstein-Vlasov system/kinetic theory, Living Rev. Relativ. 5 (2002), 2002–7, 33 pp. (electronic). | DOI | MR | Zbl

[3] H. Andréasson, D. Fajman, and M. Thaller, Static solutions to the Einstein-Vlasov system with a nonvanishing cosmological constant, SIAM J. Math. Anal. 47 (2015), no. 4, 2657–2688. | DOI | MR | Zbl

[4] —, Models for self-gravitating photon shells and geons, Ann. Henri Poincaré 18 (2017), no. 2, 681–705. | DOI | MR | Zbl

[5] H. Andréasson, M. Kunze, and G. Rein, Existence of axially symmetric static solutions of the Einstein-Vlasov system, Comm. Math. Phys. 308 (2011), no. 1, 23–47. | DOI | MR | Zbl

[6] —, Rotating, stationary, axially symmetric spacetimes with collisionless matter, Comm. Math. Phys. 329 (2014), no. 2, 787–808. | DOI | MR | Zbl

[7] C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 2, 101–118. | DOI | MR | Zbl

[8] L. Bieri and N. Zipser, Extensions of the stability theorem of the Minkowski space in general relativity, AMS/IP Studies in Advanced Mathematics, vol. 45, American Mathematical Society, Providence, RI, International Press, Cambridge, MA, 2009. | DOI | Zbl

[9] L. Bigorgne, Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions, , 2017. | arXiv

[10] S.-H. Choi, S.-Y. Ha, and H. Lee, Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data, J. Differential Equations 250 (2011), no. 1, 515–550. | DOI | MR | Zbl

[11] Y. Choquet-Bruhat, Problème de Cauchy pour le système intégro-différentiel d’Einstein-Liouville, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 3, 181–201. | DOI | Numdam | Zbl

[12] —, The null condition and asymptotic expansions for the Einstein equations, Ann. Phys. 9 (2000), no. 3-5, 258–266, Journées Relativistes 99 (Weimar). | DOI | MR | Zbl

[13] D. Christodoulou and S. Klainerman, The non-linear stability of the Minkowski space, Princeton Mathematical Series, Princeton NJ, 1993. | DOI | Zbl

[14] P. T. Chruściel and E. Delay, Existence of non-trivial, vacuum, asymptotically simple spacetimes, Classical Quantum Gravity 19 (2002), no. 9, L71–L79. | DOI | MR | Zbl

[15] J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189. | DOI | MR | Zbl

[16] J. Corvino and R. M. Schoen, On the asymptotics for the vacuum Einstein constraint equations, J. Differential Geom. 73 (2006), no. 2, 185–217. | DOI | MR | Zbl

[17] M. Dafermos, A note on the collapse of small data self-gravitating massless collisionlessmatter, J. Hyperbolic Differ. Equ. 3 (2006), no. 4, 589–598. | DOI | MR | Zbl

[18] M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 421–432. | DOI | Zbl

[19] J. Ehlers, Survey of general relativity theory, pp. 1–125, Springer Netherlands, Dordrecht, 1973. | DOI

[20] D. Fajman, The nonvacuum Einstein flow on surfaces of negative curvature and nonlinear stability, Comm. Math. Phys. 353 (2017), 905–961. | DOI | MR | Zbl

[21] D. Fajman, J. Joudioux, and J. Smulevici, A vector field method for relativistic transport equations with applications, , to appear in Analysis and P.D.E., 2015. | arXiv | DOI | MR | Zbl

[22] —, Sharp asymptotics for small data solutions of the Vlasov-Nordström system in three dimensions, , 2017. | arXiv

[23] —, The stability of the Minkowski space for the Einstein-Vlasov system, , 2017. | arXiv

[24] H. Friedrich, Cauchy problems for the conformal vacuum field equations in general relativity, Comm. Math. Phys. 91 (1983), no. 4, 445–472. | DOI | MR | Zbl

[25] S. Friedrich, Global Small Solutions of the Vlasov-Nordström System, , 2004. | arXiv

[26] P. Germain, Global existence for coupled Klein-Gordon equations with different speeds, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 6, 2463–2506 (2012). | DOI | Numdam | MR | Zbl

[27] R. T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. | DOI | Zbl

[28] R. T. Glassey and W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys. 113 (1987), no. 2, 191–208. | DOI | MR | Zbl

[29] Y. Guo, A. D. Ionescu, and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D, Ann. of Math. (2) 183 (2016), no. 2, 377–498. | DOI | MR | Zbl

[30] Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Comm. Math. Phys. 271 (2007), no. 2, 489–509. | DOI | MR | Zbl

[31] H. Hwang, A. D. Rendall, and J. J. L. Velázquez, Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data, Archive for Rational Mechanics and Analysis 200 (2011), no. 1, 313–360 (English). | DOI | MR | Zbl

[32] A. D. Ionescu and B. Pausader, Global solutions of quasilinear systems of Klein-Gordon equations in 3D, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 11, 2355–2431. | DOI | MR | Zbl

[33] F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), no. 1, 29–51. | DOI | Zbl

[34] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. | DOI | MR | Zbl

[35] —, 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., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326.

[36] —, Remark on the asymptotic behavior of the Klein-Gordon equation in ${ℝ}^{n+1}$, Comm. Pure Appl. Math. 46 (1993), no. 2, 137–144. | DOI | MR | Zbl

[37] S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in $3$D, Comm. Pure Appl. Math. 49 (1996), no. 3, 307–321. | DOI | Zbl

[38] P. G. LeFloch and Y. Ma, The global nonlinear stability of Minkowski space for self-gravitating massive fields, , 2015. | arXiv | DOI | MR | Zbl

[39] M. Lemou, F. Méhats, and P. Raphaël, Orbital stability of spherical galactic models, Invent. Math. 187 (2012), no. 1, 145–194. | DOI | MR | Zbl

[40] H. Lindblad and I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys. 256 (2005), no. 1, 43–110. | DOI | MR | Zbl

[41] —, The global stability of Minkowski space-time in harmonic gauge, Ann. of Math. (2) 171 (2010), no. 3, 1401–1477. | DOI | MR | Zbl

[42] H. Lindblad and M. Taylor, Global stability of Minkowski space for the Einstein–Vlasov system in the harmonic gauge, , 2017. | arXiv | DOI | MR | Zbl

[43] J. Loizelet, Solutions globales des équations d’Einstein-Maxwell, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 3, 565–610. | DOI

[44] G. Rein, Static solutions of the spherically symmetric Vlasov-Einstein system, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 559–570. | DOI | MR | Zbl

[45] G. Rein and A. D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys. 150 (1992), no. 3, 561–583. | DOI | MR | Zbl

[46] —, Smooth static solutions of the spherically symmetric Vlasov-Einstein system, Ann. Inst. H. Poincaré Phys. Théor. 59 (1993), no. 4, 383–397. | Numdam | Zbl

[47] —, Erratum: “Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data” [Comm. Math. Phys. 150 (1992), no. 3, 561–583], Comm. Math. Phys. 176 (1996), no. 2, 475–478. | Zbl

[48] A. D. Rendall, The Einstein-Vlasov system, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 231–250. | DOI | Zbl

[49] H. Ringström, On the topology and future stability of the universe, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2013. | DOI | Zbl

[50] O. Sarbach and T. Zannias, Tangent bundle formulation of a charged gas, American Institute of Physics Conference Series, vol. 1577, 2014, , pp. 192–207. | arXiv | DOI

[51] R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. | DOI | MR | Zbl

[52] J. Smulevici, Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE 2 (2016), no. 2, 2:11. | DOI | MR | Zbl

[53] J. Speck, The nonlinear stability of the trivial solution to the Maxwell-Born-Infeld system, J. Math. Phys. 53 (2012), no. 8, 083703, 83. | DOI | MR | Zbl

[54] J. M. Stewart, Non-equilibrium relativistic kinetic theory, pp. 1–113, Springer Berlin Heidelberg, Berlin, Heidelberg, 1971. | DOI

[55] M. Taylor, The global nonlinear stability of Minkowski space for the massless Einstein–Vlasov system, Ann. PDE 3 (2017), no. 1, 3:9. | DOI | MR | Zbl

[56] Q. Wang, An intrinsic hyperboloid approach for Einstein Klein-Gordon equations, . | arXiv | DOI | MR | Zbl

[57] Xuecheng Wang, Decay estimates for the 3d relativistic and non-relativistic Vlasov-Poisson systems, , 2018. | arXiv

[58] —, Propagation of regularity and long time behavior of 3d massive relativistic transport equation I: Vlasov-Nordström system, , 2018. | arXiv

[59] —, Propagation of regularity and long time behavior of the 3D massive relativistic transport equation II: Vlasov-Maxwell system, , 2018. | arXiv

[60] E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. | DOI | MR | Zbl

[61] G. Wolansky, Static solutions of the Vlasov-Einstein system, Arch. Ration. Mech. Anal. 156 (2001), no. 3, 205–230. | DOI | MR | Zbl

[62] W. W. Y. Wong, A commuting-vector-field approach to some dispersive estimates, , 2017. | arXiv | DOI | MR