Global behavior of small data solutions for the 2D Dirac–Klein-Gordon system

Dong, Shijie; Li, Kuijie; Ma, Yue; Yuan, Xu

doi:10.1090/tran/9011

Global behavior of small data solutions for the 2D Dirac–Klein-Gordon system
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by Shijie Dong, Kuijie Li, Yue Ma and Xu Yuan;

Trans. Amer. Math. Soc. 377 (2024), 649-695

DOI: https://doi.org/10.1090/tran/9011

Published electronically: October 11, 2023

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Abstract:

In this paper, we are interested in the two-dimensional Dirac–Klein-Gordon system, which is a basic model in particle physics. We investigate the global behavior of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold: (1) we show sharp time decay for the pointwise estimates of the solutions, which implies the asymptotic stability of this system; (2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. Our result is valid for general small, high-regular initial data, and in particular, there is no restriction on the support of the initial data.

References

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597–618. MR 1856402, DOI 10.1007/s002220100165
Serge Alinhac, Hyperbolic partial differential equations, Universitext, Springer, Dordrecht, 2009. MR 2524198, DOI 10.1007/978-0-387-87823-2
Alain Bachelot, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. H. Poincaré Phys. Théor. 48 (1988), no. 4, 387–422 (French, with English summary). MR 969173
Ioan Bejenaru and Sebastian Herr, On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2445–2467. MR 3668064, DOI 10.4171/JEMS/721
Ioan Bejenaru and Sebastian Herr, The cubic Dirac equation: small initial data in $H^{\frac 12}(\Bbb R^2)$, Comm. Math. Phys. 343 (2016), no. 2, 515–562. MR 3477346, DOI 10.1007/s00220-015-2508-4
Nikolaos Bournaveas, Low regularity solutions of the Dirac Klein-Gordon equations in two space dimensions, Comm. Partial Differential Equations 26 (2001), no. 7-8, 1345–1366. MR 1855281, DOI 10.1081/PDE-100106136
N. Bournaveas, A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension, J. Funct. Anal. 173 (2000), no. 1, 203–213. MR 1760283, DOI 10.1006/jfan.1999.3559
Timothy Candy and Sebastian Herr, Conditional large initial data scattering results for the Dirac-Klein-Gordon system, Forum Math. Sigma 6 (2018), Paper No. e9, 55. MR 3816948, DOI 10.1017/fms.2018.8
John M. Chadam and Robert T. Glassey, On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions, Arch. Rational Mech. Anal. 54 (1974), 223–237. MR 369952, DOI 10.1007/BF00250789
X. Chen, Global stability of Minkowski spacetime for a spin-1/2 field, Preprint, arXiv:2201.08280.
Yvonne Choquet-Bruhat, Solutions globales des équations de Maxwell-Dirac-Klein-Gordon (masses nulles), C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 2, 153–158 (French, with English summary). MR 610307
Piero D’Ancona, Damiano Foschi, and Sigmund Selberg, Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions, J. Hyperbolic Differ. Equ. 4 (2007), no. 2, 295–330. MR 2329387, DOI 10.1142/S0219891607001148
Piero D’Ancona, Damiano Foschi, and Sigmund Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 877–899. MR 2341835, DOI 10.4171/JEMS/100
Shijie Dong, Asymptotic behavior of the solution to the Klein-Gordon-Zakharov model in dimension two, Comm. Math. Phys. 384 (2021), no. 1, 587–607. MR 4252885, DOI 10.1007/s00220-021-04003-3
Shijie Dong, Global solution to the wave and Klein-Gordon system under null condition in dimension two, J. Funct. Anal. 281 (2021), no. 11, Paper No. 109232, 29. MR 4316723, DOI 10.1016/j.jfa.2021.109232
Shijie Dong, Philippe G. LeFloch, and Zoe Wyatt, Global evolution of the $\rm U(1)$ Higgs Boson: nonlinear stability and uniform energy bounds, Ann. Henri Poincaré 22 (2021), no. 3, 677–713. MR 4226448, DOI 10.1007/s00023-020-00955-9
Shijie Dong and Kuijie Li, Global solution to the cubic Dirac equation in two space dimensions, J. Differential Equations 331 (2022), 192–222. MR 4435916, DOI 10.1016/j.jde.2022.05.022
S. Dong and Y. Ma, Global existence and scattering of the Klein-Gordon-Zakharov system in two space dimensions, Preprint, arXiv:2111.00244. To appear in Peking Mathematical Journal.
Shijie Dong, Yue Ma, and Xu Yuan, Asymptotic behavior of 2D wave-Klein-Gordon coupled system under null condition, Bull. Sci. Math. 187 (2023), Paper No. 103313, 43. MR 4621970, DOI 10.1016/j.bulsci.2023.103313
S. Dong and Z. Wyatt, Asymptotic stability for the Dirac-Klein-Gordon system in two space dimensions, Preprint, arXiv:2105.13780, to appear in Annales de l’Institut Henri Poincaré C. Analyse Non Linéaire.
Senhao Duan and Yue Ma, Global solutions of wave-Klein–Gordon systems in $2+1$ dimensional space-time with strong couplings in divergence form, SIAM J. Math. Anal. 54 (2022), no. 3, 2691–2726. MR 4414496, DOI 10.1137/20M1377229
S. Duan, Y. Ma, and W. Zhang, Nonlinear stability of the totally geodesic wave maps in anisotropic manifolds, Preprint, arXiv:2204.12525.
Vladimir Georgiev, Decay estimates for the Klein-Gordon equation, Comm. Partial Differential Equations 17 (1992), no. 7-8, 1111–1139. MR 1179280, DOI 10.1080/03605309208820879
Axel Grünrock and Hartmut Pecher, Global solutions for the Dirac-Klein-Gordon system in two space dimensions, Comm. Partial Differential Equations 35 (2010), no. 1, 89–112. MR 2748619, DOI 10.1080/03605300903296306
Zihua Guo, Kenji Nakanishi, and Shuxia Wang, Small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry, Math. Res. Lett. 21 (2014), no. 4, 733–755. MR 3275645, DOI 10.4310/MRL.2014.v21.n4.a8
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
Alexandru D. Ionescu and Benoit Pausader, On the global regularity for a wave-Klein-Gordon coupled system, Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 933–986. MR 3952698, DOI 10.1007/s10114-019-8413-6
Alexandru D. Ionescu and Benoît Pausader, The Einstein-Klein-Gordon coupled system: global stability of the Minkowski solution, Annals of Mathematics Studies, vol. 213, Princeton University Press, Princeton, NJ, 2022. MR 4422074
S. Klainerman, 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. MR 837683
S. Klainerman, Remark on the asymptotic behavior of the Klein-Gordon equation in $\textbf {R}^{n+1}$, Comm. Pure Appl. Math. 46 (1993), no. 2, 137–144. MR 1199196, DOI 10.1002/cpa.3160460202
Sergiu Klainerman, Qian Wang, and Shiwu Yang, Global solution for massive Maxwell-Klein-Gordon equations, Comm. Pure Appl. Math. 73 (2020), no. 1, 63–109. MR 4033890, DOI 10.1002/cpa.21864
Philippe G. LeFloch and Yue Ma, The hyperboloidal foliation method, Series in Applied and Computational Mathematics, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. MR 3362362
P. G. LeFloch and Y. Ma, Nonlinear stability of self-gravitating massive fields, Preprint, arXiv:1712.10045.
Jiongyue Li and Yunlong Zang, A vector field method for some nonlinear Dirac models in Minkowski spacetime, J. Differential Equations 273 (2021), 58–82. MR 4184660, DOI 10.1016/j.jde.2020.11.043
T. Ozawa, K. Tsutaya, and Y. Tsutsumi, Normal form and global solutions for the Klein-Gordon-Zakharov equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 12 (1995), no. 4, 459–503 (English, with English and French summaries). MR 1341412, DOI 10.1016/S0294-1449(16)30156-1
Tohru Ozawa, Kimitoshi Tsutaya, and Yoshio Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z. 222 (1996), no. 3, 341–362. MR 1400196, DOI 10.1007/PL00004540
Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256, DOI 10.1002/cpa.3160380516
Jacques C. H. Simon and Erik Taflin, The Cauchy problem for nonlinear Klein-Gordon equations, Comm. Math. Phys. 152 (1993), no. 3, 433–478. MR 1213298, DOI 10.1007/BF02096615
Christopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR 1715192
A. Stingo, Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein-Gordon system in two space dimension, with mildly decaying Cauchy data, Mem. Amer. Math. Soc., To appear, Preprint, arXiv:1810.10235.
Yoshio Tsutsumi, Global solutions for the Dirac-Proca equations with small initial data in $3+1$ space time dimensions, J. Math. Anal. Appl. 278 (2003), no. 2, 485–499. MR 1974020, DOI 10.1016/S0022-247X(02)00662-5
Xuecheng Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not. IMRN 21 (2015), 10801–10846. MR 3456028, DOI 10.1093/imrn/rnv010
Yu Xi Zheng, Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations, Comm. Math. Phys. 151 (1993), no. 1, 67–87. MR 1201656, DOI 10.1007/BF02096749