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July  2019, 18(4): 1711-1734. doi: 10.3934/cpaa.2019081

## Scattering results for Dirac Hartree-type equations with small initial data

 1 Korea Institute for Advanced Study, Seoul 20455, Korea 2 Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 54896, Korea

Received  May 2018 Revised  September 2018 Published  January 2019

Fund Project: This work was partially supported by NRF (NRF-2015R1D1A1A09057795) and by German Science Foundation (IRTG 2235)

We consider the Dirac equations with cubic Hartree-type nonlinearity which are derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data global well-posedness and scattering results in the full scaling subcritical regularity regime. The strategy of the proof relies on the localized Strichartz estimates and bilinear estimates in $V^2$ spaces, together with the use of the null structure that the nonlinear term exhibits. This result is shown to be almost optimal in the sense that the iteration method based on Duhamel's formula fails over the supercritical range.

Citation: Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
##### References:
 [1] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in $H^1(\Bbb R^3)$, Comm. Math. Phys., 335 (2015), 43-82. doi: 10.1007/s00220-014-2164-0. Google Scholar [2] I. Bejenaru and S. Herr, On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 19 (2017), 2445-2467. doi: 10.4171/JEMS/721. Google Scholar [3] J. M. Chadam and R. T. Glassey, On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions, J. Math. Anal. Appl., 53 (1976), 495-507. doi: 10.1016/0022-247X(76)90087-1. Google Scholar [4] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688. Google Scholar [5] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. Google Scholar [6] P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112. doi: 10.1080/03605300701743749. Google Scholar [7] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 9 (2007), 877-899. doi: 10.4171/JEMS/100. Google Scholar [8] J. a.-P. Dias and M. Figueira, On the existence of weak solutions for a nonlinear time dependent Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 149-158. doi: 10.1017/S030821050002401X. Google Scholar [9] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar [10] M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space", [Ann. I. H. Poincaré–AN 26 (3) (2009) 917–941], Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 971–972. doi: 10.1016/j.anihpc.2010.01.006. Google Scholar [11] S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137. doi: 10.1016/j.na.2013.11.023. Google Scholar [12] S. Herr and A. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037. Google Scholar [13] S. Herr and C. Yang, Critical well-posedness and scattering results for fractional Hartree-type equations, Differential Integral Equations, 31 (2018), 701-714. Google Scholar [14] H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-De Vries, Nonlinear SchrÖdinger, Wave and Schródinger Maps., Basel: Birkhäuser/Springer, 2014. Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic hartree equations of critical type, Mathematical Physics, Analysis and Geometry, 10 (2007), 43-64. doi: 10.1007/s11040-007-9020-9. Google Scholar [16] S. Machihara and K. Tsutaya, Scattering theory for the Dirac equation with a non-local term, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 867-878. doi: 10.1017/S0308210507000479. Google Scholar [17] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988 (electronic). doi: 10.1137/S0036141001385307. Google Scholar [18] M. Nakamura and K. Tsutaya, Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation, Nonlinear Anal., 75 (2012), 3531-3542. doi: 10.1016/j.na.2012.01.012. Google Scholar [19] F. Pusateri, Modified scattering for the boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x. Google Scholar [20] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. Google Scholar [21] A. Tesfahun, Small data scattering for cubic Dirac equation with Hartree type nonlinearity in $\mathbb R ^{1+3}$, arXiv e-prints.Google Scholar [22] C. Yang, Small data scattering of semirelativistic hartree equation, Nonlinear Analysis, 178 (2019), 41-55. doi: 10.1016/j.na.2018.07.003. Google Scholar

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##### References:
 [1] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in $H^1(\Bbb R^3)$, Comm. Math. Phys., 335 (2015), 43-82. doi: 10.1007/s00220-014-2164-0. Google Scholar [2] I. Bejenaru and S. Herr, On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 19 (2017), 2445-2467. doi: 10.4171/JEMS/721. Google Scholar [3] J. M. Chadam and R. T. Glassey, On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions, J. Math. Anal. Appl., 53 (1976), 495-507. doi: 10.1016/0022-247X(76)90087-1. Google Scholar [4] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688. Google Scholar [5] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. Google Scholar [6] P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112. doi: 10.1080/03605300701743749. Google Scholar [7] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS), 9 (2007), 877-899. doi: 10.4171/JEMS/100. Google Scholar [8] J. a.-P. Dias and M. Figueira, On the existence of weak solutions for a nonlinear time dependent Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 149-158. doi: 10.1017/S030821050002401X. Google Scholar [9] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar [10] M. Hadac, S. Herr and H. Koch, Erratum to "Well-posedness and scattering for the KP-II equation in a critical space", [Ann. I. H. Poincaré–AN 26 (3) (2009) 917–941], Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 971–972. doi: 10.1016/j.anihpc.2010.01.006. Google Scholar [11] S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137. doi: 10.1016/j.na.2013.11.023. Google Scholar [12] S. Herr and A. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037. Google Scholar [13] S. Herr and C. Yang, Critical well-posedness and scattering results for fractional Hartree-type equations, Differential Integral Equations, 31 (2018), 701-714. Google Scholar [14] H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-De Vries, Nonlinear SchrÖdinger, Wave and Schródinger Maps., Basel: Birkhäuser/Springer, 2014. Google Scholar [15] E. Lenzmann, Well-posedness for semi-relativistic hartree equations of critical type, Mathematical Physics, Analysis and Geometry, 10 (2007), 43-64. doi: 10.1007/s11040-007-9020-9. Google Scholar [16] S. Machihara and K. Tsutaya, Scattering theory for the Dirac equation with a non-local term, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 867-878. doi: 10.1017/S0308210507000479. Google Scholar [17] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988 (electronic). doi: 10.1137/S0036141001385307. Google Scholar [18] M. Nakamura and K. Tsutaya, Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation, Nonlinear Anal., 75 (2012), 3531-3542. doi: 10.1016/j.na.2012.01.012. Google Scholar [19] F. Pusateri, Modified scattering for the boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x. Google Scholar [20] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. Google Scholar [21] A. Tesfahun, Small data scattering for cubic Dirac equation with Hartree type nonlinearity in $\mathbb R ^{1+3}$, arXiv e-prints.Google Scholar [22] C. Yang, Small data scattering of semirelativistic hartree equation, Nonlinear Analysis, 178 (2019), 41-55. doi: 10.1016/j.na.2018.07.003. Google Scholar
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