# American Institute of Mathematical Sciences

November  2021, 20(11): 3683-3702. doi: 10.3934/cpaa.2021126

## Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

 Department of Mathematics, Jeonbuk National University, Jeonju 54896, Republic of Korea

* Corresponding author

Received  November 2020 Revised  June 2021 Published  November 2021 Early access  July 2021

Fund Project: The author is supported by NRF-2021R1I1A3A04035040(Republic of Korea)

In this paper, we consider the Cauchy problem of $d$-dimension Hartree type Dirac equation with nonlinearity $c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle$, where $c\in \mathbb R\setminus\{0\}$, $0 < \gamma < d$($d = 2,3$). Our aim is to show the local well-posedness in $H^s$ for $s > \frac{\gamma-1}2$ with mass-supercritical cases($1 < \gamma Citation: Kiyeon Lee. Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3683-3702. doi: 10.3934/cpaa.2021126 ##### References:  [1] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in$H^1(\mathbb{R}^3)$, Commun. Math. Phys., 335 (2015), 48-82. doi: 10.1007/s00220-014-2164-0. Google Scholar [2] N. Bournaveas, T. Candy and S. Machihara, A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discrete Contin. Dyn. Syst., 34 (2014), 2693-2701. doi: 10.3934/dcds.2014.34.2693. Google Scholar [3] T. Candy and S. Herr, Transference of bilinear restriction estimates to quadratic variation norms and the Dirac-Klein-Gordon system, Anal. Partial Differ. Equ., 5 (2018), 1171-1240. doi: 10.2140/apde.2018.11.1171. Google Scholar [4] T. Candy and S. Herr, Conditional large initial data scattering results for the Dirac-Klein-Gordon system, Forum Math., 6 (2018), 55. doi: 10.1017/fms. 2018.8. Google Scholar [5] 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), 495-597. doi: 10.1016/0022-247X(76)90087-1. Google Scholar [6] Y. Cho, K. Lee and T. Ozawa, Small data scattering of 2d Hartree type Dirac equations, preprint. Google Scholar [7] 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 [8] Y. Cho and T. Ozawa, On radial solutions of semi-relativistic Hartree equations, Discrete Contin. Dyn. Syst., 1 (2008), 71-82. doi: 10.3934/dcdss.2008.1.71. Google Scholar [9] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Euro. Math. Soc., 9 (2007), 877-899. doi: 10.4171/JEMS/100. Google Scholar [10] 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 [11] H. Huh and S. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, Commun. Partial Differ. Equ., 41 (2016), 375-397. doi: 10.1080/03605302.2015.1132730. Google Scholar [12] S. Machihara and K. Tsutaya, Scattering theory for the Dirac equation with a non-local term, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 139 (2009), 867-878. doi: 10.1017/S0308210507000479. Google Scholar [13] L. Molinet, Je an-Claude Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307. Google Scholar [14] 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 [15] S. Selberg, Anisotropic bilinear$L^2$estimates related to the 3D wave equation, Int. Math. Res. Not., 9 (2008), rnn 107. doi: 10.1093/imrn/rnn107. Google Scholar [16] S. Selberg, Bilinear Fourier restriction estimates related to the$2$D wave equation, Adv. Differ. Equ., 16 (2011), 667-690. Google Scholar [17] A. Tesfahun, Long-time behavior of solutions to cubic Dirac equation with Hartree type nonlinearity in$\mathbb{R}^{1+2}$, Int. Math. Res. Not., 19 (2020), 6489-6538. doi: 10.1093/imrn/rny217. Google Scholar [18] X. Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not., 21 (2015), 10801-10846. doi: 10.1093/imrn/rnv010. Google Scholar [19] C. Yang, Scattering results for Dirac Hartree-type equations with small initial data, Commun. Pure. Appl. Anal., 18 (2019), 1711-1734. doi: 10.3934/cpaa.2019081. Google Scholar show all references ##### References:  [1] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in$H^1(\mathbb{R}^3)$, Commun. Math. Phys., 335 (2015), 48-82. doi: 10.1007/s00220-014-2164-0. Google Scholar [2] N. Bournaveas, T. Candy and S. Machihara, A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discrete Contin. Dyn. Syst., 34 (2014), 2693-2701. doi: 10.3934/dcds.2014.34.2693. Google Scholar [3] T. Candy and S. Herr, Transference of bilinear restriction estimates to quadratic variation norms and the Dirac-Klein-Gordon system, Anal. Partial Differ. Equ., 5 (2018), 1171-1240. doi: 10.2140/apde.2018.11.1171. Google Scholar [4] T. Candy and S. Herr, Conditional large initial data scattering results for the Dirac-Klein-Gordon system, Forum Math., 6 (2018), 55. doi: 10.1017/fms. 2018.8. Google Scholar [5] 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), 495-597. doi: 10.1016/0022-247X(76)90087-1. Google Scholar [6] Y. Cho, K. Lee and T. Ozawa, Small data scattering of 2d Hartree type Dirac equations, preprint. Google Scholar [7] 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 [8] Y. Cho and T. Ozawa, On radial solutions of semi-relativistic Hartree equations, Discrete Contin. Dyn. Syst., 1 (2008), 71-82. doi: 10.3934/dcdss.2008.1.71. Google Scholar [9] P. D'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Euro. Math. Soc., 9 (2007), 877-899. doi: 10.4171/JEMS/100. Google Scholar [10] 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 [11] H. Huh and S. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, Commun. Partial Differ. Equ., 41 (2016), 375-397. doi: 10.1080/03605302.2015.1132730. Google Scholar [12] S. Machihara and K. Tsutaya, Scattering theory for the Dirac equation with a non-local term, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 139 (2009), 867-878. doi: 10.1017/S0308210507000479. Google Scholar [13] L. Molinet, Je an-Claude Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307. Google Scholar [14] 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 [15] S. Selberg, Anisotropic bilinear$L^2$estimates related to the 3D wave equation, Int. Math. Res. Not., 9 (2008), rnn 107. doi: 10.1093/imrn/rnn107. Google Scholar [16] S. Selberg, Bilinear Fourier restriction estimates related to the$2$D wave equation, Adv. Differ. Equ., 16 (2011), 667-690. Google Scholar [17] A. Tesfahun, Long-time behavior of solutions to cubic Dirac equation with Hartree type nonlinearity in$\mathbb{R}^{1+2}$, Int. Math. Res. Not., 19 (2020), 6489-6538. doi: 10.1093/imrn/rny217. Google Scholar [18] X. Wang, On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Not., 21 (2015), 10801-10846. doi: 10.1093/imrn/rnv010. Google Scholar [19] C. Yang, Scattering results for Dirac Hartree-type equations with small initial data, Commun. Pure. Appl. Anal., 18 (2019), 1711-1734. doi: 10.3934/cpaa.2019081. Google Scholar Well-posedness results for semi-relativistic and Dirac equations  Author(s) Equations dimension$ H^s(\mathbb{R}^d)  |x|^{- {\gamma}} $Cho–Ozawa(2006, [7]) S-R$ d \ge 2 $LWP for$ s> \frac{ {\gamma}}2-  0< {\gamma}< d $Cho–Ozawa (2008, [8]) S-R$ d \ge 2 $GWP for$ s\ge \frac12 $in radial case$ 0< {\gamma}<\frac{2d-1}{d} $Machihara–Tsutaya (2009, [12]) Dirac$ d=3 $LWP for$ s>\frac{ {\gamma}}6 +\frac12  2< {\gamma}<3 $Pusateri (2014, [14]) S-R$ d =3 $Modified scattering$ {\gamma} =1 $Bournaveas–Candy–Machihara (2014, [2]) CSD$ d =2 $LWP for$ s>\frac14  {\gamma} =1 $Herr–Lenzmann (2014, [10]) S-R$ d =3 $LWP for$ s> \frac14  {\gamma} =1 $Cho–Lee–Ozawa (2020, [6]) Dirac$ d=2 $GWP for$ s> {\gamma}-1  1< {\gamma}<2 $ Author(s) Equations dimension$ H^s(\mathbb{R}^d)  |x|^{- {\gamma}} $Cho–Ozawa(2006, [7]) S-R$ d \ge 2 $LWP for$ s> \frac{ {\gamma}}2-  0< {\gamma}< d $Cho–Ozawa (2008, [8]) S-R$ d \ge 2 $GWP for$ s\ge \frac12 $in radial case$ 0< {\gamma}<\frac{2d-1}{d} $Machihara–Tsutaya (2009, [12]) Dirac$ d=3 $LWP for$ s>\frac{ {\gamma}}6 +\frac12  2< {\gamma}<3 $Pusateri (2014, [14]) S-R$ d =3 $Modified scattering$ {\gamma} =1 $Bournaveas–Candy–Machihara (2014, [2]) CSD$ d =2 $LWP for$ s>\frac14  {\gamma} =1 $Herr–Lenzmann (2014, [10]) S-R$ d =3 $LWP for$ s> \frac14  {\gamma} =1 $Cho–Lee–Ozawa (2020, [6]) Dirac$ d=2 $GWP for$ s> {\gamma}-1  1< {\gamma}<2 $ [1] Hartmut Pecher. 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