# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021126
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## 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 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($1 < \gamma<d$) and mass-critical case(${\gamma} = 1$) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be $C^3$ at the origin for $H^s$ with $s < \frac{\gamma-1}2$.

Citation: Kiyeon Lee. Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions. Communications on Pure &amp; Applied Analysis, 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

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##### 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 $s> {\gamma}-1$ $1< {\gamma}<2$
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