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<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 & 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

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N. BournaveasT. 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

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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

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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

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Y. Cho, K. Lee and T. Ozawa, Small data scattering of 2d Hartree type Dirac equations, preprint.  Google Scholar

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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

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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'AnconaD. 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

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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. MolinetJe 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. BournaveasT. 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'AnconaD. 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. MolinetJe 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

Table 1.  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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