June  2022, 42(6): 3065-3076. doi: 10.3934/dcds.2022008

Local well-posedness for the Maxwell-Dirac system in temporal gauge

Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20 42119 Wuppertal, Germany

Received  June 2021 Revised  November 2021 Published  June 2022 Early access  February 2022

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in 3+1 dimensions:
$ \begin{align*} \partial^{\mu} F_{\mu \nu} & = - \langle \psi, \alpha_{\nu} \psi \rangle \ \\ -i \alpha^{\mu} \partial_{\mu} \psi & = A_{\mu} \alpha^{\mu} \psi \, , \end{align*} $
where
$ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu} $
, and
$ \alpha^{\mu} $
are the 4x4 Dirac matrices. We assume the temporal gauge
$ A_0 = 0 $
and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system.
Citation: Hartmut Pecher. Local well-posedness for the Maxwell-Dirac system in temporal gauge. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3065-3076. doi: 10.3934/dcds.2022008
References:
[1]

P. d'AnconaD. Foschi and S. Selberg, Atlas of products for wave-Sobolev spaces on $\mathbb{R}^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.  doi: 10.1090/S0002-9947-2011-05250-5.

[2]

P. d'AnconaD. Foschi and S. Selberg, Null structure and almost optimal local well-posedness of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839.  doi: 10.1353/ajm.0.0118.

[3]

P. d'Ancona and S. Selberg, Global well-posedness of the Maxwell-Dirac system in two space dimensions, J. Funct. Anal., 260 (2011), 2300-2365.  doi: 10.1016/j.jfa.2010.12.010.

[4]

N. Bournaveas, Local existence for the Maxwell-Dirac equations in three space dimensions, Comm. Partial Differential Equations, 21 (1996), 693-720.  doi: 10.1080/03605309608821204.

[5]

H. Huh and S.-J. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, Comm. Partial Differential Equations, 41 (2016), 375-397.  doi: 10.1080/03605302.2015.1132730.

[6]

S. Klainerman and M. Machedon, (with appendices by J. Bourgain and D. Tataru), Remark on strichartz-type inequalities, Internat. Math. Res. Notices, (1996), 201–220. doi: 10.1155/S1073792896000153.

[7]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44.  doi: 10.1215/S0012-7094-94-07402-4.

[8]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.  doi: 10.1142/S0219199702000634.

[9]

N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger, Int. Math. Res. No., 13 (2003), 697-734.  doi: 10.1155/S107379280320310X.

[10]

N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations, Commun. Math. Phys., 243 (2003), 123-136.  doi: 10.1007/s00220-003-0951-0.

[11]

S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn 107, 63 pp. doi: 10.1093/imrn/rnn107.

[12]

S. Selberg and A. Tesfahun, Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge, J. Eur. Math. Soc., 18 (2016), 1729-1752.  doi: 10.4171/JEMS/627.

[13]

S. Selberg and A. Tesfahun, Ill-posedness of the Maxwell-Dirac system below charge in space dimension three and lower, Nonlinear Differential Equations Appl., 28 (2021), Paper No. 42, 20 pp. doi: 10.1007/s00030-021-00703-w.

[14]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. 

[15]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Diff. Equ., 189 (2003), 366-382. 

[16]

T. Tao, Multilinear weighted convolution of $L^2$-functions and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 838-908.  doi: 10.1353/ajm.2001.0035.

show all references

References:
[1]

P. d'AnconaD. Foschi and S. Selberg, Atlas of products for wave-Sobolev spaces on $\mathbb{R}^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.  doi: 10.1090/S0002-9947-2011-05250-5.

[2]

P. d'AnconaD. Foschi and S. Selberg, Null structure and almost optimal local well-posedness of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839.  doi: 10.1353/ajm.0.0118.

[3]

P. d'Ancona and S. Selberg, Global well-posedness of the Maxwell-Dirac system in two space dimensions, J. Funct. Anal., 260 (2011), 2300-2365.  doi: 10.1016/j.jfa.2010.12.010.

[4]

N. Bournaveas, Local existence for the Maxwell-Dirac equations in three space dimensions, Comm. Partial Differential Equations, 21 (1996), 693-720.  doi: 10.1080/03605309608821204.

[5]

H. Huh and S.-J. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, Comm. Partial Differential Equations, 41 (2016), 375-397.  doi: 10.1080/03605302.2015.1132730.

[6]

S. Klainerman and M. Machedon, (with appendices by J. Bourgain and D. Tataru), Remark on strichartz-type inequalities, Internat. Math. Res. Notices, (1996), 201–220. doi: 10.1155/S1073792896000153.

[7]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44.  doi: 10.1215/S0012-7094-94-07402-4.

[8]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.  doi: 10.1142/S0219199702000634.

[9]

N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger, Int. Math. Res. No., 13 (2003), 697-734.  doi: 10.1155/S107379280320310X.

[10]

N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations, Commun. Math. Phys., 243 (2003), 123-136.  doi: 10.1007/s00220-003-0951-0.

[11]

S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn 107, 63 pp. doi: 10.1093/imrn/rnn107.

[12]

S. Selberg and A. Tesfahun, Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge, J. Eur. Math. Soc., 18 (2016), 1729-1752.  doi: 10.4171/JEMS/627.

[13]

S. Selberg and A. Tesfahun, Ill-posedness of the Maxwell-Dirac system below charge in space dimension three and lower, Nonlinear Differential Equations Appl., 28 (2021), Paper No. 42, 20 pp. doi: 10.1007/s00030-021-00703-w.

[14]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. 

[15]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Diff. Equ., 189 (2003), 366-382. 

[16]

T. Tao, Multilinear weighted convolution of $L^2$-functions and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 838-908.  doi: 10.1353/ajm.2001.0035.

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