A note on the Chern-Simons-Dirac equations in the Coulomb gauge

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July 2014, 34(7): 2693-2701. doi: 10.3934/dcds.2014.34.2693

A note on the Chern-Simons-Dirac equations in the Coulomb gauge

Nikolaos Bournaveas ^1, , Timothy Candy ^2, and Shuji Machihara ^3,

1.	Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JE, United Kingdom
2.	Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
3.	Department of Mathematics, Faculty of Education, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City 338-8570, Japan

Received June 2013 Revised October 2013 Published December 2013

Abstract
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We prove that the Chern-Simons-Dirac equations in the Coulomb gauge are locally well-posed from initial data in $H^s$ with $s>\frac{1}{4}$. To study nonlinear Wave or Dirac equations at this regularity generally requires the presence of null structure. The novel point here is that we make no use of the null structure of the system. Instead we exploit the additional elliptic structure in the Coulomb gauge together with the bilinear Strichartz estimates of Klainerman-Tataru.

Keywords: Chern-Simons-Dirac, well-posedness, bilinear estimates, Coulomb gauge..

Mathematics Subject Classification: Primary: 35Q41, 35A0.

Citation: Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693

References:

[1]	N. Bournaveas, Low regularity solutions of the Chern-Simons-Higgs equations in the Lorentz gauge, Electron. J. Differential Equations, 2009, 10 pp.
[2]	N. Bournaveas, T. Candy and S. Machihara, Local and global well-posedness for the Chern-Simons-Dirac system in one dimension, Differential Integral Equations, 25 (2012), 699-718.
[3]	S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. (2), 99 (1974), 48-69. doi: 10.2307/1971013.
[4]	Y. M. Cho, J. W. Kim and D. H. Park, Fermionic vortex solutions in Chern-Simons electrodynamics, Phys. Rev. D (3), 45 (1992), 3802-3806. doi: 10.1103/PhysRevD.45.3802.
[5]	S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Physical Review Letters, 48 (1982), 975-978. doi: 10.1103/PhysRevLett.48.975.
[6]	H. Huh, Cauchy problem for the fermion field equation coupled with the Chern-Simons gauge, Lett. Math. Phys., 79 (2007), 75-94. doi: 10.1007/s11005-006-0118-y.
[7]	_______, Local and global solutions of the Chern-Simons-Higgs system, J. Funct. Anal., 242 (2007), 526-549. doi: 10.1016/j.jfa.2006.09.009.
[8]	_______, Global solutions and asymptotic behaviors of the Chern-Simons-Dirac equations in $\mathbbR^{1+1}$, J. Math. Anal. Appl., 366 (2010), 706-713. doi: 10.1016/j.jmaa.2009.12.055.
[9]	_______, Towards the Chern-Simons-Higgs equation with finite energy, Discrete Contin. Dyn. Syst., 30 (2011), 1145-1159. doi: 10.3934/dcds.2011.30.1145.
[10]	H. Huh and S.-J. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, preprint, arXiv:1209.3841, (2012).
[11]	S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295. doi: 10.1142/S0219199702000634.
[12]	S. Klainerman and D. Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116. doi: 10.1090/S0894-0347-99-00282-9.
[13]	H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math., 118 (1996), 1-16. doi: 10.1353/ajm.1996.0002.
[14]	B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, preprint, arXiv:1212.1476, (2012). doi: 10.1093/imrn/rnt161.
[15]	A. Lopez and E. Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B, 44 (1991), 5246-5262. doi: 10.1103/PhysRevB.44.5246.
[16]	S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Contin. Dyn. Syst., 33 (2013), 2531-2546. doi: 10.3934/dcds.2013.33.2531.

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References:

[1]	N. Bournaveas, Low regularity solutions of the Chern-Simons-Higgs equations in the Lorentz gauge, Electron. J. Differential Equations, 2009, 10 pp.
[2]	N. Bournaveas, T. Candy and S. Machihara, Local and global well-posedness for the Chern-Simons-Dirac system in one dimension, Differential Integral Equations, 25 (2012), 699-718.
[3]	S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. (2), 99 (1974), 48-69. doi: 10.2307/1971013.
[4]	Y. M. Cho, J. W. Kim and D. H. Park, Fermionic vortex solutions in Chern-Simons electrodynamics, Phys. Rev. D (3), 45 (1992), 3802-3806. doi: 10.1103/PhysRevD.45.3802.
[5]	S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Physical Review Letters, 48 (1982), 975-978. doi: 10.1103/PhysRevLett.48.975.
[6]	H. Huh, Cauchy problem for the fermion field equation coupled with the Chern-Simons gauge, Lett. Math. Phys., 79 (2007), 75-94. doi: 10.1007/s11005-006-0118-y.
[7]	_______, Local and global solutions of the Chern-Simons-Higgs system, J. Funct. Anal., 242 (2007), 526-549. doi: 10.1016/j.jfa.2006.09.009.
[8]	_______, Global solutions and asymptotic behaviors of the Chern-Simons-Dirac equations in $\mathbbR^{1+1}$, J. Math. Anal. Appl., 366 (2010), 706-713. doi: 10.1016/j.jmaa.2009.12.055.
[9]	_______, Towards the Chern-Simons-Higgs equation with finite energy, Discrete Contin. Dyn. Syst., 30 (2011), 1145-1159. doi: 10.3934/dcds.2011.30.1145.
[10]	H. Huh and S.-J. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, preprint, arXiv:1209.3841, (2012).
[11]	S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295. doi: 10.1142/S0219199702000634.
[12]	S. Klainerman and D. Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116. doi: 10.1090/S0894-0347-99-00282-9.
[13]	H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math., 118 (1996), 1-16. doi: 10.1353/ajm.1996.0002.
[14]	B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, preprint, arXiv:1212.1476, (2012). doi: 10.1093/imrn/rnt161.
[15]	A. Lopez and E. Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B, 44 (1991), 5246-5262. doi: 10.1103/PhysRevB.44.5246.
[16]	S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Contin. Dyn. Syst., 33 (2013), 2531-2546. doi: 10.3934/dcds.2013.33.2531.

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