# American Institute of Mathematical Sciences

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

 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

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.
Citation: Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete & Continuous Dynamical Systems - A, 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 (). Google Scholar [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. Google Scholar [3] S. S. Chern and J. Simons, Characteristic forms and geometric invariants,, Ann. of Math. (2), 99 (1974), 48. doi: 10.2307/1971013. Google Scholar [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. doi: 10.1103/PhysRevD.45.3802. Google Scholar [5] S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories,, Physical Review Letters, 48 (1982), 975. doi: 10.1103/PhysRevLett.48.975. Google Scholar [6] H. Huh, Cauchy problem for the fermion field equation coupled with the Chern-Simons gauge,, Lett. Math. Phys., 79 (2007), 75. doi: 10.1007/s11005-006-0118-y. Google Scholar [7] _______, Local and global solutions of the Chern-Simons-Higgs system,, J. Funct. Anal., 242 (2007), 526. doi: 10.1016/j.jfa.2006.09.009. Google Scholar [8] _______, Global solutions and asymptotic behaviors of the Chern-Simons-Dirac equations in $\mathbbR^{1+1}$,, J. Math. Anal. Appl., 366 (2010), 706. doi: 10.1016/j.jmaa.2009.12.055. Google Scholar [9] _______, Towards the Chern-Simons-Higgs equation with finite energy,, Discrete Contin. Dyn. Syst., 30 (2011), 1145. doi: 10.3934/dcds.2011.30.1145. Google Scholar [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, (2012). Google Scholar [11] S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations,, Commun. Contemp. Math., 4 (2002), 223. doi: 10.1142/S0219199702000634. Google Scholar [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. doi: 10.1090/S0894-0347-99-00282-9. Google Scholar [13] H. Lindblad, Counterexamples to local existence for semi-linear wave equations,, Amer. J. Math., 118 (1996), 1. doi: 10.1353/ajm.1996.0002. Google Scholar [14] B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger,, preprint, (2012). doi: 10.1093/imrn/rnt161. Google Scholar [15] A. Lopez and E. Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories,, Phys. Rev. B, 44 (1991), 5246. doi: 10.1103/PhysRevB.44.5246. Google Scholar [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. doi: 10.3934/dcds.2013.33.2531. Google Scholar

show all references

##### References:
 [1] N. Bournaveas, Low regularity solutions of the Chern-Simons-Higgs equations in the Lorentz gauge,, Electron. J. Differential Equations, 2009 (). Google Scholar [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. Google Scholar [3] S. S. Chern and J. Simons, Characteristic forms and geometric invariants,, Ann. of Math. (2), 99 (1974), 48. doi: 10.2307/1971013. Google Scholar [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. doi: 10.1103/PhysRevD.45.3802. Google Scholar [5] S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories,, Physical Review Letters, 48 (1982), 975. doi: 10.1103/PhysRevLett.48.975. Google Scholar [6] H. Huh, Cauchy problem for the fermion field equation coupled with the Chern-Simons gauge,, Lett. Math. Phys., 79 (2007), 75. doi: 10.1007/s11005-006-0118-y. Google Scholar [7] _______, Local and global solutions of the Chern-Simons-Higgs system,, J. Funct. Anal., 242 (2007), 526. doi: 10.1016/j.jfa.2006.09.009. Google Scholar [8] _______, Global solutions and asymptotic behaviors of the Chern-Simons-Dirac equations in $\mathbbR^{1+1}$,, J. Math. Anal. Appl., 366 (2010), 706. doi: 10.1016/j.jmaa.2009.12.055. Google Scholar [9] _______, Towards the Chern-Simons-Higgs equation with finite energy,, Discrete Contin. Dyn. Syst., 30 (2011), 1145. doi: 10.3934/dcds.2011.30.1145. Google Scholar [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, (2012). Google Scholar [11] S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations,, Commun. Contemp. Math., 4 (2002), 223. doi: 10.1142/S0219199702000634. Google Scholar [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. doi: 10.1090/S0894-0347-99-00282-9. Google Scholar [13] H. Lindblad, Counterexamples to local existence for semi-linear wave equations,, Amer. J. Math., 118 (1996), 1. doi: 10.1353/ajm.1996.0002. Google Scholar [14] B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger,, preprint, (2012). doi: 10.1093/imrn/rnt161. Google Scholar [15] A. Lopez and E. Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories,, Phys. Rev. B, 44 (1991), 5246. doi: 10.1103/PhysRevB.44.5246. Google Scholar [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. doi: 10.3934/dcds.2013.33.2531. Google Scholar
 [1] Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389 [2] Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199 [3] Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669 [4] Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531 [5] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [6] Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193 [7] Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605 [8] Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737 [9] Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002 [10] Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259 [11] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [12] Elissar Nasreddine. Well-posedness for a model of individual clustering. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2647-2668. doi: 10.3934/dcdsb.2013.18.2647 [13] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 [14] Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1 [15] David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113 [16] Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527 [17] A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469 [18] Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 [19] Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145 [20] Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053

2018 Impact Factor: 1.143