# American Institute of Mathematical Sciences

July  2014, 13(4): 1669-1683. doi: 10.3934/cpaa.2014.13.1669

## Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge

 1 Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, United States 2 Department of Mathematics, University of California, San Diego (UCSD), La Jolla, CA 92093-0112, United States

Received  December 2013 Revised  January 2014 Published  February 2014

We consider the Maxwell-Klein-Gordon equation in 2D in the Coulomb gauge. We establish local well-posedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution $\phi$ of the gauged Klein-Gordon equation. The main tool for handling the wave equations is the product estimate established by D'Ancona, Foschi, and Selberg. Due to low regularity, we are unable to use the conventional approaches to handle the elliptic variable $A_0$, so we provide a new approach.
Citation: Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
##### References:
 [1] Magdalena Czubak, Local wellposedness for the $2+1$-dimensional monopole equation, Anal. PDE, 3 (2010), 151-174. doi: 10.2140/apde.2010.3.151. [2] Piero D'Ancona, Damiano Foschi, and Sigmund Selberg, Product estimates for wave-Sobolev spaces in $2+1$ and $1+1$ dimensions, In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 125-150. Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/526/10379. [3] Viktor Grigoryan and Andrea R. Nahmod, Almost critical well-posedness for nonlinear wave equation with $Q_{\mu\nu}$ null forms in 2D, arXiv:1307.6194. [4] Markus Keel, Tristan Roy, and Terence Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621. doi: 10.3934/dcds.2011.30.573. [5] Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268. doi: 10.1002/cpa.3160460902. [6] Sergiu Klainerman and Matei Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. [7] Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 1209-1215, Warsaw, 1984. PWN. [8] Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces $H_{s,\delta}$, Internat. Math. Res. Notices, 17 (1996), 853-865. doi: 10.1155/S1073792896000529. [9] Sergiu Klainerman and Sigmund Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295, doi: 10.1142/S0219199702000634. [10] Sergiu Klainerman and Daniel 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. [11] Matei Machedon and Jacob Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations, J. Amer. Math. Soc., 17 (2004), 297-359. doi: 10.1090/S0894-0347-03-00445-4. [12] Vincent Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296. doi: 10.1063/1.524669. [13] Hartmut Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, arXiv:1308.1598. [14] Martin Schwarz, Jr, Global solutions of Maxwell-Higgs on Minkowski space, J. Math. Anal. Appl., 229 (1999), 426-440. doi: 10.1006/jmaa.1998.6164. [15] Sigmund Selberg, Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. [16] Sigmund Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations, 27 (2002), 1183-1227. doi: 10.1081/PDE-120004899. [17] Sigmund Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236. [18] Yi Zhou, Local existence with minimal regularity for nonlinear wave equations, Amer. J. Math., 119 (1997), 671-703.

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##### References:
 [1] Magdalena Czubak, Local wellposedness for the $2+1$-dimensional monopole equation, Anal. PDE, 3 (2010), 151-174. doi: 10.2140/apde.2010.3.151. [2] Piero D'Ancona, Damiano Foschi, and Sigmund Selberg, Product estimates for wave-Sobolev spaces in $2+1$ and $1+1$ dimensions, In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 125-150. Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/526/10379. [3] Viktor Grigoryan and Andrea R. Nahmod, Almost critical well-posedness for nonlinear wave equation with $Q_{\mu\nu}$ null forms in 2D, arXiv:1307.6194. [4] Markus Keel, Tristan Roy, and Terence Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621. doi: 10.3934/dcds.2011.30.573. [5] Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268. doi: 10.1002/cpa.3160460902. [6] Sergiu Klainerman and Matei Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. [7] Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 1209-1215, Warsaw, 1984. PWN. [8] Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces $H_{s,\delta}$, Internat. Math. Res. Notices, 17 (1996), 853-865. doi: 10.1155/S1073792896000529. [9] Sergiu Klainerman and Sigmund Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295, doi: 10.1142/S0219199702000634. [10] Sergiu Klainerman and Daniel 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. [11] Matei Machedon and Jacob Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations, J. Amer. Math. Soc., 17 (2004), 297-359. doi: 10.1090/S0894-0347-03-00445-4. [12] Vincent Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296. doi: 10.1063/1.524669. [13] Hartmut Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, arXiv:1308.1598. [14] Martin Schwarz, Jr, Global solutions of Maxwell-Higgs on Minkowski space, J. Math. Anal. Appl., 229 (1999), 426-440. doi: 10.1006/jmaa.1998.6164. [15] Sigmund Selberg, Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. [16] Sigmund Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations, 27 (2002), 1183-1227. doi: 10.1081/PDE-120004899. [17] Sigmund Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236. [18] Yi Zhou, Local existence with minimal regularity for nonlinear wave equations, Amer. J. Math., 119 (1997), 671-703.
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