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 & 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,, \emph{Anal. PDE}, 3 (2010), 151. 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 \emph{Nonlinear partial differential equations and hyperbolic wave phenomena}, (2010), 125. 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,, \emph{Discrete Contin. Dyn. Syst.}, 30 (2011), 573. doi: 10.3934/dcds.2011.30.573.

[5]

Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem,, \emph{Comm. Pure Appl. Math.}, 46 (1993), 1221. doi: 10.1002/cpa.3160460902.

[6]

Sergiu Klainerman and Matei Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, \emph{Duke Math. J.}, 74 (1994), 19.

[7]

Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations,, In \emph{Proceedings of the International Congress of Mathematicians, (1983), 1209.

[8]

Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces $H_{s,\delta}$,, \emph{Internat. Math. Res. Notices}, 17 (1996), 853. doi: 10.1155/S1073792896000529.

[9]

Sergiu Klainerman and Sigmund Selberg, Bilinear estimates and applications to nonlinear wave equations,, \emph{Commun. Contemp. Math.}, 4 (2002), 223. doi: 10.1142/S0219199702000634.

[10]

Sergiu Klainerman and Daniel Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$,, \emph{J. Amer. Math. Soc.}, 12 (1999), 93. 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,, \emph{J. Amer. Math. Soc.}, 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4.

[12]

Vincent Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$-dimensional spacetime,, \emph{J. Math. Phys.}, 21 (1980), 2291. 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,, \emph{J. Math. Anal. Appl.}, 229 (1999), 426. 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, (1999).

[16]

Sigmund Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1183. doi: 10.1081/PDE-120004899.

[17]

Sigmund Selberg, On an estimate for the wave equation and applications to nonlinear problems,, \emph{Differential Integral Equations}, 15 (2002), 213.

[18]

Yi Zhou, Local existence with minimal regularity for nonlinear wave equations,, \emph{Amer. J. Math.}, 119 (1997), 671.

show all references

References:
[1]

Magdalena Czubak, Local wellposedness for the $2+1$-dimensional monopole equation,, \emph{Anal. PDE}, 3 (2010), 151. 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 \emph{Nonlinear partial differential equations and hyperbolic wave phenomena}, (2010), 125. 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,, \emph{Discrete Contin. Dyn. Syst.}, 30 (2011), 573. doi: 10.3934/dcds.2011.30.573.

[5]

Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem,, \emph{Comm. Pure Appl. Math.}, 46 (1993), 1221. doi: 10.1002/cpa.3160460902.

[6]

Sergiu Klainerman and Matei Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, \emph{Duke Math. J.}, 74 (1994), 19.

[7]

Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations,, In \emph{Proceedings of the International Congress of Mathematicians, (1983), 1209.

[8]

Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces $H_{s,\delta}$,, \emph{Internat. Math. Res. Notices}, 17 (1996), 853. doi: 10.1155/S1073792896000529.

[9]

Sergiu Klainerman and Sigmund Selberg, Bilinear estimates and applications to nonlinear wave equations,, \emph{Commun. Contemp. Math.}, 4 (2002), 223. doi: 10.1142/S0219199702000634.

[10]

Sergiu Klainerman and Daniel Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$,, \emph{J. Amer. Math. Soc.}, 12 (1999), 93. 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,, \emph{J. Amer. Math. Soc.}, 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4.

[12]

Vincent Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$-dimensional spacetime,, \emph{J. Math. Phys.}, 21 (1980), 2291. 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,, \emph{J. Math. Anal. Appl.}, 229 (1999), 426. 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, (1999).

[16]

Sigmund Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1183. doi: 10.1081/PDE-120004899.

[17]

Sigmund Selberg, On an estimate for the wave equation and applications to nonlinear problems,, \emph{Differential Integral Equations}, 15 (2002), 213.

[18]

Yi Zhou, Local existence with minimal regularity for nonlinear wave equations,, \emph{Amer. J. Math.}, 119 (1997), 671.

[1]

Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034

[2]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573

[3]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

[4]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[5]

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

[6]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321

[7]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903

[8]

Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061

[9]

Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036

[10]

Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005

[11]

Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030

[12]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261

[13]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

[14]

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

[15]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135

[16]

Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097

[17]

Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271

[18]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

[19]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[20]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]