# American Institute of Mathematical Sciences

November  2016, 15(6): 2203-2219. doi: 10.3934/cpaa.2016034

## Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

 1 Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal

Received  December 2015 Revised  July 2016 Published  September 2016

The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.
Citation: 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
##### References:
 [1] P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions,, \emph{Contemporary Math.}, 526 (2010), 125. doi: 10.1090/conm/526/10379. Google Scholar [2] M. Beals, Self-spreading of singularities for solutions to semilinear wave equations,, \emph{Annals Math.}, 118 (1983), 187. doi: 10.2307/2006959. Google Scholar [3] M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 1669. doi: 10.3934/cpaa.2014.13.1669. Google Scholar [4] S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities,, \emph{Int. Math. Res. Notices}, 5 (1996), 201. doi: 10.1155/S1073792896000153. Google Scholar [5] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, \emph{Duke Math. J.}, 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4. Google Scholar [6] M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations,, \emph{J. AMS}, 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4. Google Scholar [7] V. 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. Google Scholar [8] H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge,, \emph{Adv. Diff. Equ.}, 19 (2014), 359. Google Scholar [9] H. Pecher, Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge,, \emph{Adv. Diff. Equ.}, 20 (2015), 1009. Google Scholar [10] S. Selberg, Multilinear Space-time Estimates and Applications to Local Exisatence Theory for Nonlinear Wave Equations,, PhD. Thesis Princeton, (1999). Google Scholar [11] S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation,, \emph{Int. Math. Res. Not.}, (2008). Google Scholar [12] S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge,, \emph{Comm. PDE}, 35 (2010), 1029. doi: 10.1080/03605301003717100. Google Scholar [13] T. Tao, Multilinear weighted convolutions of $L^2$-functions and applications to non-linear dispersive equations,, \emph{Amer. J. Math.}, 123 (2001), 838. Google Scholar [14] T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm,, \emph{J. Diff. Equ.}, 189 (2003), 366. doi: 10.1016/S0022-0396(02)00177-8. Google Scholar [15] J. Yuan, Global solutions of two coupled Maxwell systems in the temporal gauge,, \emph{Discr. Cont. Dyn. Syst.}, 36 (2016), 1709. doi: 10.3934/dcds.2016.36.1709. Google Scholar

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##### References:
 [1] P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions,, \emph{Contemporary Math.}, 526 (2010), 125. doi: 10.1090/conm/526/10379. Google Scholar [2] M. Beals, Self-spreading of singularities for solutions to semilinear wave equations,, \emph{Annals Math.}, 118 (1983), 187. doi: 10.2307/2006959. Google Scholar [3] M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 1669. doi: 10.3934/cpaa.2014.13.1669. Google Scholar [4] S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities,, \emph{Int. Math. Res. Notices}, 5 (1996), 201. doi: 10.1155/S1073792896000153. Google Scholar [5] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, \emph{Duke Math. J.}, 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4. Google Scholar [6] M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations,, \emph{J. AMS}, 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4. Google Scholar [7] V. 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. Google Scholar [8] H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge,, \emph{Adv. Diff. Equ.}, 19 (2014), 359. Google Scholar [9] H. Pecher, Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge,, \emph{Adv. Diff. Equ.}, 20 (2015), 1009. Google Scholar [10] S. Selberg, Multilinear Space-time Estimates and Applications to Local Exisatence Theory for Nonlinear Wave Equations,, PhD. Thesis Princeton, (1999). Google Scholar [11] S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation,, \emph{Int. Math. Res. Not.}, (2008). Google Scholar [12] S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge,, \emph{Comm. PDE}, 35 (2010), 1029. doi: 10.1080/03605301003717100. Google Scholar [13] T. Tao, Multilinear weighted convolutions of $L^2$-functions and applications to non-linear dispersive equations,, \emph{Amer. J. Math.}, 123 (2001), 838. Google Scholar [14] T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm,, \emph{J. Diff. Equ.}, 189 (2003), 366. doi: 10.1016/S0022-0396(02)00177-8. Google Scholar [15] J. Yuan, Global solutions of two coupled Maxwell systems in the temporal gauge,, \emph{Discr. Cont. Dyn. Syst.}, 36 (2016), 1709. doi: 10.3934/dcds.2016.36.1709. Google Scholar
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