American Institute of Mathematical Sciences

August  2011, 30(3): 573-621. doi: 10.3934/dcds.2011.30.573

Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

 1 Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455 2 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, United States 3 Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095

Received  May 2010 Revised  December 2010 Published  March 2011

We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of Klainerman-Machedon [24] on finite energy data $s \geq 1$, and Eardley-Moncrief [11] for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the $L^2_x$ norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.
Citation: 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
References:
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Tao, A refined global well-posedness for the Schrodinger equations with derivative,, SIAM J. Math., 34 (2002), 64. doi: 10.1137/S0036141001394541. Google Scholar [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\R$ and $\T$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications,, J. Funct. Anal., 211 (2004), 173. doi: 10.1016/S0022-1236(03)00218-0. Google Scholar [9] J. Colliander, M. Keel, G. Staffilani, H. Takoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrodinger on $\R^2$,, Disc. Cont. Dynam. Systems A, 21 (2008), 665. doi: 10.3934/dcds.2008.21.665. Google Scholar [10] S. Cuccagna, On the local existence for the Maxwell Klein Gordon system in $\R^{3+1}$,, Comm. Partial Differential Equations, 24 (1999), 851. doi: 10.1080/03605309908821449. Google Scholar [11] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in $\R^{3+1}$,, Comm. Math. Phys., 83 (1982), 171. doi: 10.1007/BF01976040. Google Scholar [12] D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations,, Les Annales Scientifiques de l'Ecole Normale Superieure, 33 (2000), 211. doi: 10.1016/S0012-9593(00)00109-9. Google Scholar [13] I. Gallagher and F. Planchon, On global solutions to a defocusing semi-linear wave equation,, Revista Mat. Iberoamericana, 19 (2003), 161. Google Scholar [14] L. Kapitanski, Weak and yet weaker solutions of semilinear wave equations,, Comm. Partial Differential Equations, 19 (1994), 1629. doi: 10.1080/03605309408821067. Google Scholar [15] M. Keel and T. Tao, Endpoint strichartz estimates,, Amer. Math. J., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar [16] M. Keel and T. Tao, Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data,, Internat. Math. Res. Not., 21 (1998), 1117. doi: 10.1155/S107379289800066X. Google Scholar [17] C. Kenig, G. Ponce and L. Vega, Global well-posedness for semi-linear wave equations,, Comm. Partial Differential Equations, 25 (2000), 1741. doi: 10.1080/03605300008821565. Google Scholar [18] S. Klainerman, On the regularity of classical field theories in Minkowski space-time $\R^{3+1}$,, Prog. in Nonlin. Diff. Eq. and their Applic., 29 (1997), 113. Google Scholar [19] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem,, Comm. Pure Appl. Math., 46 (1993), 1221. doi: 10.1002/cpa.3160460902. Google Scholar [20] S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in $\R^{3+1}$,, Ann. of Math., 142 (1995), 39. doi: 10.2307/2118611. Google Scholar [21] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications,, Duke Math J., 81 (1995), 99. doi: 10.1215/S0012-7094-95-08109-5. Google Scholar [22] S. Klainerman and M. Machedon, Remark on Strichartz-type inequalities,, With appendices by Jean Bourgain and Daniel Tataru. Internat. Math. Res. Notices, 5 (1996), 201. doi: 10.1155/S1073792896000153. Google Scholar [23] S. Klainerman and M. Machedon, Estimates for null forms and the spaces $H_{s,\delta}$,, Internat. Math. Res. Notices, 17 (1996), 853. doi: 10.1155/S1073792896000529. Google Scholar [24] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4. Google Scholar [25] S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories,, Diff. and Integral Eq., 10 (1997), 1019. Google Scholar [26] 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 [27] S. Klainerman and D. Tataru, On the optimal 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 [28] S. Klainerman, I. Rodnianski and T. Tao, A physical approach to wave equation bilinear estimate,, Dedicated to the memory of Thomas H. Wolff, 87 (2002), 299. doi: 10.1007/BF02868479. Google Scholar [29] H. Lindblad and C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, J. Funct. Anal., 130 (1995), 357. doi: 10.1006/jfan.1995.1075. Google Scholar [30] M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations,, J. Amer. Math. Soc., 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4. Google Scholar [31] T. Roy, Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation On $\R^{3}$,, Discrete Contin. Dyn. Syst., 24 (2009), 1307. doi: 10.3934/dcds.2009.24.1307. Google Scholar [32] T. Roy, Global well-posedness for the radial defocusing cubic wave equation and for rough data,, Elec. J. Diff. Eq., 166 (2007), 1. Google Scholar [33] S. Selberg, "Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Princeton University Thesis, (1999). Google Scholar [34] S. Selberg, Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in 1+4 dimensions,, Communications in PDE, 27 (2002), 1183. doi: 10.1081/PDE-120004899. Google Scholar [35] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970). Google Scholar [36] C. D. Sogge, "Lectures on Nonlinear Wave Equations,", Monographs in Analysis II, (1995). Google Scholar [37] T. Tao, Multilinear weighted convolution of $L^2_x$ functions, and applications to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839. doi: 10.1353/ajm.2001.0035. Google Scholar [38] T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS regional conference series in mathematics, (2006). Google Scholar [39] K. Uhlenbeck, Connections with $L^p$ bounds on curvature,, Comm. Math. Phys., 83 (1982), 31. doi: 10.1007/BF01947069. Google Scholar

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References:
 [1] J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", AMS Publications, (1999). Google Scholar [2] H. Bahouri and J-Y. Chemin, On global well-posedness for defocusing cubic wave equation,, Int. Math. Res. Not., (2006). Google Scholar [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for KdV in Sobolev spaces of negative index,, Elec. J. Diff. Eq., 2001 (2001), 1. Google Scholar [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for the Schrodinger equations with derivative,, SIAM J. Math., 33 (2001), 649. doi: 10.1137/S0036141001384387. Google Scholar [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation,, Math. Res. Letters, 9 (2002), 659. Google Scholar [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for the Schrodinger equations with derivative,, SIAM J. Math., 34 (2002), 64. doi: 10.1137/S0036141001394541. Google Scholar [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\R$ and $\T$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications,, J. Funct. Anal., 211 (2004), 173. doi: 10.1016/S0022-1236(03)00218-0. Google Scholar [9] J. Colliander, M. Keel, G. Staffilani, H. Takoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrodinger on $\R^2$,, Disc. Cont. Dynam. Systems A, 21 (2008), 665. doi: 10.3934/dcds.2008.21.665. Google Scholar [10] S. Cuccagna, On the local existence for the Maxwell Klein Gordon system in $\R^{3+1}$,, Comm. Partial Differential Equations, 24 (1999), 851. doi: 10.1080/03605309908821449. Google Scholar [11] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in $\R^{3+1}$,, Comm. Math. Phys., 83 (1982), 171. doi: 10.1007/BF01976040. Google Scholar [12] D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations,, Les Annales Scientifiques de l'Ecole Normale Superieure, 33 (2000), 211. doi: 10.1016/S0012-9593(00)00109-9. Google Scholar [13] I. Gallagher and F. Planchon, On global solutions to a defocusing semi-linear wave equation,, Revista Mat. Iberoamericana, 19 (2003), 161. Google Scholar [14] L. Kapitanski, Weak and yet weaker solutions of semilinear wave equations,, Comm. Partial Differential Equations, 19 (1994), 1629. doi: 10.1080/03605309408821067. Google Scholar [15] M. Keel and T. Tao, Endpoint strichartz estimates,, Amer. Math. J., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar [16] M. Keel and T. Tao, Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data,, Internat. Math. Res. Not., 21 (1998), 1117. doi: 10.1155/S107379289800066X. Google Scholar [17] C. Kenig, G. Ponce and L. Vega, Global well-posedness for semi-linear wave equations,, Comm. Partial Differential Equations, 25 (2000), 1741. doi: 10.1080/03605300008821565. Google Scholar [18] S. Klainerman, On the regularity of classical field theories in Minkowski space-time $\R^{3+1}$,, Prog. in Nonlin. Diff. Eq. and their Applic., 29 (1997), 113. Google Scholar [19] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem,, Comm. Pure Appl. Math., 46 (1993), 1221. doi: 10.1002/cpa.3160460902. Google Scholar [20] S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in $\R^{3+1}$,, Ann. of Math., 142 (1995), 39. doi: 10.2307/2118611. Google Scholar [21] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications,, Duke Math J., 81 (1995), 99. doi: 10.1215/S0012-7094-95-08109-5. Google Scholar [22] S. Klainerman and M. Machedon, Remark on Strichartz-type inequalities,, With appendices by Jean Bourgain and Daniel Tataru. Internat. Math. Res. Notices, 5 (1996), 201. doi: 10.1155/S1073792896000153. Google Scholar [23] S. Klainerman and M. Machedon, Estimates for null forms and the spaces $H_{s,\delta}$,, Internat. Math. Res. Notices, 17 (1996), 853. doi: 10.1155/S1073792896000529. Google Scholar [24] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4. Google Scholar [25] S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories,, Diff. and Integral Eq., 10 (1997), 1019. Google Scholar [26] 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 [27] S. Klainerman and D. Tataru, On the optimal 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 [28] S. Klainerman, I. Rodnianski and T. Tao, A physical approach to wave equation bilinear estimate,, Dedicated to the memory of Thomas H. Wolff, 87 (2002), 299. doi: 10.1007/BF02868479. Google Scholar [29] H. Lindblad and C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, J. Funct. Anal., 130 (1995), 357. doi: 10.1006/jfan.1995.1075. Google Scholar [30] M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations,, J. Amer. Math. Soc., 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4. Google Scholar [31] T. Roy, Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation On $\R^{3}$,, Discrete Contin. Dyn. Syst., 24 (2009), 1307. doi: 10.3934/dcds.2009.24.1307. Google Scholar [32] T. Roy, Global well-posedness for the radial defocusing cubic wave equation and for rough data,, Elec. J. Diff. Eq., 166 (2007), 1. Google Scholar [33] S. Selberg, "Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Princeton University Thesis, (1999). Google Scholar [34] S. Selberg, Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in 1+4 dimensions,, Communications in PDE, 27 (2002), 1183. doi: 10.1081/PDE-120004899. Google Scholar [35] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970). Google Scholar [36] C. D. Sogge, "Lectures on Nonlinear Wave Equations,", Monographs in Analysis II, (1995). Google Scholar [37] T. Tao, Multilinear weighted convolution of $L^2_x$ functions, and applications to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839. doi: 10.1353/ajm.2001.0035. Google Scholar [38] T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS regional conference series in mathematics, (2006). Google Scholar [39] K. Uhlenbeck, Connections with $L^p$ bounds on curvature,, Comm. Math. Phys., 83 (1982), 31. doi: 10.1007/BF01947069. Google Scholar
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