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:
[1]

J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", AMS Publications, (1999).

[2]

H. Bahouri and J-Y. Chemin, On global well-posedness for defocusing cubic wave equation,, Int. Math. Res. Not., (2006).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

I. Gallagher and F. Planchon, On global solutions to a defocusing semi-linear wave equation,, Revista Mat. Iberoamericana, 19 (2003), 161.

[14]

L. Kapitanski, Weak and yet weaker solutions of semilinear wave equations,, Comm. Partial Differential Equations, 19 (1994), 1629. doi: 10.1080/03605309408821067.

[15]

M. Keel and T. Tao, Endpoint strichartz estimates,, Amer. Math. J., 120 (1998), 955. doi: 10.1353/ajm.1998.0039.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories,, Diff. and Integral Eq., 10 (1997), 1019.

[26]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations,, Commun. Contemp. Math., 4 (2002), 223. doi: 10.1142/S0219199702000634.

[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.

[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.

[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.

[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.

[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.

[32]

T. Roy, Global well-posedness for the radial defocusing cubic wave equation and for rough data,, Elec. J. Diff. Eq., 166 (2007), 1.

[33]

S. Selberg, "Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Princeton University Thesis, (1999).

[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.

[35]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970).

[36]

C. D. Sogge, "Lectures on Nonlinear Wave Equations,", Monographs in Analysis II, (1995).

[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.

[38]

T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS regional conference series in mathematics, (2006).

[39]

K. Uhlenbeck, Connections with $L^p$ bounds on curvature,, Comm. Math. Phys., 83 (1982), 31. doi: 10.1007/BF01947069.

show all references

References:
[1]

J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", AMS Publications, (1999).

[2]

H. Bahouri and J-Y. Chemin, On global well-posedness for defocusing cubic wave equation,, Int. Math. Res. Not., (2006).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

I. Gallagher and F. Planchon, On global solutions to a defocusing semi-linear wave equation,, Revista Mat. Iberoamericana, 19 (2003), 161.

[14]

L. Kapitanski, Weak and yet weaker solutions of semilinear wave equations,, Comm. Partial Differential Equations, 19 (1994), 1629. doi: 10.1080/03605309408821067.

[15]

M. Keel and T. Tao, Endpoint strichartz estimates,, Amer. Math. J., 120 (1998), 955. doi: 10.1353/ajm.1998.0039.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories,, Diff. and Integral Eq., 10 (1997), 1019.

[26]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations,, Commun. Contemp. Math., 4 (2002), 223. doi: 10.1142/S0219199702000634.

[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.

[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.

[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.

[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.

[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.

[32]

T. Roy, Global well-posedness for the radial defocusing cubic wave equation and for rough data,, Elec. J. Diff. Eq., 166 (2007), 1.

[33]

S. Selberg, "Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Princeton University Thesis, (1999).

[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.

[35]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970).

[36]

C. D. Sogge, "Lectures on Nonlinear Wave Equations,", Monographs in Analysis II, (1995).

[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.

[38]

T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS regional conference series in mathematics, (2006).

[39]

K. Uhlenbeck, Connections with $L^p$ bounds on curvature,, Comm. Math. Phys., 83 (1982), 31. doi: 10.1007/BF01947069.

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