## Duke Mathematical Journal

### Global well-posedness for the Maxwell–Klein–Gordon equation in $4+1$ dimensions: Small energy

#### Abstract

We prove that the critical Maxwell–Klein–Gordon equation on ${\mathbb{R}}^{4+1}$ is globally well-posed for smooth initial data which are small in the energy norm. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.

#### Article information

Source
Duke Math. J., Volume 164, Number 6 (2015), 973-1040.

Dates
First available in Project Euclid: 17 April 2015

https://projecteuclid.org/euclid.dmj/1429282677

Digital Object Identifier
doi:10.1215/00127094-2885982

Mathematical Reviews number (MathSciNet)
MR3336839

Zentralblatt MATH identifier
1329.35209

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 70S15: Yang-Mills and other gauge theories

#### Citation

Krieger, Joachim; Sterbenz, Jacob; Tataru, Daniel. Global well-posedness for the Maxwell–Klein–Gordon equation in $4+1$ dimensions: Small energy. Duke Math. J. 164 (2015), no. 6, 973--1040. doi:10.1215/00127094-2885982. https://projecteuclid.org/euclid.dmj/1429282677

#### References

• [1] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
• [2] P. Bizoń and Z. Tabor, On blowup of Yang-Mills fields, Phys. Rev. D (3) 64 (2001), no. 12, art. no. 121701.
• [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc., Providence, 1999.
• [4] T. Cazenave, J. Shatah, and A. S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincaré Phys. Th. 68 (1998), 315–349.
• [5] S. Cuccagna, On the local existence for the Maxwell-Klein-Gordon system in $R^{3+1}$, Comm. Partial Differential Equations 24 (1999), 851–867.
• [6] D. M. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space, I: Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), 171–191.
• [7] M. Keel, T. Roy, and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst. 30 (2011), 573–621.
• [8] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.
• [9] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J. 74 (1994), 19–44.
• [10] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J. 81 (1995), 99–133.
• [11] S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Differential Integral Equations 10 (1997), 1019–1030.
• [12] S. Klainerman and M. Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. J. 87 (1997), 553–589.
• [13] S. Klainerman and I. Rodnianski, On the global regularity of wave maps in the critical Sobolev norm, Int. Math. Res. Not. IMRN 2001, no. 13, 655–677.
• [14] S. Klainerman and I. Rodnianski, Improved local well-posedness for quasilinear wave equations in dimension three, Duke Math. J. 117 (2003), 1–124.
• [15] S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math. 4 (2002), 223–295.
• [16] S. Klainerman and D. Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$, J. Amer. Math. Soc. 12 (1999), 93–116.
• [17] J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, EMS Monogr. Math., Eur. Math. Soc., Zürich, 2012.
• [18] J. Krieger and J. Sterbenz, Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space, Mem. Amer. Math. Soc. 223 (2013), no. 1047.
• [19] 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), 297–359.
• [20] A. Nahmod, A. Stefanov, and K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom. 11 (2003), 49–83.
• [21] I. Rodnianski and T. Tao, Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions, Comm. Math. Phys. 251 (2004), 377–426.
• [22] S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations 27 (2002), 1183–1227.
• [23] J. Shatah and M. Struwe, The Cauchy problem for wave maps, Int. Math. Res. Not. IMRN 2002, no. 11, 555–571.
• [24] H. F. Smith and D. Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2) 162 (2005), 291–366.
• [25] J. Sterbenz, Global regularity and scattering for general non-linear wave equations, II: $(4+1)$ dimensional Yang-Mills equations in the Lorentz gauge, Amer. J. Math. 129 (2007), 611–664.
• [26] J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys. 298 (2010), 139–230.
• [27] T. Tao, Global regularity of wave maps, I: Small critical Sobolev norm in high dimension, Int. Math. Res. Not. IMRN 2001, no. 6, 299–328.
• [28] T. Tao, Global regularity of wave maps, II: Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443–544.
• [29] D. Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), 349–376.
• [30] D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), 37–77.
• [31] D. Tataru, Rough solutions for the wave maps equation, Amer. J. Math. 127 (2005), 293–377.
• [32] M. E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Math. Surveys Monogr. 81, Amer. Math. Soc., Providence, 2000.
• [33] K. K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), 31–42.