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

@article{Keel2009GlobalWO, title={Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm}, author={Markus Keel and Tristan Roy and Terence Tao}, journal={arXiv: Analysis of PDEs}, year={2009} }

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 \cite{kl-mac:mkg} on finite energy data $s \geq 1$, and Eardley-Moncrief \cite{eardley} 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…

## 39 Citations

### Global well-posedness for the energy critical massive Maxwell-Klein-Gordon equation with small data

- Mathematics
- 2016

In this paper we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $\mathbb{R}^{1+4}$ for data with small energy. The results…

### Local well-posedness for the (n+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

- Mathematics
- 2016

This is an extension of the paper [14] by the author for the 2+1 dimensional Maxwell-Klein-Gordon equations in temporal gauge to the n+1 dimensional situation for $n \ge 3$. They are shown to be…

### Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data

- MathematicsAnnals of PDE
- 2019

In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on $$\mathbb {R}^{1+d}$$R1+d$$(d \ge 4)$$(d≥4) for data with…

### Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge

- Mathematics
- 2013

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…

### Local Well-Posedness of the (4 + 1)-Dimensional Maxwell–Klein–Gordon Equation at Energy Regularity

- Mathematics
- 2015

This paper is the first part of a trilogy [22, 23] dedicated to a proof of global well-posedness and scattering of the $$(4+1)$$(4+1)-dimensional mass-less Maxwell–Klein–Gordon equation (MKG) for any…

### Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

- Mathematics
- 2015

This article constitutes the final and main part of a three-paper sequence (Ann PDE, 2016. doi:10.1007/s40818-016-0006-4; Oh and Tataru, 2015. arXiv:1503.01561), whose goal is to prove global…

### Local well-posedness for low regularity data for the higher dimensional Maxwell-Klein-Gordon system in Lorenz gauge

- MathematicsJournal of Mathematical Physics
- 2018

The Cauchy problem for the Maxwell-Klein-Gordon equations in the Lorenz gauge in n space dimensions (n ≥ 4) is shown to be locally well-posed for low regularity (large) data. The result relies on the…

### Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equations in Lorenz gauge

- Mathematics
- 2017

The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in $n$ space dimensions ($n \ge 2$) is locally well-posed for low regularity data, in two and three space dimensions even for…

### Numerical Integrators for Maxwell-Klein-Gordon and Maxwell-Dirac Systems in Highly to Slowly Oscillatory Regimes

- Physics, Mathematics
- 2017

Maxwell-Klein-Gordon (MKG) and Maxwell-Dirac (MD) systems physically describe the mutual interaction of moving relativistic particles with their self-generated electromagnetic field. Solving these…

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