# 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}
}
• Published 9 October 2009
• Mathematics
• arXiv: Analysis of PDEs
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…

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## References

SHOWING 1-10 OF 43 REFERENCES

### Finite energy solutions of the Yang-Mills equations in $\mathbb{R}^{3+1}$

• Mathematics
• 1995
Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the

### Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations

• Mathematics
• 2003
This paper contains a detailed study of the local-in-time regularity properties of the Maxwell–Klein–Gordon (MKG) equations. The MKG equations represent a physical model for the interaction of a spin

### ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

ABSTRACT We prove that the Maxwell-Klein-Gordon system on relative to the Coulomb gauge is locally well-posed for initial data in for all ϵ > 0. This builds on previous work by Klainerman and

### Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$

• Mathematics
• 2001
The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to

• Mathematics
• 2008