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… 

Figures from this paper

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

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

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

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

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

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

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

  • H. Pecher
  • Mathematics
    Journal 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

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

On global behavior of solutions of the Maxwell-Klein-Gordon equations

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

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

References

SHOWING 1-10 OF 43 REFERENCES

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

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

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 $\T$

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

Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$

The initial value problem for the cubic defocusing nonlinear Schrodinger equation $i \partial_t u + \Delta u = |u|^2 u$ on theplane is shown to be globally well-posed for initial data in $H^s

Global Well-Posedness for Schrödinger Equations with Derivative

We prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in Hs for s>2/3, for small L2 data. The result follows from an application of the

Adapted Linear-Nonlinear Decomposition And Global Well-posedness For Solutions To The Defocusing Cubic Wave Equation On $\mathbb{R}^{3}$

We prove global well-posedness for the defocusing cubic wave equation with data in $H^{s} \times H^{s-1}$, $1>s>{13/18}$. The main task is to estimate the variation of an almost conserved quantity on

Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data

We prove local and global existence from large, rough initial data for a wave map between 1+1 dimensional Minkowski space and an analytic manifold. Included here is global existence for large data in

A Refined Global Well-Posedness Result for Schrödinger Equations with Derivative

In this paper we prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in Hs for $s > \frac12$ for data small in L2 . To understand the

Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R^2

The initial value problem for the cubic defocusing nonlinear Schr\"odinger equation $i \partial_t u + \Delta u = |u|^2 u$ on the plane is shown to be globally well-posed for initial data in $H^s