Maxwell-Klein-Gordon equations

The Maxwell-Klein-Gordon equation is the special case of the Yang-Mills-Higgs equation when the Lie group G is just the circle U(1), and there is no potential energy term $V(\phi )$. Thus A is now purely imaginary, and $\phi$ is complex.

Overview

The Maxwell-Klein-Gordon equation is the Maxwell equation coupled with a massless Klein-Gordon equation (i.e. a free wave equation). If the scalar field $\phi$ is set to 0, the equation collapses to the linear Maxwell equations, which are basically a vector-valued variant of the free wave equation.

As with Yang-Mills, the three standard gauges are the Lorenz gauge, the Coulomb gauge, and the temporal gauge. The Lorenz gauge is most natural from a co-ordinate free viewpoint, but is difficult to work with technically. In principle the temporal gauge is the easiest to work with, being local in space, but in practice the Coulomb gauge is preferred because the null form structure of Maxwell-Klein-Gordon is most apparent in this gauge.

In the Coulomb gauge, MKG has the schematic form

$\Delta A_{0}=O(\phi \phi _{t})+O(\Phi ^{3})$ $\Box A=\nabla ^{-1}Q(\phi ,\phi )$ $\Box \phi =Q(\nabla ^{-1}A,\phi )+O((A_{0})_{t}\phi )+O(A_{0}\phi _{t})+O(\Phi ^{3})$

where $O(\Phi ^{3})$ denotes terms that are cubic in $(A_{0},A,\phi )$. Unfortunately, the equation for the A_0 portion of the Coloumb gauge is elliptic, which generates some low frequency issues. However, if we ignore the A_0 terms and the cubic terms then we reduce to the model equation

$\Box A=\nabla ^{-1}Q(\phi ,\phi )$ $\Box \phi =Q(\nabla ^{-1}A,\phi )$

which is slightly better than the corresponding model for Yang-Mills.

MKG has the advantage over YM that the Coulomb gauge is easily constructed globally using Riesz transforms, so there are less technical issues involved with this gauge.

Maxwell-Klein-Gordon on R

• Scaling is s_c = -1/2.
• LWP can be shown in the temporal gauge for s>1/2 by energy estimates. For s<1/2 one begins to have difficulty interpreting the solution even in the distributional sense, but this might be avoidable, perhaps by a good choice of gauge. (The Coulomb gauge seems to have some technical difficulties however).
• GWP is easy to show in the temporal gauge for s \geq 1 by energy methods and Hamiltonian conservation. Presumably one can improve the s \geq 1 constraint substantially.

Maxwell-Klein-Gordon on R²

• Scaling is s_c = 0.
• Heuristically, one expects X^{s,\delta} methods to give LWP for s > 1/4, but we do not know if this has been done rigorously.
  • Strichartz estimates give s > 1/2 (PoSi1993), while energy methods give s>1.
• GWP is known for smooth data in the temporal gauge (Mc1980).
  • This should extend to s \geq 1 and probably below, but we do not know if this is in the literature.

Maxwell-Klein-Gordon on R³

• Scaling is s_c = 1/2.
• LWP for s>1/2 in the Coulomb Gauge MaStz-p
  • For the model equation, LWP fails for s < 3/4 MaStz-p. Thus the MKG result exploits additional structure in the MKG equation which is not present in the model equation.
  • For s>3/4 this was proven in the Coloumb gauge in Cu1999.
  • For s\geq1 this was proven in the Coulomb and Temporal gauges in KlMa1994.
  • For s>1 this follows (in any of the three gauges) from Strichartz estimates PoSi1993
  • For s>3/2 this follows (in any of the three gauges) from energy estimates.
  • There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorenz gauge.
  • The endpoint s=1/2 looks extremely difficult, even for the model equation. Perhaps things would be easier if one only had to deal with the null form $\nabla ^{-1}Q(\phi ,\phi )$, as this is slightly smoother than $Q(\nabla ^{-1}A,\phi )$.
• GWP for s>7/8 in the Coloumb gauge KeTa-p.
  • For s\geq 1 this was proven in KlMa1997.
  • For smooth data this was proven in EaMc1982.
• For physical applications it is of interest to study MKG when the scalar field $\phi$ propagates with a strictly slower velocity than the electromagnetic field A. In this case one cannot exploit the null form estimates; nevertheless, the estimates are more favourable, mainly because the two light cones are now transverse. Indeed, one has GWP for s\geq1 in all three standard gauges Tg2000. The local and global theory for this equation may well be improvable.
• In the nonrelativistic limit this equation converges to a Maxwell-Poisson system MasNa2003