Maxwell-Schrodinger system

Maxwell-Schrodinger system in $R^{3}$

This system is a partially non-relativistic analogue of the Maxwell-Klein-Gordon system, coupling a U(1) connection $A_{a}\,$ with a complex scalar field u; it is thus an example of a wave-Schrodinger system. The Lagrangian density is

$\int F^{ab}F_{ab}+2\Im {\overline {u}}D_{t}u-{\overline {D_{j}u}}D_{j}u\ dxdt$

giving rise to the system of PDE

$iu_{t}=D_{j}uD_{j}u/2+A_{0}a\,$ $F_{ab}=J_{b}\,$

where the current density $J_{b}\,$ is given by

$J=|u|^{2};J_{j}=-Im{{\underline {u}}D_{j}u}\,$

As with the MKG system, there is a gauge invariance for the connection; one can place A in the Lorentz, Coulomb, or Temporal gauges (other choices are of course possible).

Let us place u in $H^{s}\,$, and A in $H^{\sigma }H^{sigma-1}\,.$ The lack of scale invariance makes it difficult to exactly state what the critical regularity would be, but it seems to be $s=\sigma =1/2\,.$

• In the Lorentz and Temporal gauges, one has LWP for $s\geq 5/3\,$ and $s-1\leq \sigma \leq s+1,(5s-2)/3$ [NkrWad-p]
  • For smooth data ($s=\sigma >5/2\,$) in the Lorentz gauge this is in NkTs1986 (this result works in all dimensions)
• Global weak solutions are known in the energy class ($s=\sigma =1\,$) in the Lorentz and Coulomb gauges GuoNkSr1996. GWP is still open however.
• Modified wave operators have been constructed in the Coulomb gauge in the case of vanishing magnetic field in [GiVl-p3], [GiVl-p5]. No smallness condition is needed on the data at infinity.
  • A similar result for small data is in Ts1993
• In one dimension, GWP in the energy class is known Ts1995
• In two dimensions, GWP for smooth solutions is known TsNk1985