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 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 x 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 >= 5/3 and s-1 <= sigma <= 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