Maxwell-Schrodinger system in R3
This system is a partially non-relativistic analogue of the Maxwell-Klein-Gordon system, coupling a U(1) connection with a complex scalar field u; it is thus an example of a wave-Schrodinger system. The Lagrangian density is
giving rise to the system of PDE
where the current density is given by
As with the MKG system, there is a gauge invariance for the connection; one can place A in the Lorenz, Coulomb, or Temporal gauges (other choices are of course possible).
Let us place u in , and A in The lack of scale invariance makes it difficult to exactly state what the critical regularity would be, but it seems to be
- GWP in the energy space s = σ = 1 in the Coulomb gauge was established by Bejenaru and Tataru in 2007. The argument also gives a priori estimates when s > 1 / 2,σ = 1 and LWP when s > 3 / 4,σ = 1.
- 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