Maxwell-Schrodinger system

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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

$\int F^{ab} F_{ab} + 2 \Im \overline{u} D_t u - \overline{D_j u} D_j u\ dx dt$

giving rise to the system of PDE

$iu_t = D_j u D_j u / 2 + A_0 a\,$

$\partial^aF_{ab} = J_b\,$

where the current density $J_b\,$ is given by

$J= |u|^2; J_j= -Im{\underline{u}D_ju}\,$

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 $s = \sigma = 1/2\,.$

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.
- In the Lorentz and Temporal gauges, LWP for $s \ge 5/3\,$ and $s-1 \le \sigma \le s+1, (5s-2)/3$ was established in 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 were constructed in the energy class ( $s=\sigma=1\,$ ) in the Lorentz and Coulomb gauges GuoNkSr1996.
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