Maxwell-Schrodinger system

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Maxwell-Schrodinger system in

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 in the Coulomb gauge was established by Bejenaru and Tataru in 2007. The argument also gives a priori estimates when and LWP when .
    • In the Lorenz and Temporal gauges, LWP for and was established in NkrWad-p
    • For smooth data () in the Lorenz gauge this is in NkTs1986 (this result works in all dimensions)
    • Global weak solutions were constructed in the energy class () in the Lorenz 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