# Maxwell-Schrodinger system

### Maxwell-Schrodinger system in ${\displaystyle R^{3}}$

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

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

giving rise to the system of PDE

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

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

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

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

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

• GWP in the energy space ${\displaystyle s=\sigma =1}$ in the Coulomb gauge was established by Bejenaru and Tataru in 2007. The argument also gives a priori estimates when ${\displaystyle s>1/2,\sigma =1}$ and LWP when ${\displaystyle s>3/4,\sigma =1}$.
• In the Lorenz and Temporal gauges, LWP for ${\displaystyle s\geq 5/3\,}$ and ${\displaystyle s-1\leq \sigma \leq s+1,(5s-2)/3}$ was established in NkrWad-p
• For smooth data (${\displaystyle s=\sigma >5/2\,}$) in the Lorenz gauge this is in NkTs1986 (this result works in all dimensions)
• Global weak solutions were constructed in the energy class (${\displaystyle s=\sigma =1\,}$) 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