Difference between revisions of "Schrodinger:Maxwell-Schrodinger system"

From DispersiveWiki
Jump to navigationJump to search
 
(Cleaning bibliographic references)
Line 18: Line 18:
  
 
* In the Lorentz and Temporal gauges, one has LWP for s >= 5/3 and s-1 <= sigma <= s+1, (5s-2)/3 [NkrWad-p]
 
* 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 [[references:NkTs1986 NkTs1986]] (this result works in all dimensions)
+
** For smooth data (s=sigma > 5/2) in the Lorentz gauge this is in [[Bibliography#NkTs1986|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 [[references:GuoNkSr1996 GuoNkSr1996]]. GWP is still open however.
+
* Global weak solutions are known in the energy class (s=sigma=1) in the Lorentz and Coulomb gauges [[Bibliography#GuoNkSr1996|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.
 
* 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 [[references:Ts1993 Ts1993]]
+
** A similar result for small data is in [[Bibliography#Ts1993|Ts1993]]
* In one dimension, GWP in the energy class is known [[references:Ts1995 Ts1995]]
+
* In one dimension, GWP in the energy class is known [[Bibliography#Ts1995|Ts1995]]
* In two dimensions, GWP for smooth solutions is known [[references:TsNk1985 TsNk1985]]
+
* In two dimensions, GWP for smooth solutions is known [[Bibliography#TsNk1985|TsNk1985]]
  
 
<div class="MsoNormal" style="text-align: center"><center>
 
<div class="MsoNormal" style="text-align: center"><center>

Revision as of 20:27, 28 July 2006

Maxwell-Schrodinger system in

This system is a partially non-relativistic analogue of the [wave:mkg Maxwell-Klein-Gordon system]., coupling a U(1) connection Aa with a complex scalar field u. The Lagrangian density is

\int Fab Fab + 2 Im u D u - Dj u Dj u

giving rise to the system of PDE

i ut = Dj u Dj u/2 + A u
da Fab = Jb

where the current density Jb is given by

J = |u|^2; Jj = - Im u Dj 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