Schrodinger:Maxwell-Schrodinger system: Difference between revisions

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===Maxwell-Schrodinger system in <math>R^3</math>===
#REDIRECT [[Maxwell-Schrodinger system]]
 
This system is a partially non-relativistic analogue of the [wave:mkg Maxwell-Klein-Gordon system]., coupling a U(1) connection A<sub><font face="Symbol">a</font></sub> with a complex scalar field u. The Lagrangian density is
 
<center>\int F<sup><font face="Symbol">ab</font></sup> F<sub><font face="Symbol">ab</font></sub> + 2 Im <u>u</u> D u - D<sub>j</sub> u D<sup>j</sup> u</center>
 
giving rise to the system of PDE
 
<center>i u<sub>t</sub> = D<sub>j</sub> u D<sup>j</sup> u/2 + A u <br /> d<sup><font face="Symbol">a</font></sup> F<sub><font face="Symbol">ab</font></sub> = J<sub><font face="Symbol">b</font></sub></center>
 
where the current density J<sub><font face="Symbol">b</font></sub> is given by
 
<center>J<sub></sub> = |u|^2; J<sub>j</sub> = - Im <u>u</u> D<sub>j</sub> u</center>
 
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 [[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 [[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.
** A similar result for small data is in [[Bibliography#Ts1993|Ts1993]]
* In one dimension, GWP in the energy class is known [[Bibliography#Ts1995|Ts1995]]
* In two dimensions, GWP for smooth solutions is known [[Bibliography#TsNk1985|TsNk1985]]
 
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[[Category:Equations]]

Latest revision as of 03:50, 29 July 2006