|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| ===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 [[references: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.
| |
| * 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]]
| |
| * In one dimension, GWP in the energy class is known [[references:Ts1995 Ts1995]]
| |
| * In two dimensions, GWP for smooth solutions is known [[references:TsNk1985 TsNk1985]]
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| [[Category:Equations]]
| |
