Maxwell-Schrodinger system: Difference between revisions
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===Maxwell-Schrodinger system in <math>R^3</math>=== | ===Maxwell-Schrodinger system in <math>R^3</math>=== | ||
This system is a partially non-relativistic analogue of the [[Maxwell-Klein-Gordon system]], coupling a U(1) connection A<sub><font face="Symbol">a</font></sub> with a complex scalar field u; it is thus an example of a [[ | This system is a partially non-relativistic analogue of the [[Maxwell-Klein-Gordon system]], coupling a U(1) connection A<sub><font face="Symbol">a</font></sub> with a complex scalar field u; it is thus an example of a [[wave-Schrodinger system]]. The Lagrangian density is | ||
<center>\int F | <center><math>\int F^{ab} F_{ab} + 2 \Im \overline{u} D_t u - \overline{D_j u} D_j u\ dx dt</math> | ||
</center> | |||
giving rise to the system of PDE | giving rise to the system of PDE | ||
<center> | <center><math> iu_t = D_j u D_j u / 2 + A_0 a</math></center> | ||
<center><math>F_{ab} = J_b</math></center> | |||
where the current density J<sub><font face="Symbol">b</font></sub> is given by | where the current density J<sub><font face="Symbol">b</font></sub> is given by | ||
| Line 13: | Line 15: | ||
<center>J<sub></sub> = |u|^2; J<sub>j</sub> = - Im <u>u</u> D<sub>j</sub> u</center> | <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). | As with the [[MKG|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. | 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. | ||
Revision as of 17:03, 30 July 2006
Maxwell-Schrodinger system in
This system is a partially non-relativistic analogue of the Maxwell-Klein-Gordon system, coupling a U(1) connection Aa 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 Jb is given by
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
