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

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* 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 [[ | + | ** 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 [[ | + | * 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 [[ | + | ** A similar result for small data is in [[Bibliography#Ts1993|Ts1993]] |

− | * In one dimension, GWP in the energy class is known [[ | + | * In one dimension, GWP in the energy class is known [[Bibliography#Ts1995|Ts1995]] |

− | * In two dimensions, GWP for smooth solutions is known [[ | + | * In two dimensions, GWP for smooth solutions is known [[Bibliography#TsNk1985|TsNk1985]] |

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## 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 A_{a} with a complex scalar field u. The Lagrangian density is

^{ab}F

_{ab}+ 2 Im

__u__D u - D

_{j}u D

^{j}u

giving rise to the system of PDE

_{t}= D

_{j}u D

^{j}u/2 + A u

d

^{a}F

_{ab}= J

_{b}

where the current density J_{b} is given by

_{}= |u|^2; J

_{j}= - Im

__u__D

_{j}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