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 [[Wave-Schrodinger system]]. The Lagrangian density is
This system is a partially non-relativistic analogue of the [[Maxwell-Klein-Gordon system]], coupling a U(1) connection <math>A_a\,</math> with a complex scalar field u; it is thus an example of a [[wave-Schrodinger system]]. 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>
<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>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>
<center><math> iu_t = D_j u D_j u / 2 + A_0 a\,</math></center>  
<center><math>\partial^aF_{ab} = J_b\,</math></center>


where the current density J<sub><font face="Symbol">b</font></sub> is given by
where the current density <math>J_b\,</math> is given by


<center>J<sub></sub> = |u|^2; J<sub>j</sub> = - Im <u>u</u> D<sub>j</sub> u</center>
<center><math>J= |u|^2; J_j= -Im{\underline{u}D_ju}\,</math></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 Lorenz, 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 <math>H^s\,</math>, and A in <math>H^\sigma H^{\sigma-1}\,.</math> The lack of scale invariance makes it difficult to exactly state what the critical regularity would be, but it seems to be <math>s = \sigma = 1/2\,.</math>


* In the Lorentz and Temporal gauges, one has LWP for s >= 5/3 and s-1 <= sigma <= s+1, (5s-2)/3 [NkrWad-p]
* GWP in the energy space <math>s=\sigma=1</math> in the Coulomb gauge was established by Bejenaru and Tataru in 2007.  The argument also gives a priori estimates when <math> s>1/2, \sigma=1</math> and LWP when <math> s>3/4, \sigma=1</math>.
** For smooth data (s=sigma > 5/2) in the Lorentz gauge this is in [[Bibliography#NkTs1986|NkTs1986]] (this result works in all dimensions)
** In the Lorenz and Temporal gauges, LWP for <math>s \ge 5/3\,</math> and <math>s-1 \le \sigma \le s+1, (5s-2)/3</math> was established in [[NkrWad-p]]
* 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.
** For smooth data (<math>s=\sigma > 5/2\,</math>) in the Lorenz gauge this is in [[NkTs1986]] (this result works in all dimensions)
* 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.
** Global weak solutions were constructed in the energy class (<math>s=\sigma=1\,</math>) in the Lorenz and Coulomb gauges [[GuoNkSr1996]].  
** A similar result for small data is in [[Bibliography#Ts1993|Ts1993]]
* 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.
* In one dimension, GWP in the energy class is known [[Bibliography#Ts1995|Ts1995]]
** A similar result for small data is in [[Ts1993]]
* In two dimensions, GWP for smooth solutions is known [[Bibliography#TsNk1985|TsNk1985]]
* In one dimension, GWP in the energy class is known [[Ts1995]]
* In two dimensions, GWP for smooth solutions is known [[TsNk1985]]


[[Category:Equations]]
[[Category:Equations]]
[[Category:Wave]]
[[Category:Wave]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]

Latest revision as of 13:07, 3 January 2009

Maxwell-Schrodinger system in

This system is a partially non-relativistic analogue of the Maxwell-Klein-Gordon system, coupling a U(1) connection 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 is given by

As with the MKG system, there is a gauge invariance for the connection; one can place A in the Lorenz, Coulomb, or Temporal gauges (other choices are of course possible).

Let us place u in , and A in The lack of scale invariance makes it difficult to exactly state what the critical regularity would be, but it seems to be

  • GWP in the energy space in the Coulomb gauge was established by Bejenaru and Tataru in 2007. The argument also gives a priori estimates when and LWP when .
    • In the Lorenz and Temporal gauges, LWP for and was established in NkrWad-p
    • For smooth data () in the Lorenz gauge this is in NkTs1986 (this result works in all dimensions)
    • Global weak solutions were constructed in the energy class () in the Lorenz and Coulomb gauges GuoNkSr1996.
  • 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