Maxwell-Schrodinger system: Difference between revisions

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(Changed "Lorentz gauge" to "Lorenz gauge" as this is named after the danish physicist Ludvig Lorenz)
 
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<center><math>J= |u|^2; J_j= -Im{\underline{u}D_ju}\,</math></center>
<center><math>J= |u|^2; J_j= -Im{\underline{u}D_ju}\,</math></center>


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).
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 <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>
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>


* 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>.
* 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>.
** In the Lorentz 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]]
** 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]]
** For smooth data (<math>s=\sigma > 5/2\,</math>) in the Lorentz gauge this is in [[NkTs1986]] (this result works in all dimensions)
** For smooth data (<math>s=\sigma > 5/2\,</math>) in the Lorenz gauge this is in [[NkTs1986]] (this result works in all dimensions)
** Global weak solutions were constructed in the energy class (<math>s=\sigma=1\,</math>) in the Lorentz and Coulomb gauges [[GuoNkSr1996]].  
** Global weak solutions were constructed in the energy class (<math>s=\sigma=1\,</math>) 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.
* 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]]
** A similar result for small data is in [[Ts1993]]

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