Cubic NLS on 2d manifolds: Difference between revisions

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==Cubic NLS on RxT and T2==
In this page we discuss the [[cubic NLS]] on various two-dimensional domains (other than [[cubic NLS on R2|on R^2]]).  in all cases the [[critical]] regularity
is <math>s_c = 0</math>, thus this is a [[mass-critical NLS]].


* Scaling is <math>s_c\,</math>= 0.
==Cubic NLS on the torus T^2==
* For <math>R \times T</math> one has LWP for <math>s>0\,</math> [[TkTz-p2]].
 
* For <math> T \times T</math> one has LWP for <math>s>0\,</math> [[Bo1993]].
* One has LWP for <math>s>0\,</math> [[Bo1993]].
* In the defocussing case one has GWP for <math>s>1\,</math> in both cases by Hamiltonian conservation.
* In the defocussing case one has GWP for <math>s>1\,</math> in by Hamiltonian conservation.
** On <math>T \times T</math> one can improve this to <math>s > 2/3\,</math> by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).
** One can improve this to <math>s > 2/3\,</math> by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).
* In the focusing case on <math>T \times T</math> one has blowup for data close to the ground state, with a blowup rate of <math>(T^* -t )^{-1}\,</math> [[BuGdTz-p]]
* In the focusing case one has blowup for data close to the ground state, with a blowup rate of <math>(T^* -t )^{-1}\,</math> [[BuGdTz-p]]
* The <math>H^k\,</math> norm grows like <math>O(t^{2(k-1)+})\,</math> as long as the <math>H^1\,</math> norm stays bounded.
 
==Cubic NLS on the cylinder <math>R \times T</math>==
 
* One has LWP for <math>s>0\,</math> [[TkTz-p2]].


==Cubic NLS on the sphere S^2==
==Cubic NLS on the sphere S^2==


* If instead one considers the sphere <math>S^2\,</math> then uniform local well-posedness fails for <math>3/20 < s < 1/4\,</math> [[BuGdTz2002]], [[Ban-p]], but holds for <math>s>1/4\,</math> [[BuGdTz-p7]].
* Uniform local well-posedness fails for <math>3/20 < s < 1/4\,</math> [[BuGdTz2002]], [[Ban2004a]], but holds for <math>s>1/4\,</math> [[BuGdTz-p7]].
** For <math>s >1/2\,</math> this is in [[BuGdTz-p3]].
** For <math>s >1/2\,</math> this is in [[BuGdTz-p3]].
** These results for the sphere can mostly be generalized to other Zoll manifolds.
** These results for the sphere can mostly be generalized to other Zoll manifolds.
==Cubic NLS on bounded domains==
See [[BuGdTz-p]]. Sample results: blowup solutions exist close to the ground state, with a blowup rate of <math>(T-t)^{-1}\,</math>. If the domain is a disk then uniform LWP fails for <math>1/5 < s < 1/3\,</math>, while for a square one has LWP for all <math>s>0\,.</math> In general domains one has LWP for <math>s>2.</math>.


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

Latest revision as of 01:04, 18 July 2009

In this page we discuss the cubic NLS on various two-dimensional domains (other than on R^2). in all cases the critical regularity is , thus this is a mass-critical NLS.

Cubic NLS on the torus T^2

  • One has LWP for Bo1993.
  • In the defocussing case one has GWP for in by Hamiltonian conservation.
    • One can improve this to by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).
  • In the focusing case one has blowup for data close to the ground state, with a blowup rate of BuGdTz-p
  • The norm grows like as long as the norm stays bounded.

Cubic NLS on the cylinder

  • One has LWP for TkTz-p2.

Cubic NLS on the sphere S^2

  • Uniform local well-posedness fails for BuGdTz2002, Ban2004a, but holds for BuGdTz-p7.
    • For this is in BuGdTz-p3.
    • These results for the sphere can mostly be generalized to other Zoll manifolds.

Cubic NLS on bounded domains

See BuGdTz-p. Sample results: blowup solutions exist close to the ground state, with a blowup rate of . If the domain is a disk then uniform LWP fails for , while for a square one has LWP for all In general domains one has LWP for .