Cubic NLS on 2d manifolds: Difference between revisions
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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]]. | |||
==Cubic NLS on the torus T^2== | |||
* | * One has LWP for <math>s>0\,</math> [[Bo1993]]. | ||
* In the defocussing case one has GWP for <math>s>1\,</math> in | * In the defocussing case one has GWP for <math>s>1\,</math> in by Hamiltonian conservation. | ||
** | ** 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 by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).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 | * 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== | ||
* | * 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 .
