Cubic NLS on 2d manifolds

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 $s_{c}=0$, thus this is a mass-critical NLS.

Cubic NLS on the torus T^2

• One has LWP for $s>0\,$ Bo1993.
• In the defocussing case one has GWP for $s>1\,$ in by Hamiltonian conservation.
  • One can improve this to $s>2/3\,$ by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).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 $(T^{*}-t)^{-1}\,$ BuGdTz-p
• The $H^{k}\,$ norm grows like $O(t^{2(k-1)+})\,$ as long as the $H^{1}\,$ norm stays bounded.

Cubic NLS on the cylinder $R\times T$

• One has LWP for $s>0\,$ TkTz-p2.

Cubic NLS on the sphere S^2

• Uniform local well-posedness fails for $3/20<s<1/4\,$ BuGdTz2002, Ban2004a, but holds for $s>1/4\,$ BuGdTz-p7.
  • For $s>1/2\,$ 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 $(T-t)^{-1}\,$. If the domain is a disk then uniform LWP fails for $1/5<s<1/3\,$, while for a square one has LWP for all $s>0\,.$ In general domains one has LWP for $s>2.$.