# 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 sc = 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 2/3\,$ by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).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..