Cubic NLS on 2d manifolds

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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 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 2/3\,$ by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).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 BuGdTz2002, Ban2004a, but holds for 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 $s > 2.$ .