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

## Cubic NLS on the torus T^2

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2/3\,}$ by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).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

## Cubic NLS on the cylinder ${\displaystyle R\times T}$

• One has LWP for ${\displaystyle s>0\,}$ TkTz-p2.

## Cubic NLS on the sphere S^2

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