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 , 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    • One can improve this to by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).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 .