NLS on manifolds and obstacles

From DispersiveWiki

The NLS has been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in BuGdTz-p3, while for smooth three-dimensional compact surfaces and one has LWP in for , together with weak solutions in BuGdTz-p3. In the special case of a sphere one has LWP in for and BuGdTz-p3.

  • For the cubic equation on two-dimensional surfaces one has LWP in for BuGdTz-p3
  • For one has GWP Vd1984, OgOz1991 and regularity BrzGa1980.
  • For uniform ill-posedness can be obtained by adapting the argument in BuGdTz2002 or CtCoTa-p.

A key tool here is the development of Strichartz estimates on curved space. For general manifolds one has all the Strichartz estimates (locally in time), but with a loss of derivatives, see BuGdTz-p3. (This though compares favorably to Sobolev embedding, which would require a loss of derivatives.) When the manifold is flat outside of a compact set and obeys a non-trapping condition, the optimal Strichartz estimates (locally in time) were obtained in StTt-p.
When instead the manifold is decaying outside of a compact set and obeys a non-trapping condition, the Strichartz estimates (locally in time) with an epsilon loss were obtained by Burq Bu-p3; in the special case of estimates on , and for non-trapping asymptotically conic manifolds, the epsilon was removed in HslTaWun-p.

Outside of a non-trapping obstacle (with Dirichlet boundary conditions), the known results are as follows.

  • If then one has GWP in H^1 assuming a coercivity condition (e.g. if the nonlinearity is defocusing) BuGdTz-p4.
    • Note there is a loss compared with the non-obstacle theory, where one expects the condition to be .
    • The same is true for the endpoint if the energy is sufficiently small BuGdTz-p4.
    • If then the flow map is Lipschitz BuGdTz-p4
    • For this is in BrzGa1980, Vd1984, OgOz1991
  • If then one has GWP in BuGdTz-p4
    • For GWP for smooth data is in Jor1961
    • Again, in the non-obstacle theory one would expect
    • if then one also has strong uniqueness in the class BuGdTz-p4

On a domain in , with Dirichlet boundary conditions, the results are as follows.

  • Local well-posedness in for can be obtained by energy methods.
  • In two dimensions when , global well-posedness in the energy class (assuming energy less than the ground state, in the focusing case) is in BrzGa1980, Vd1984, OgOz1991, Ca1989.More precise asymptotics of a minimal energy blowup solution in the focusing case are in BuGdTz-p, Ban-p3
  • When blowup can occur in the focusing case Kav1987

Specific manifolds and equations

  • Improved results are known for the cubic NLS for certain special manifolds, such as spheres, cylinders, and torii.
  • The quintic NLS has also been studied on several special manifolds, such as the circle.
  • GWP and scattering for defocusing NLS on Schwarzchild manifolds for radial data is in LabSf1999