NLS on manifolds and obstacles: Difference between revisions

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The NLS has been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in <math>H^1\,</math> [BuGdTz-p3], while for smooth three-dimensional compact surfaces and <math>p=3\,</math> one has LWP in <math>H^s\,</math> for <math>s>1\,</math>, together with weak solutions in H<sup>1</sup> [BuGdTz-p3]. In the special case of a sphere one has LWP in <math>H^{d/2 + 1/2}\,</math> for <math>d \le 3\,</math> and <math>p < 5\,</math> [BuGdTz-p3].
The NLS has been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in <math>H^1\,</math> [BuGdTz-p3], while for smooth three-dimensional compact surfaces and <math>p=3\,</math> one has LWP in <math>H^s\,</math> for <math>s>1\,</math>, together with weak solutions in <math>H^1\,</math> [BuGdTz-p3]. In the special case of a sphere one has LWP in <math>H^{d/2 + 1/2}\,</math> for <math>d \le 3\,</math> and <math>p < 5\,</math> [BuGdTz-p3].


<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>For the cubic equation on two-dimensional surfaces one has LWP in <math>H^s\,</math> for <math>s > 1/2\,</math> [BuGdTz-p3]
<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>For the cubic equation on two-dimensional surfaces one has LWP in <math>H^s\,</math> for <math>s > 1/2\,</math> [BuGdTz-p3]

Revision as of 18:49, 7 August 2006

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]

oFor one has GWP Vd1984, OgOz1991 and regularity BrzGa1980

oFor 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