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 <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]. | 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]]. | ||
* For the cubic equation on two-dimensional surfaces one has LWP in <math>H^s\,</math> for <math>s > 1/2\,</math> [[BuGdTz-p3]] | |||
* For <math>s \ge 1\,</math> one has GWP [[Vd1984]], [[OgOz1991]] and regularity [[BrzGa1980]]. | |||
* For <math>s < 0\,</math> 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 <math>L^q_t L^r_x\,</math> Strichartz estimates (locally in time), but with a loss of <math>1/q\,</math> derivatives, see [BuGdTz-p3]. (This though compares favorably to Sobolev embedding, which would require a loss of <math>2/q\,</math> 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]]. <br /> 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 <math>L^4\,</math> estimates on <math>R^3\,</math>, and for non-trapping asymptotically conic manifolds, the epsilon was removed in [[HslTaWun-p]]. | A key tool here is the development of Strichartz estimates on curved space. For general manifolds one has all the <math>L^q_t L^r_x\,</math> Strichartz estimates (locally in time), but with a loss of <math>1/q\,</math> derivatives, see [BuGdTz-p3]. (This though compares favorably to Sobolev embedding, which would require a loss of <math>2/q\,</math> 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]]. <br /> 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 <math>L^4\,</math> estimates on <math>R^3\,</math>, and for non-trapping asymptotically conic manifolds, the epsilon was removed in [[HslTaWun-p]]. | ||
Revision as of 17:09, 12 July 2007
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.
- If then one has GWP in 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
