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 | 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) | |||
Outside of a non-trapping obstacle (with Dirichlet boundary conditions), the known results are as follows. | Outside of a non-trapping obstacle (with Dirichlet boundary conditions), the known results are as follows. | ||
* If <math>(p-1)(d-2) < 2\,</math> then one has GWP in H^1 assuming a coercivity condition (e.g. if the nonlinearity is defocusing) [BuGdTz-p4]. | * If <math>(p-1)(d-2) < 2\,</math> 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 <math>(p-1)(d-2) < 4\,</math>. | ** Note there is a loss compared with the non-obstacle theory, where one expects the condition to be <math>(p-1)(d-2) < 4\,</math>. | ||
** The same is true for the endpoint <math>d=3, p=3\,</math> if the energy is sufficiently small [BuGdTz-p4]. | ** The same is true for the endpoint <math>d=3, p=3\,</math> if the energy is sufficiently small [[BuGdTz-p4]]. | ||
** If <math>d \le 4\,</math> then the flow map is Lipschitz [BuGdTz-p4] | ** If <math>d \le 4\,</math> then the flow map is Lipschitz [[BuGdTz-p4]] | ||
** For <math>d=2, p \le 3\,</math> this is in [[ | ** For <math>d=2, p \le 3\,</math> this is in [[BrzGa1980]], [[Vd1984]], [[OgOz1991]] | ||
* If <math>p < 1 + 2/d\,</math> then one has GWP in <math>L^2\,</math> [BuGdTz-p4] | * If <math>p < 1 + 2/d\,</math> then one has GWP in <math>L^2\,</math> [[BuGdTz-p4]] | ||
** For <math>d=3\,</math> GWP for smooth data is in [[ | ** For <math>d=3\,</math> GWP for smooth data is in [[Jor1961]] | ||
** Again, in the non-obstacle theory one would expect <math>p < 1 + 4/d\,</math> | ** Again, in the non-obstacle theory one would expect <math>p < 1 + 4/d\,</math> | ||
** if <math>p < 1 + 1/d\,</math> then one also has strong uniqueness in the class <math>L^2\,</math> [BuGdTz-p4] | ** if <math>p < 1 + 1/d\,</math> then one also has strong uniqueness in the class <math>L^2\,</math> [[BuGdTz-p4]] | ||
On a domain in <math>R^d\,</math>, with Dirichlet boundary conditions, the results are as follows. | On a domain in <math>R^d\,</math>, with Dirichlet boundary conditions, the results are as follows. | ||
* Local well-posedness in <math>H^s\,</math> for <math>s > d/2\,</math> can be obtained by energy methods. | * Local well-posedness in <math>H^s\,</math> for <math>s > d/2\,</math> can be obtained by energy methods. | ||
* In two dimensions when <math>p \le 3\,</math>, global well-posedness in the energy class (assuming energy less than the ground state, in the <math>p=3\,</math> focusing case) is in [[ | * In two dimensions when <math>p \le 3\,</math>, global well-posedness in the energy class (assuming energy less than the ground state, in the <math>p=3\,</math> focusing case) is in [[BrzGa1980]], [[Vd1984]], [[OgOz1991]], [[Ca1989]].More precise asymptotics of a minimal energy blowup solution in the focusing <math>p=3\,</math> case are in [[BuGdTz-p]], [[Ban-p3]] | ||
* When <math>p > 1 + 4/d\,</math> blowup can occur in the focusing case [[ | * When <math>p > 1 + 4/d\,</math> blowup can occur in the focusing case [[Kav1987]] | ||
== Specific manifolds and equations == | == Specific manifolds and equations == | ||
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* Improved results are known for the [[cubic NLS]] for certain special manifolds, such as spheres, cylinders, and torii. | * 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. | * 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 [[ | * GWP and scattering for defocusing NLS on Schwarzchild manifolds for radial data is in [[LabSf1999]] | ||
[[Category:Equations]] | [[Category:Equations]] | ||
[[Category:Schrodinger]] | [[Category:Schrodinger]] | ||
Latest 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
