NLS on manifolds and obstacles: Difference between revisions
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<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] | ||
<span style="mso-fareast-font-family: "Courier New""><font face=""Courier New""><span style="mso-list: Ignore">o</span></font></span>For <math>s \ge 1\,</math> one has GWP [[ | <span style="mso-fareast-font-family: "Courier New""><font face=""Courier New""><span style="mso-list: Ignore">o</span></font></span>For <math>s \ge 1\,</math> one has GWP [[Vd1984]], [[OgOz1991]] and regularity [[BrzGa1980]] | ||
<span style="mso-fareast-font-family: "Courier New""><font face=""Courier New""><span style="mso-list: Ignore">o</span></font></span>For <math>s < 0\,</math> uniform ill-posedness can be obtained by adapting the argument in [[ | <span style="mso-fareast-font-family: "Courier New""><font face=""Courier New""><span style="mso-list: Ignore">o</span></font></span>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] | ||
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** 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] | ||
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* 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]] | ||
Revision as of 14:33, 10 August 2006
The NLS has been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1\,} [BuGdTz-p3], while for smooth three-dimensional compact surfaces and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p=3\,} one has LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s\,} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>1\,} , together with weak solutions in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1\,} [BuGdTz-p3]. In the special case of a sphere one has LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2 + 1/2}\,} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \le 3\,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p < 5\,} [BuGdTz-p3].
·For the cubic equation on two-dimensional surfaces one has LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s\,} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1/2\,} [BuGdTz-p3]
oFor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1\,} one has GWP Vd1984, OgOz1991 and regularity BrzGa1980
oFor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s < 0\,} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^q_t L^r_x\,}
Strichartz estimates (locally in time), but with a loss of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/q\,}
derivatives, see [BuGdTz-p3]. (This though compares favorably to Sobolev embedding, which would require a loss of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2/q\,}
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^4\,}
estimates on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3\,}
, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (p-1)(d-2) < 2\,}
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (p-1)(d-2) < 4\,} .
- The same is true for the endpoint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=3, p=3\,} if the energy is sufficiently small [BuGdTz-p4].
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \le 4\,} then the flow map is Lipschitz [BuGdTz-p4]
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2, p \le 3\,} this is in BrzGa1980, Vd1984, OgOz1991
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p < 1 + 2/d\,}
then one has GWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,}
[BuGdTz-p4]
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=3\,} GWP for smooth data is in Jor1961
- Again, in the non-obstacle theory one would expect Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p < 1 + 4/d\,}
- if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p < 1 + 1/d\,} then one also has strong uniqueness in the class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} [BuGdTz-p4]
On a domain in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^d\,} , with Dirichlet boundary conditions, the results are as follows.
- Local well-posedness in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s\,} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > d/2\,} can be obtained by energy methods.
- In two dimensions when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p \le 3\,} , global well-posedness in the energy class (assuming energy less than the ground state, in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p=3\,} focusing case) is in BrzGa1980, Vd1984, OgOz1991, Ca1989.More precise asymptotics of a minimal energy blowup solution in the focusing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p=3\,} case are in [BuGdTz-p], [Ban-p3]
- When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p > 1 + 4/d\,} 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
