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]
* 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]].


<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For <math>s \ge 1\,</math> one has GWP [[Bibliography#Vd1984|Vd1984]], [[Bibliography#OgOz1991|OgOz1991]] and regularity [[Bibliography#BrzGa1980|BrzGa1980]]
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]].
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><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 [[Bibliography#BuGdTz2002|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]


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 [[Bibliography#BrzGa1980|BrzGa1980]], [[Bibliography#Vd1984|Vd1984]], [[Bibliography#OgOz1991|OgOz1991]]
** 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 [[Bibliography#Jor1961|Jor1961]]
** 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 [[Bibliography#BrzGa1980|BrzGa1980]], [[Bibliography#Vd1984|Vd1984]], [[Bibliography#OgOz1991|OgOz1991]], [[Bibliography#Ca1989|Ca1989]].More precise asymptotics of a minimal energy blowup solution in the focusing <math>p=3\,</math> case are in [BuGdTz-p], [Ban-p3]
* 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 [[Bibliography#Kav1987|Kav1987]]
* 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 [[Bibliography#LabSf1999|LabSf1999]]
* 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 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
  • 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 \ge 1\,} one has GWP Vd1984, OgOz1991 and regularity BrzGa1980.
  • 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 < 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 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
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