NLS on manifolds and obstacles

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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 $H^1\,$ BuGdTz-p3, while for smooth three-dimensional compact surfaces and $p=3\,$ one has LWP in $H^s\,$ for $s>1\,$ , together with weak solutions in $H^1\,$ BuGdTz-p3. In the special case of a sphere one has LWP in $H^{d/2 + 1/2}\,$ for $d \le 3\,$ and $p < 5\,$ BuGdTz-p3.

For the cubic equation on two-dimensional surfaces one has LWP in $H^s\,$ for $s > 1/2\,$ BuGdTz-p3
For $s \ge 1\,$ one has GWP Vd1984, OgOz1991 and regularity BrzGa1980.
For $s < 0\,$ 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 $L^q_t L^r_x\,$ Strichartz estimates (locally in time), but with a loss of $1/q\,$ derivatives, see BuGdTz-p3. (This though compares favorably to Sobolev embedding, which would require a loss of $2/q\,$ 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 $L^4\,$ estimates on $R^3\,$ , 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 $(p-1)(d-2) < 4\,$ .
- The same is true for the endpoint $d=3, p=3\,$ if the energy is sufficiently small BuGdTz-p4.
- If $d \le 4\,$ then the flow map is Lipschitz BuGdTz-p4
- For $d=2, p \le 3\,$ this is in BrzGa1980, Vd1984, OgOz1991
If then one has GWP in BuGdTz-p4
- For $d=3\,$ GWP for smooth data is in Jor1961
- Again, in the non-obstacle theory one would expect $p < 1 + 4/d\,$
- if $p < 1 + 1/d\,$ then one also has strong uniqueness in the class $L^2\,$ BuGdTz-p4

On a domain in $R^d\,$ , with Dirichlet boundary conditions, the results are as follows.

Local well-posedness in $H^s\,$ for $s > d/2\,$ can be obtained by energy methods.
In two dimensions when $p \le 3\,$ , global well-posedness in the energy class (assuming energy less than the ground state, in the $p=3\,$ focusing case) is in BrzGa1980, Vd1984, OgOz1991, Ca1989.More precise asymptotics of a minimal energy blowup solution in the focusing $p=3\,$ case are in BuGdTz-p, Ban-p3
When $p > 1 + 4/d\,$ blowup can occur in the focusing case Kav1987

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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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Categories: Equations | Schrodinger