NLS on manifolds and obstacles

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 $d\leq 3\,$ 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]

oFor $s\geq 1\,$ one has GWP Vd1984, OgOz1991 and regularity BrzGa1980

oFor $s<0\,$ 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 $L_{t}^{q}L_{x}^{r}\,$ 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 $(p-1)(d-2)<2\,$ 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 $(p-1)(d-2)<4\,$.
  • The same is true for the endpoint $d=3,p=3\,$ if the energy is sufficiently small [BuGdTz-p4].
  • If $d\leq 4\,$ then the flow map is Lipschitz [BuGdTz-p4]
  • For $d=2,p\leq 3\,$ this is in BrzGa1980, Vd1984, OgOz1991
• If $p<1+2/d\,$ then one has GWP in $L^{2}\,$ [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 $p\leq 3\,$, 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

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