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 H¹ [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 H¹ [BuGdTz-p3]. In the special case of a sphere one has LWP in H^{d/2 + 1/2} for d³3 and p < 5 [BuGdTz-p3].

·For the cubic equation on two-dimensional surfaces one has LWP in H^s for s > ½ [BuGdTz-p3]

oFor s >= 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^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 (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 <= 4 then the flow map is Lipschitz [BuGdTz-p4]
  • For d=2, p <= 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 <=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