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

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

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

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