We obtain the Strichartz inequality ∫01 ∫M|u(t, z)|4 dt(z)dt ≤ C ||u (0) || H1/4(M)4 for any smooth three-dimensional Riemannian manifold (M, g) which is asymptotically conic at infinity and nontrapping, where u is a solution to the Schrödinger equation iut + (1/2)ΔMu = 0. The exponent H1/4(M) is sharp, by scaling considerations. In particular our result covers asymptotically flat nontrapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product U(t, z′, z″) := u(t, z′)u(t, z″) of the solution with itself. We also use smoothing estimates for Schrödinger solutions including one (proved here) with weight r-1 at infinity and with the gradient term involving only one angular derivative.
- Asymptotically conic manifolds
- Interaction morawetz inequality
- Scattering metrics
- Smoothing estimates
- Strichartz estimates
ASJC Scopus subject areas
- Applied Mathematics