## Abstract

We obtain the Strichartz inequality ∫_{0}^{1} ∫_{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 iu_{t} + (1/2)Δ_{M}u = 0. The exponent H^{1/4}(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.

Original language | English (US) |
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Pages (from-to) | 157-205 |

Number of pages | 49 |

Journal | Communications in Partial Differential Equations |

Volume | 30 |

Issue number | 1-3 |

DOIs | |

State | Published - 2005 |

### Funding

J.W. is supported in part by NSF grant DMS-0323021. The authors thank an anonymous referee for helpful comments on the manuscript. A.H. is supported in part by an Australian Research Council Fellowship. T.T. is a Clay Prize Fellow and is supported in part by grants from the Packard Foundation.

## Keywords

- Asymptotically conic manifolds
- Interaction morawetz inequality
- Scattering metrics
- Smoothing estimates
- Strichartz estimates

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics