A strichartz inequality for the schrödinger equation on nontrapping asymptotically conic manifolds

Andrew Hassell, Terence Tao, Jared Wunsch*

*Corresponding author for this work

Research output: Contribution to journalArticle

29 Scopus citations

Abstract

We obtain the Strichartz inequality ∫01M|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.

Original languageEnglish (US)
Pages (from-to)157-205
Number of pages49
JournalCommunications in Partial Differential Equations
Volume30
Issue number1-3
DOIs
StatePublished - Jan 1 2005

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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