Author： HASSELL ANDREW
Publisher： Taylor & Francis Ltd
ISSN： 0360-5302
Source： Communications in Partial Differential Equations, Vol.30, Iss.2, 2005-01, pp. : 157-205
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Abstract
We obtain the Strichartz inequality 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.
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