TY - JOUR

T1 - A strichartz inequality for the schrödinger equation on nontrapping asymptotically conic manifolds

AU - Hassell, Andrew

AU - Tao, Terence

AU - Wunsch, Jared

N1 - Funding Information:
J.W. is supported in part by NSF grant DMS-0323021. The authors thank an anonymous referee for helpful comments on the manuscript.
Funding Information:
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.

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

KW - Asymptotically conic manifolds

KW - Interaction morawetz inequality

KW - Scattering metrics

KW - Smoothing estimates

KW - Strichartz estimates

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U2 - 10.1081/PDE-200044482

DO - 10.1081/PDE-200044482

M3 - Article

AN - SCOPUS:17444367698

VL - 30

SP - 157

EP - 205

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 1-3

ER -