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 -