A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds

Details A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds

2003-12-11

arxiv.org/abs/math/0312225v2

Mathematics

Analysis of PDEs

43 pages, 1 figure; minor corrections to earlier version

Scientific paper

We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schr\"odinger equation $iu_t + {1/2} \Delta_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping 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 Schr\"odinger solutions including a new one proved here with weight $r^{-1}$ at infinity and with the gradient term involving only one angular derivative.

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