Sharp Strichartz estimates on nontrapping asymptotically conic manifolds

By Andrew Hassell, Terence Tao, Jared Wunsch. 2004

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We obtain the Strichartz inequalities ║u║ L q t L r x ([0,1]× M ) ≥ C║ u (0) L 2 ║( M ) for any smooth n -dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where u is a solution to the Schrodinger equation iu t + 1/2 Δ M u = 0, and 2 q, r ≥ ∞ are admissible Strichartz exponents (2/ q + n/r = n /2). This corresponds with the estimates available for Euclidean space (except for the endpoint ( q, r ) = (2, 2 n/n -2) when n 2). These estimates imply existence theorems for semi-linear Schrodinger equations on M , by adapting arguments from Cazenave and Weissler and Kato. This result improves on our previous result, which was an L 4 t,x Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically flat manifolds.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fb58f5fde82bc8eee07039b0799d0fccf0fc54e

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Citations in the corpus

Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “We also draw the reader’s attention to the related papers [1, 6, 11, 27, 54, 58, 64, 67], as well as the references therein.”
Strichartz estimates for long range perturbations (2005, background): “Rd with the metric gij = δij the unit d× d matrix) the solutions of (1.1)-(1.2) enjoy the (local in time) Strichartz estimates (1.3) ‖u‖Lp([0,1];Lq(Rd)) ≤ C‖u0‖L2(Rd) , where (1.4) 2 p + d q = d 2 , p ≥ 2, (p, q) 6= (2,∞) .”