Details

Autor(en) / Beteiligte

• Hassell, Andrew
• Tao, Terence
• Wunsch, Jared

Titel

Sharp Strichartz Estimates on Nontrapping Asymptotically Conic Manifolds

Ist Teil von

• American journal of mathematics, 2006-08, Vol.128 (4), p.963-1024

Ort / Verlag

Baltimore, MD: Johns Hopkins University Press

Erscheinungsjahr

2006

Link zum Volltext

Quelle

Project MUSE

Beschreibungen/Notizen

• We obtain the Strichartz inequalities$\left\ u \right\_{L_t^q L_x^r ([0,1] \times M)} \le C\left\ {u(0)} \right\_{L^2 (M)}$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 Schrödinger equation$iu_t + {1 \over 2}\Delta _{\rm{M}} u = {\rm{0}}$, and 2 < q,$r \le \infty $are admissible Strichartz exponents$({2 \over 9} + {n \over r} = {n \over 2})$. This corresponds with the estimates available for Euclidean space (except for the endpoint$(q,r) = (2,{{2n} \over {n - 2}})$when n > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on M, by adapting arguments from Cazenave and Weissler and Kato. This result improves on our previous result, which was an$L_{t,x}^4 $. 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.