Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

Abstract

We obtain the Strichartz inequalities $$\Vert u{\Vert }_{L_t^q{L}_x^r([0,1]\times M)}\le C\Vert u(0){\Vert }_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $i{u}_t+1/2{\Delta }_{M}u=0$, and $2<q,r\le \infty$ are admissible Strichartz exponents ($\frac{2}{q}+\frac{n}{r}=\frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r)=(2,\frac{2n}{n-2})$ when $n>2$). These estimates imply existence theorems for semi-linear Schrödinger equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}. This result improves on our previous result in \cite{HTW}, which was an $L_{t,x}^4$ Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.

50 pages, 2 figures

References in corpus (5)

• Global Strichartz estimates for nontrapping perturbations of the Laplacian
• Semiclassical estimates in asymptotically Euclidean scattering
• Determining Asymptotics of Magnetic Fields from Fixed Energy Scattering Data
• Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations
• Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation

Cited by in corpus (3)

• Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications
• Decay estimates for the wave and Dirac equations with a magnetic potential
• Cutoff Resolvent Estimates and the Semilinear Schrödinger Equation