Dispersive Estimates for Manifolds with One Trapped Orbit
نویسنده
چکیده
For a large class of complete, non-compact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator:
منابع مشابه
Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations
We construct an invariant manifold of periodic orbits for a class of non-linear Schrödinger equations. Using standard ideas of the theory of center manifolds, we rederive the results of Soffer and Weinstein ([SW1], [SW2]) on the large time asymptotics of small solutions (scattering theory).
متن کاملWeighted Strichartz Estimates for Radial Schrödinger Equation on Non-compact Manifolds
We prove global weighted Strichartz estimates for radial solutions of linear Schrödinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose vol...
متن کاملStrichartz Estimates for the Wave Equation on Manifolds with Boundary
Strichartz estimates are well established on flat Euclidean space, where M = R and gij = δij . In that case, one can obtain a global estimate with T = ∞; see for example Strichartz [27], Ginibre and Velo [9], Lindblad and Sogge [16], Keel and Tao [14], and references therein. However, for general manifolds phenomena such as trapped geodesics and finiteness of volume can preclude the development...
متن کاملThe incompressible Navier-Stokes equations on non-compact manifolds
We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We shall also discuss the uniqueness of Leray weak solutions in the two dimensional case.
متن کاملACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE
A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...
متن کامل