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Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

By Nicolas Burq, Colin Guillarmou and Andrew Hassell


Abstract. In [21], Doi proved that the L 2 t H 1 2 x local smoothing effect for Schrödinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L 1 → L ∞ dispersive estimates still hold without loss for e it ∆ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension. The influence of the geometry on the behaviour of solutions of linear or non linear partial differential equations has been widely studied recently, and especially in the context of wave or Schrödinger equations. In particular, the understanding of the smoothing effect for the Schrödinger flow and Strichartz type estimates has been related to the global behaviour of the geodesic flow on the manifold (see for example the works by Doi [21] and Burq [11]). Let us recall that for the Laplacian ∆ on a d-dimensional non-compact Riemannian manifold (M, g), the local smoothing effect for bounded time t ∈ [0, T] and Schrödinger waves u = eit∆u0: M × R → C is the estimate ||χe it ∆ u0| | L2 ((0,T);H1/2(M)) ≤ CT ||u0| | L2 (M), ∀u0 ∈ L 2 (M) where CT> 0 is a constant depending a priori on T and χ is a compactly supported smooth function (the asumption on χ can of course be weakened in many cases, e.g for M = Rd) [19]. In other words, although the solution is only L2 in space uniformly in time, it is actually half a derivative better (locally) in an L2-in-time sense. For its description in geometric setting, the picture now is fairly complete: the so called “nontrapping condition ” stating roughly that every geodesic maximally extended goes to infinity, is known to be essentially necessary and sufficient (modulo reasonable conditions near infinity) [11]. Another tool for analyzing non-linear Schrödinger equations is the family of so-called Strichartz estimates introduced by [39]: for Schrödinger waves on Euclidean space Rd with initial data u0, (0.1) ||e it ∆ u0| | L p ((0,T);L q (R d)) ≤ CT ||u0| | L 2 (R d) if p, q ≥ 2, 2 d p

Year: 2013
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