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Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics. (English) Zbl 1206.58009
The authors study the Strichartz estimates for the Schrödinger equation posed on manifolds with hyperbolic trapped geodesics. There are two fundamental estimates which reflect the behavior of solutions of linear or nonlinear wave or Schrödinger equations: local smoothing effect and Strichartz type estimates.
For the description of the local smoothing effect in geometric settings, it is known that the “nontrapping condition” is essentially necessary and sufficient [see S.-i. Doi, Math. Ann. 318, No. 2, 355–389 (2000; Zbl 0969.35029) and N. Burq, Duke Math. J. 123, No. 2, 403–427 (2004; Zbl 1061.35024)].
In contrast, this is not clear for the Strichartz estimates. The Strichartz estimates are a family of space time integrability estimates of the form
\[ \|e^{it\Delta} u_0\|_{L^p_t((0,T); L^q(M))}\leq C_T \|u_0\|_{L^2(M)}. \]
Global-in-time Strichartz estimates (with \(T=\infty\)) are known for the flat Laplacian on \(\mathbb{R}^n\), while the local-in-time Strichartz estimates are known in several geometric situations where the manifold is non-trapping (asymptotically Euclidean, conic or hyperbolic). On the other hand, it is clear that such a global-in- time estimate cannot hold on compact manifolds or non-compact manifolds in the presence of elliptic (stable) non-degenerate periodic orbits of the geodesic flow. Moreover, there do exist examples of trapping manifolds where a certain Strichartz estimate is true (\(p=q=4\) and \(M=\mathbb{R}\times\mathbb{T}\) by H. Takaoka and N. Tzvetkov [J. Funct. Anal. 182, No. 2, 427–442 (2001; Zbl 0976.35085)]).
In this paper, the authors prove Strichartz estimates (without loss) for some examples of Riemannian manifolds where trapping does occur (and consequently loss is unavoidable for the smoothing effect), but the trapped set is hyperbolic and of sufficiently small fractal dimension.

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B65 Smoothness and regularity of solutions to PDEs
53C22 Geodesics in global differential geometry
Full Text: DOI
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