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arXiv · 1201.2464

Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

Hans Christianson
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Abstract

We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schrödiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $β>0$ loss. This is in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension.

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Hans Christianson. 2012-01-12. Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping. https://doi.org/10.1007/s00220-013-1805-z

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