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

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

2012-01-12

arxiv.org/abs/1201.2464v1

Mathematics

Analysis of PDEs

Scientific paper

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 Schr\"odiner 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 $\beta>0$ 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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