Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications

Details Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications

2004-09-21

arxiv.org/abs/math/0409379v1

Mathematics

Analysis of PDEs

2 figures

Scientific paper

We prove smoothing estimates for Schr\"odinger equations $i\partial_t \phi+\partial_x (a(x) \partial_x \phi) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.

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