article

Title: On the size of divergence sets for the Schroedinger equation with radial data

Authors: Jonathan Bennett and Keith M. Rogers

Issue: Volume 61 (2012), Issue 1, 1-13

Abstract:

$We consider the Schr\"odinger equation $ i\partial_tu + \Delta u = 0$ with initial data in $H^s(\mathbb{R}^n)$. A classical problem is to identify the exponents $s$ for which $u(\cdot,t)$ converges almost everywhere to the initial data as $t$ tends to zero. In one spatial dimension, Carleson proved that the convergence is guaranteed when $s = \frac{1}{4}$, and Dahlberg and Kenig proved that divergence can occur on a set of nonzero Lebesgue measure when $s < \frac{1}{4}$. In higher dimensions Prestini deduced the same conclusions when restricting attention to radial data. We refine this by proving that the Hausdorff dimension of the divergence sets can be at most $n-\frac{1}{2}$ for radial data in $H^{1/4}(\mathbb{R}^n)$, and this is sharp.$