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On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data

Thomas Simon
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Abstract

Consider an inviscid Burgers equation whose initial data is a Levy a-stable process Z with a > 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/a, as soon as a is close to 1. This gives a negative answer to a conjecture of Janicki and Woyczynski. Along the way, we contradict a recent conjecture of Z. Shi about the lower tails of integrated stable processes.

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Thomas Simon. 2007-02-21. On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data. https://doi.org/10.1007/s10955-008-9508-0

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