Title: Conservation Laws Driven by Lévy Stable and Linnik Diffusions: Shock Dissolution and Hausdorff Dimension
Speaker: James Austrow (Case Western Reserve University)
Advisor: Wojbor Woyczynski (Professor, Case Western Reserve University)
Abstract: We study two problems involving stochastic Burgers flows. The first concerns the solutions of fractal conservation laws, as in the inviscid Burgers equation modified by a non-local operator representing the action of a Lévy α-stable diffusion. Previous work has established that for α < 1 such laws create and preserve shocks for initial data which are bounded, odd, and convex on the positive half line. Furthermore, recent numeric results suggest the conjecture that these shocks disappear after a finite time. We offer additional support for the conjecture by computing and tabulating the shock dissolution time for a range of α parameter values. The second problem concerns the simpler inviscid Burgers equation with stochastic initial data. Earlier work investigates the solution of these equations when the initial data is an α-stable motion and estimates a relationship between α and the Hausdorff dimension of the regular points of the solution. We study the analogous problem for Linnik diffusion-type initial data.