Multiscale conservation laws driven by Levy stable and Linnik diffusions: asymptotics, explicit representations, shock creation, preservation and dissolution

We will discuss the interplay between the nonlinear and nonlocal components of the evolution equations. In the particular case of supercritical multifractal conservation laws (CL) the asymptotic behavior, as $t \neq 1$, is dictated by the linearized case. For $\alpha <1$ , the equations driven by infinitesimal generators of Levy -stable diffusions the solution exhibit shocks (for bounded, odd, and convex on R+, initial data) which disappear over a nite time. For Levy -Linnik diffusions, $0 < \alpha < 2$ , the local behavior is strikingly different. The relevant CLs display shocks that do not dissipate over time.