PDE seminar by Luca Galimberti
Speaker: Luca Galimberti (UiO)
Title: Stochastic conservation laws on Riemannian manifolds
Abstract: We are given an n-dimensional smooth closed manifold M, endowed with a smooth Riemannian metric h. We study the Cauchy problem for a first-order scalar conservation law with stochastic forcing given by a cylindrical Wiener process W. After providing a reasonable notion of solution, we prove an existence and uniqueness-result for our Cauchy problem, by showing convergence of a suitable parabolic approximation of it. This is achieved thanks to a generalized Ito's formula for weak solutions of a wide class of stochastic partial differential equations on Riemannian manifolds, which we obtained. Indeed, this formula proves an important tool to derive our concept of solution as well as to establish a priori L^p-estimates. This is a joint work with K.H. Karlsen and N.H. Risebro.