Well-posedness for Stochastic Continuity Equations with Rough Coefficients.

Well-posedness for Stochastic Continuity Equations with Rough Coefficients.

-
Sam Punshon-Smith , University of Maryland
Fine Hall 322

According to the theory of Diperna/ Lions, the continuity equation associated to a Sobolev (or BV) vector field with bounded divergence has a unique weak solution in L^p. Under the addition of white in time stochastic perturbations to the characteristics of the continuity equation, it is known that uniqueness can be obtained under a relaxation of the regularity conditions and the requirement of bounded divergence.  In this talk, we will consider the general stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients and discuss conditions under which the equation has a unique solution. Using the renormalization approach of DiPerna/Lions we will present a proof of uniqueness of solutions to the stochastic transport with additive noise and a drift in L^q_t L^p_x, satisfying the subcritical Ladyzhenskaya–Prodi–Serrin criterion 2/q + d/p < 1.