Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition

Abstract

We consider the stochastic divergence-free continuity equations with Ladyzhenskaya-Prodi-Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of strong uniqueness, in the probabilistic sense, relies on stochastic characteristic method and the generalized Itô-Wentzell-Kunita formula. The stability property for the unique solution is proved with respect to the initial data. Moreover, a persistence result is established by a representation formula.

Stochastic continuity equations–A general uniqueness result

“…The proof of the above lemma follows from minor modification of the arguments in Neves, Olivera [16], see Lemma 2.1 on that paper.…”

“…Here we follow our strategy introduced in [16], which is to say, under the Ladyzhenskaya-Prodi-Serrin condition for b, we are allowed to compose the solution u to the continuity equation with the flow X. Then, we bring the stochastic flow, in fact its inverse, with all its space derivatives on the test function.…”

“…In particular, the ones that have an intrinsic perturbation from a stochastic term, see [2], [4], [9], [10], [12], [15], and [16]. In fact, the intrinsic random perturbation of the PDE is strongly related to stochastic differential equations, more precisely stochastic flows.…”

“…In fact, the intrinsic random perturbation of the PDE is strongly related to stochastic differential equations, more precisely stochastic flows. In this direction, the reader is addressed to the notion of stochastic balance laws introduced in [16].…”

Existence and Smoothness of the Density for the Stochastic Continuity Equation

“…The stochastic continuity equation with random perturbation given by a standard Brownian noise has been first studied in the celebrated works by Kunita [8], [9]. More recent works on these topics are, among others, [1], [6], [7] [12] and [13].…”