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.