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

TL;DR

This paper establishes well-posedness for stochastic divergence-free continuity equations under Ladyzhenskaya-Prodi-Serrin conditions, highlighting the role of stochastic methods and stability analysis, despite potential non-uniqueness in the deterministic case.

Contribution

It introduces a stochastic characteristic approach and generalized Ito-Ventzel-Kunita formula to prove well-posedness and stability for these equations.

Findings

Proved well-posedness under Ladyzhenskaya-Prodi-Serrin condition

Demonstrated stability of solutions with respect to initial data

Showed potential failure of uniqueness in the deterministic setting

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 uniqueness realies on stochastic characteristic method and the generalized Ito-Ventzel-Kunita formula. Moreover, we prove a stability property for the solution with respect to the initial datum.