Well-posedness for the stochastic electrokinetic flow

TL;DR

This paper proves the global existence and uniqueness of solutions for a stochastic electrokinetic flow model in both 2D and 3D, including cases without small initial data restrictions.

Contribution

It establishes the first comprehensive results on well-posedness for the stochastic Nernst-Planck-Navier-Stokes system with multiplicative noise in bounded domains.

Findings

Global weak martingale solutions exist in 2D and 3D.

Unique maximal strong pathwise solutions are proven to exist.

In 2D, solutions are global without small initial data constraints.

Abstract

We consider the stochastic electrokinetic flow in a smooth bounded domain $\mathcal{D}$, modelled by a Nernst-Planck-Navier-Stokes system with a blocking boundary conditions for ionic species concentrations, perturbed by multiplicative noise. Several results are established in this paper. In both $2d$ and $3d$ cases, we establish the global existence of weak martingale solution which is weak in both PDEs and probability sense, and also the existence and uniqueness of the maximal strong pathwise solution which is strong in PDEs and probability sense. Particularly, we show that the maximal pathwise solution is global one in $2d$ case without the restriction of smallness of initial data.