Existence and stability of nonequilibrium steady states of Nernst–Planck–Navier–Stokes systems

We consider the Nernst-Planck-Navier-Stokes system in a bounded domain of ${\mathbb {R}}^d$, $d=2,3$ with general nonequilibrium Dirichlet boundary conditions for ionic concentrations. prove existence smooth steady state solutions and present sufficient condition terms only data that guarantees these have nonzero fluid velocity. show time evolving are ultimately uniformly, independently their initial size. In addition, we one dimensional states currents they globally nonlinearly stable as three periodic strip, if sufficiently weak.