Global Smooth Solution for Navier–Stokes/Poisson–Nernst–Planck System in \({\mathbb {R}}^{2}\)

Abstract

In this paper, we show global smoothness of solutions to the Navier–Stokes/Poisson–Nernst–Planck system for the transport and diffusion of ions in electrolyte solutions. For the multi-ionic species model, the key step to obtain global smoothness is to enhance the regularity of ions density and the velocity field by using the \(L^\infty L^p\)-estimate of the charge density, which is from a clear energy-dissipation equality. As their direct consequence, utilizing Duhamel’s principle, we obtain global smoothness for the multi-ionic species case in \({\mathbb {R}}^2\). Moreover, the decay rate of solutions for the coupled Navier–Stokes/Poisson–Nernst–Planck system with two-ionic species is given at the end of this paper.