Global Solutions of the Nernst-Planck-Euler Equations

TL;DR

This paper establishes the global existence and uniqueness of solutions for the coupled Nernst-Planck and Euler equations in two dimensions, including convergence from Navier-Stokes to Euler solutions as viscosity vanishes.

Contribution

It proves the global existence of weak and smooth solutions for the Nernst-Planck-Euler system in 2D, including convergence results from Navier-Stokes to Euler equations.

Findings

Global existence of weak solutions in L^p for vorticity

Global existence and uniqueness of smooth solutions

Convergence of Navier-Stokes solutions to Euler solutions as viscosity approaches zero

Abstract

We consider the initial value problem for the Nernst-Planck equations coupled to the incompressible Euler equations in $\mathbb T^2$. We prove global existence of weak solutions for vorticity in $L^p$. We also obtain global existence and uniqueness of smooth solutions. We show that smooth solutions of the Nernst-Planck-Navier-Stokes equations converge to solutions of the Nernst-Planck-Euler equations as viscosity tends to zero. All the results hold for large data.