Existence of $C^{1,\alpha}$ Singular Solutions to Euler-Nernst-Planck-Poisson System on $\mathbb{R}^3$ with Free-Moving Charges

TL;DR

This paper constructs a specific class of singular solutions with $C^{1,eta}$ regularity for a 3D electro-hydrodynamics system combining Euler and Nernst-Planck-Poisson equations, demonstrating the existence of free-moving charge singularities.

Contribution

It introduces a novel $C^{1,eta}$ blow-up solution for the coupled Euler-Nernst-Planck-Poisson system, extending previous frameworks to include free-moving charges.

Findings

Existence of $C^{1,eta}$ singular solutions in 3D electro-hydrodynamics

Construction based on special spherical Laplacian solutions

Extension of Elgindi's framework to coupled systems

Abstract

We construct a special $C^{1,\alpha}$ blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst-Planck-Poisson equation. Our construction lies on the framework established in Elgindi [11] and relies on a special solution to variant spherical Laplacian.