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

Abstract

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

Global Solutions of the Nernst-Planck-Euler Equations

“…There are very few results on the NPE system (1.1)- (1.5). A recent work [24] asserts that Hölder solutions may lose their regularity in finite time for equations which are purely inviscid (no ionic diffusivities). This work is based on the method of [15] which is applicable to C 1,α solutions of the Euler equations that are not Lipschitz.…”