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

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.