Global Existence of Large Solutions for the 3D incompressible Navier--Stokes--Poisson--Nernst--Planck Equations

TL;DR

This paper proves the global existence of large solutions for a complex 3D fluid model coupling Navier-Stokes, Poisson, and Nernst-Planck equations, without small initial data assumptions, using algebraic structure analysis.

Contribution

It establishes the global existence of solutions for the coupled system without smallness constraints on certain initial components, leveraging the system's algebraic structure.

Findings

Existence of global solutions under large initial data.

No smallness assumption on the third velocity component.

Unique solutions are guaranteed under specified initial conditions.

Abstract

This work is concerned with the global existence of large solutions to the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier--Stokes--Poisson--Nernst--Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants $c_{0}, C_{0}$ such that if the initial data satisfies \begin{align*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+\|N_{0}-P_{0}\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{C_{0}\big(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+(\|N_{0}+P_{0}\|_{\dot{B}^{-2+\frac{3}{r}}_{r,1}}+1)\exp\big\{C_{0}\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}\big\}+1\big)\Big\} \leq c_{0}, \end{align*} then the incompressible Navier--Stokes--Poisson--Nernst--Planck equations admits a unique global solution.