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Global existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes system in high space dimensions

Xiangsheng Xu

TL;DR

The paper proves the global existence of a strong solution to the initial-value NPNS system in $\mathbb{R}^N$ for $N\ge 3$ with any number of ionic species. The approach hinges on a novel scaling of the dependent variables combined with a De Giorgi iteration, yielding a uniform $L^{\infty}$ bound for $w=\sum_i c_i$ that is independent of the time horizon. This key estimate enables Calderón–Zygmund regularity for the electric potential $\phi$ and bootstraps to bounded velocity $u$, allowing the local strong solution to be extended globally. The method extends previous results for the two-species case to arbitrary $I$, providing a robust framework for global regularity of the NPNS system on unbounded domains. The results solidify the connection between weak and strong solutions and have implications for modeling ion transport in electrokinetic fluids.

Abstract

We study the existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes (NPNS) system in $\mathbb{R}^N, N\geq 3$. The system describes the electrodiffusion of ions in a viscous Newtonian fluid. A strong solution is obtained in any dimension of space without constraints on the number of species or the size of the given data.

Global existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes system in high space dimensions

TL;DR

The paper proves the global existence of a strong solution to the initial-value NPNS system in for with any number of ionic species. The approach hinges on a novel scaling of the dependent variables combined with a De Giorgi iteration, yielding a uniform bound for that is independent of the time horizon. This key estimate enables Calderón–Zygmund regularity for the electric potential and bootstraps to bounded velocity , allowing the local strong solution to be extended globally. The method extends previous results for the two-species case to arbitrary , providing a robust framework for global regularity of the NPNS system on unbounded domains. The results solidify the connection between weak and strong solutions and have implications for modeling ion transport in electrokinetic fluids.

Abstract

We study the existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes (NPNS) system in . The system describes the electrodiffusion of ions in a viscous Newtonian fluid. A strong solution is obtained in any dimension of space without constraints on the number of species or the size of the given data.
Paper Structure (3 sections, 4 theorems, 214 equations)

This paper contains 3 sections, 4 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1.1

Assume that A result in J asserts that there exists a local-in-time strong solution $(c_1,\cdots, c_I, u, \phi)$ to nsf1-nsf5 with $c_i\geq 0$. Define Then for each $L_N\geq 1$ there exist two positive numbers $C=C\left(N,z_1, \cdots, z_I, L_N\right)$ and $s_5=s_5(N, L_N)$ such that

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['wes']}
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of \ref{['cb']}
  • proof : Proof of \ref{['cb']}
  • Claim 3.1
  • ...and 2 more