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Global well-posedness and enhanced dissipation for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise

Quyuan Lin, Rongchang Liu, Weinan Wang

Abstract

In this paper, we consider the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise. By assuming the ionic species have the same diffusivity and opposite valences, we prove the global well-posedness of the system. Furthermore, we illustrate the enhanced dissipation phenomenon in the system with specific transportation noise by establishing that it enables an arbitrarily large exponential convergence rate of the solutions.

Global well-posedness and enhanced dissipation for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise

Abstract

In this paper, we consider the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise. By assuming the ionic species have the same diffusivity and opposite valences, we prove the global well-posedness of the system. Furthermore, we illustrate the enhanced dissipation phenomenon in the system with specific transportation noise by establishing that it enables an arbitrarily large exponential convergence rate of the solutions.
Paper Structure (15 sections, 17 theorems, 170 equations)

This paper contains 15 sections, 17 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. The Nernst-Planck-Navier-Stokes system
  3. NPNS with transport noise and enhanced dissipation
  4. Organization of the paper
  5. Preliminaries and main results
  6. Preliminaries
  7. Main results
  8. Analysis of the cut-off system
  9. Uniform estimates of the Galerkin system and compactness
  10. Global martingale solutions of the cut-off system
  11. Pathwise uniqueness of the cut-off system
  12. Non-negativity and global well-posedness
  13. Enhanced dissipation
  14. Energy inequality and exponential stability
  15. Enhanced dissipation

Key Result

Theorem 2.2

Let $(u_0,c_1(0),c_2(0)) \in L^2\left(\Omega; H\times L^2\times L^2\right)$ be $\mathcal{F}_0$-measurable random variables such that $c_1(0),c_2(0)\geq 0$$\mathbb P$-a.s.. Assume $b, \theta\in L_2(\mathscr{U}, L^\infty)$. Then there exists a unique global pathwise solution to the system e.w04091 in

Theorems & Definitions (33)

  • Definition 2.1: Pathwise solution
  • Theorem 2.2: Global well-posedness
  • Theorem 2.3: Enhanced dissipation
  • Theorem 3.1
  • proof
  • Definition 3.2: Martingale solution
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 23 more