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Large deviations for 2D Stochastic Chemotaxis-Navier-Stokes System

Yunfeng Chen, Xuhui Peng, Jianliang Zhai

Abstract

In this paper, we establish a large deviation principle for 2D stochastic Chemotaxis-Navier-Stokes equation perturbed by a small multiplicative noise. The main difficulties come from the lack of a suitable compact embedding into the space occupied by the solutions and the inherent complexity of equation. Finite dimensional projection arguments and introducing suitable stopping times play important roles.

Large deviations for 2D Stochastic Chemotaxis-Navier-Stokes System

Abstract

In this paper, we establish a large deviation principle for 2D stochastic Chemotaxis-Navier-Stokes equation perturbed by a small multiplicative noise. The main difficulties come from the lack of a suitable compact embedding into the space occupied by the solutions and the inherent complexity of equation. Finite dimensional projection arguments and introducing suitable stopping times play important roles.

Paper Structure

This paper contains 7 sections, 14 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Framework and Statement of the Main Results
  3. Proof of Proposition \ref{['Thm skeleton1']}
  4. Existence of Local Solutions
  5. Global existence
  6. Verification of (a) in Condition \ref{['cod']}
  7. Verification of (b) in Condition \ref{['cod']}

Key Result

Proposition 2.1

Assume and the assumptions ( H.1)-( H.5) hold. Then there exists a unique mild/variational solution $(n^\varepsilon,c^\varepsilon,u^\varepsilon)$ to the system (eq system 00). Moreover, $\mathbb{P}$-a.s., for any $T>0$,

Theorems & Definitions (21)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.2
  • Proposition 2.3
  • Theorem \oldthetheorem
  • proof
  • Lemma 2.1
  • Proposition 3.1
  • ...and 11 more