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Global well-posedness and exponential decay of strong solution for the three-dimensional inhomogeneous incompressible micropolar equations with density-dependent transport coefficients and large initial data

Peng Lu, Yuanyuan Qiao

TL;DR

This work proves global existence of strong solutions and exponential decay for the 3D inhomogeneous incompressible micropolar equations with density-dependent transport coefficients in a bounded domain, under Dirichlet boundary conditions and large initial data. The authors assume power-law density dependence for the viscosity and related coefficients, with $\alpha>1$ and $0<\beta\le(\alpha+1)/2$, and require the initial density to be linearly equivalent to a positive constant. The approach hinges on intricate energy methods, density-dependent Stokes regularity, and time- and exponential-weighted estimates to obtain uniform a priori bounds that close a bootstrap argument, ultimately establishing global well-posedness and exponential decay of $\|u\|_{H^1}$ and $\|w\|_{H^1}$. The results extend the understanding of micropolar flows with variable density and large data, providing a robust framework for stability analysis in bounded domains.

Abstract

In this paper, we consider the Dirichlet problem of three-dimensional inhomogeneous incompressible micropolar equations with density-dependent viscosity. Under the assumption that the coefficients are power functions of the density, we establish the global existence of strong solutions as long as the initial density is linear equivalent to a large constant state. There is no restriction on the size of initial velocity and micro-rotational velocity. As a by-product, we prove the exponential decay for the solution.

Global well-posedness and exponential decay of strong solution for the three-dimensional inhomogeneous incompressible micropolar equations with density-dependent transport coefficients and large initial data

TL;DR

This work proves global existence of strong solutions and exponential decay for the 3D inhomogeneous incompressible micropolar equations with density-dependent transport coefficients in a bounded domain, under Dirichlet boundary conditions and large initial data. The authors assume power-law density dependence for the viscosity and related coefficients, with and , and require the initial density to be linearly equivalent to a positive constant. The approach hinges on intricate energy methods, density-dependent Stokes regularity, and time- and exponential-weighted estimates to obtain uniform a priori bounds that close a bootstrap argument, ultimately establishing global well-posedness and exponential decay of and . The results extend the understanding of micropolar flows with variable density and large data, providing a robust framework for stability analysis in bounded domains.

Abstract

In this paper, we consider the Dirichlet problem of three-dimensional inhomogeneous incompressible micropolar equations with density-dependent viscosity. Under the assumption that the coefficients are power functions of the density, we establish the global existence of strong solutions as long as the initial density is linear equivalent to a large constant state. There is no restriction on the size of initial velocity and micro-rotational velocity. As a by-product, we prove the exponential decay for the solution.
Paper Structure (3 sections, 12 theorems, 82 equations)

This paper contains 3 sections, 12 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A priori estimates

Key Result

Theorem 1.1

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^3$. Assume that Given constants $\bar{\rho}>1$ and $C_0>1$, suppose that the initial data $(\rho_0,u_0, w_0)$ satisfies Then there exists a positive constant $M$ depending only on $C_0,\bar{\mu}, \bar{\xi}, \bar{\eta}, \bar{\lambda}, \alpha$, $\|\nabla\rho_0\|_{L^q}, \| u_0\|_{H^2}, \| w_0\|_{H^2}$ and $\Omega$ such that if then the problem

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • Lemma 5
  • Lemma 6
  • ...and 9 more