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# Dynamics of Partial Differential Equations

## Volume 21 (2024)

### Number 1

### Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity

Pages: 77 – 96

DOI: https://dx.doi.org/10.4310/DPDE.2024.v21.n1.a4

#### Authors

#### Abstract

We show the global well-posedness to the three-dimensional (3D) Cauchy problem of nonhomogeneous micropolar fluids with density-dependent viscosity and vacuum in $\mathbb{R}^3$ provided that the initial mass is sufficiently small. Moreover, we also obtain that the gradients of velocity and micro-rotational velocity converge exponentially to zero in $H^1$ as time goes to infinity. Our analysis relies heavily on delicate energy estimates and *the structural characteristic of the system under consideration*. In particular, the initial velocity and micro-rotational velocity could be arbitrarily large.

#### Keywords

nonhomogeneous micropolar fluids, global well-posedness, density-dependent viscosity, large initial velocity, large initial micro-rotational velocity

#### 2010 Mathematics Subject Classification

Primary 35Q35. Secondary 76D03.

This research was partially supported by National Natural Science Foundation of China (Nos. 11971393, 12371120).

Received 3 February 2023

Published 7 November 2023