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Anisotropic micropolar fluids subject to a uniform microtorque: the stable case

Antoine Remond-Tiedrez and Ian Tice

Vol. 17 (2024), No. 1, 41–132
Abstract

We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that when the microstructure is inertially oblate (i.e., pancake-like) this equilibrium is nonlinearly asymptotically stable.

Our proof employs a nonlinear energy method built from the natural energy dissipation structure of the problem. Numerous difficulties arise due to the dissipative-conservative structure of the problem. Indeed, the dissipation fails to be coercive over the energy, which itself is weakly coupled in the sense that, while it provides estimates for the fluid velocity and microstructure angular velocity, it only provides control of two of the six components of the microinertia tensor. To overcome these problems, our method relies on a delicate combination of two distinct tiers of energy-dissipation estimates, together with transport-like advection-rotation estimates for the microinertia. When combined with a quantitative rigidity result for the microinertia, these allow us to deduce the existence of global-in-time decaying solutions near equilibrium.

Keywords
anisotropic micropolar fluid, nonlinear asymptotic stability
Mathematical Subject Classification
Primary: 35B40, 74A60, 76A05
Secondary: 35M31, 35Q30, 76D03
Milestones
Received: 13 April 2021
Revised: 28 April 2022
Accepted: 15 June 2022
Published: 5 February 2024
Authors
Antoine Remond-Tiedrez
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Cambridge
United Kingdom
Ian Tice
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA
United States

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