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Anisotropic
micropolar fluids subject to a uniform microtorque: the
stable case
Antoine Remond-Tiedrez and Ian Tice |
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Vol. 17 (2024), No. 1, 41–132
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Abstract |
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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
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Mathematical Subject Classification
Primary: 35B40, 74A60, 76A05
Secondary: 35M31, 35Q30, 76D03
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Milestones
Received: 13 April 2021
Revised: 28 April 2022
Accepted: 15 June 2022
Published: 5 February 2024
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Authors |
| Antoine Remond-Tiedrez | |
| Department of Applied Mathematics
and Theoretical Physics University of Cambridge Cambridge United Kingdom |
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| Ian Tice | |
| Department of Mathematical
Sciences Carnegie Mellon University Pittsburgh, PA United States |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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