Anisotropic micropolar fluids subject to a uniform microtorque: the stable case
Antoine Remond-Tiedrez, Ian Tice · Analysis & PDE · 2024
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.1. Introduction 42 2. Strategy and difficulties 49 3. Notation 61 4. A priori estimates 63 5. Local well-posedness 93 6. Continuation argument 110 7. Global well-posedness and decay 119 Appendix A. Identities involving the microinertia 126 Appendix B. Analytical results 128 References 130This paper, together with the companion paper [Remond-Tiedrez and Tice 2021], provides a sharp nonlinear stability criterion for an anisotropic micropolar fluid subject to a uniform microtorque.The companion paper is concerned with the unstable regime; we tackle the stable regime here.Note to the reader: The introduction of Section 1 serves as a "shortest path" to the main result recorded in Theorem 1.2, providing the necessary physical and mathematical background to appropriately state the main result.For a more detailed discussion of the problem and the strategy employed to prove nonlinear stability, we direct the reader's attention to Section 2.