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

Antoine Remond-Tiedrez, Ian Tice

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.

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

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.

Paper Structure

This paper contains 89 sections, 93 theorems, 400 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Brief description of the model
  3. The equilibrium and its stability
  4. A brief summary of techniques and difficulties
  5. Statement of the main result
  6. Sharp nonlinear stability criterion
  7. Previous work
  8. Strategy and difficulties
  9. Perturbative formulation and overall strategy
  10. Lack of spectral gap
  11. Weak coupling
  12. A priori estimates
  13. Linear analysis
  14. $\theta$-coercivity and the two-tier energy structure.
  15. Nonlinear effects
  16. ...and 74 more sections

Key Result

Theorem 1.2

Suppose that the global assumptions of def:global_assumptions hold and let $X_{eq} = \left( u_{eq}, \omega_{eq}, J_{eq} \right) = \left( 0, \frac{\tau}{2\kappa} e_3, \mathop{\mathrm{diag}}\nolimits\left( \lambda, \lambda, \nu \right) \right)$ and $p_{eq} = 0$ be the equilibrium solution of eq:full_s Moreover the solutions satisfy the estimate Recall that the functionals present on the left-hand s

Figures (5)

  • Figure 1: Two rigid bodies with uniform density which possess an inertial axis of symmetry.
  • Figure 2: A pictorial summary of the argument carried out in \ref{['sec:no_spectral_gap']} which proves the lack of a spectral gap.
  • Figure 3: Pictorial summary of how the various pieces of the a priori estimates depend on one another. The arrows indicate the steps taken to close our scheme of a priori estimates -- c.f. \ref{['thm:a_priori']} for details. In the first pass all estimates obtained are in terms of the smallness parameter, this is indicated by the pink dashed arrows. In the second pass all estimates obtained are in terms of the initial conditions, this is indicated by the green solid arrows. Note that the seventh step of \ref{['thm:a_priori']} is omitted here since it plays an essential role in the propagation of the estimates over time, but is not essential for their closure.
  • Figure 4: How the terms in the interaction $\mathcal{I}_8$ are broken up in order to be estimated.
  • Figure 5: The strategy of the theorem for the main a priori estimates, namely \ref{['thm:a_priori']}.

Theorems & Definitions (186)

  • Definition 1.1: Global assumptions
  • Theorem 1.2: Nonlinear asymptotic stability and decay
  • Corollary 1.3: Decay rates
  • Theorem 1.4
  • Proposition 4.1: $L^p$ estimates for advection-rotation equations
  • proof
  • Lemma 4.2: $L^\infty$ estimate for $K$
  • proof
  • Lemma 4.3: $L^\infty$ estimate for $\nabla K$
  • proof
  • ...and 176 more