各向异性微极流体在均匀微力矩作用下的稳定情形
Anisotropic micropolar fluids subject to a uniform microtorque: the stable case
AI总结：
本文研究了周期域上受均匀微力矩作用的三维不可压缩粘性各向异性微极流体，证明了当微结构呈扁平状时其非平凡平衡态的非线性渐近稳定性。
摘要：
我们在周期域上研究一种三维、不可压缩、粘性、具有各向异性微结构的微极流体。在均匀微力矩作用下，该系统存在唯一的非平凡平衡态。我们证明，当微结构在惯性上呈扁平状（即类似薄饼状）时，该平衡态是非线性渐近稳定的。我们的证明采用了基于该问题自然能量耗散结构构建的非线性能量方法。由于问题的耗散-保守结构，出现了许多困难。事实上，耗散无法对能量具有强制性，而能量本身是弱耦合的，即虽然它提供了对流体速度和微结构角速度的估计，但它仅能控制微惯性张量六个分量中的两个。为了克服这些问题，我们的方法依赖于两层不同能量-耗散估计的精巧结合，以及微惯性的类传输对流-旋转估计。结合微惯性的定量刚性结果，这些方法使我们能够推导出平衡态附近整体时间衰减解的存在性。
英文摘要：
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.
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