论文标题
各向异性微极流体受到均匀微型液体：稳定的情况
Anisotropic micropolar fluids subject to a uniform microtorque: the stable case
论文作者
论文摘要
我们在周期域上研究了具有各向异性微观结构的三维，不可压缩的，粘性的，微极流体。该系统受到均匀的微型尺度的影响，该系统允许独特的非平衡平衡。我们证明，当微观结构惯性地植入（即煎饼状）时，这种平衡在非线性渐近稳定时。 我们的证明采用了一种非线性能量方法，它是根据问题的自然能量耗散结构构建的。由于该问题的耗散保守结构，出现了许多困难。的确，耗散无法在能量上强制性，而能量本身是微弱耦合的，虽然它提供了对流体速度和微结构角速度的估计，但它仅提供对微量张量的六个组成部分中的两个。为了克服这些问题，我们的方法依赖于两个不同的能量散落估计值的精致组合，以及微渗透的类似运输的对流旋转估计值。当与微文化的定量刚性相结合时，这些结合使我们能够推断出在平衡附近的全球时间衰减溶液的存在。
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.
