各向异性刚体与无界 2D 通道中 Poiseuille 流相互作用的长时间行为
Long-time behavior of an anisotropic rigid body interacting with a Poiseuille flow in an unbounded 2D channel
AI总结：
研究椭圆刚体在无界2D通道中受Poiseuille流作用的长时间动力学行为，证明弱解的全局存在性及小流速下回归平衡，首次分析了具有非平凡最终状态的流固相互作用问题。
摘要：
我们研究一个椭圆刚体在 2D 无界通道中允许垂直平移和旋转的长时间行为，该刚体在远距离处受到 Poiseuille 流的作用。流体运动由不可压缩 Navier-Stokes 方程建模，而固体运动通过 Newton 定律描述。除了固体惯性和流体动力学力之外，我们假设固体的动力学由内部弹性恢复力驱动，但没有任何结构阻尼。通过对椭圆体接近通道边界时运动的精确描述，我们证明了弱解的全局时间存在性。我们的第二个主要贡献是证明了在 Poiseuille 流幅度较小的情况下系统会回归平衡。据我们所知，这是对具有给定非平凡最终状态的流体-固体相互作用问题的首次长时间分析。
英文摘要：
We study the long-time behavior of an elliptic rigid body which is allowed to vertically translate and rotate in a 2D unbounded channel under the action of a Poiseuille flow at large distances. The motion of the fluid is modelled by the incompressible Navier-Stokes equations, while the motion of the solid is described through Newton's laws. In addition to the solid inertia and the hydrodynamic forces, we assume the dynamics of the solid is driven by internal elastic restoring forces but without any structural damping. Through a precise description of the motion of the elliptic body whenever it comes close to the channel boundaries, we prove global-in-time existence of weak solutions. Our second main contribution is a proof of return to equilibrium in case the amplitude of the Poiseuille flow is small. \black To our knowledge, this represents the first long-time analysis of fluid-solid interaction problems with a given non-trivial final state.
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