Mathematical analysis of stability ofviscous incompressible flows in several unbounded domains

粘性不可压缩流体的运动可以通过分析Navier-Stokes方程来理解。本文研究了该方程在两个重要的无界区域：开口区域和旋转体外部区域中解的稳定性。首先，孔径域是由一个孔径连接的两个分离的半空间的紧扰动。值得注意的是，即使对于斯托克斯方程，在通常的边界条件下也可能存在许多解。在本研究中，当通过孔径的通量被指定时，我们证明了唯一的整体解的存在性，并在数据较少的情况下推导出其大时间行为。其次，关于外问题，物体静止或平移的情况在许多文献中已经讨论过，而旋转的情况研究较少，因为，在参考系中，一个具有无界系数的漂移项的出现给数值计算带来了很大困难。本研究澄清了简化方程在参考系中的某些结构，从而克服了最关键的一步，最终得到了所需的定理。简而言之，我们得到了由线性部分生成的半群的时间衰减和小定常流的空间衰减，并利用它们证明了定常流的渐近稳定性。此外，我们还得到了各种Lebesgue空间中扰动的某些确定的衰减率。半群的衰变被描述为L_p-L_q估计，而定常流的衰变被理解为Lorentz空间，特别是弱-L_q空间。

The motion of a viscous incompressible fluids can be understood via analysis of the Navier-Stokes equation. The aim of this research is to study stability of solutions to this equation in the Mowing two important unbounded domains: aperture domain and exterior domain of a rotating body. First, the aperture domain is a compact perturbation of two separated half-spaces, which are connected by an aperture. It is remarkable that, even for the Stokes equation, many solutions may exist subject to usual boundary conditions. In this research, when a flux through the aperture is prescribed, we prove the existence of a unique global solution and deduce its large time behavior provided that the data are small Secondly, concerning the exterior problem, the case where the body is at rest or translating has been discussed in many papers, while the rotating case has been less studied because, in a reference frame, a drift term with unbounded coefficient appears and causes a lot of difficulties. The present research clarifies some structure of the reduced equation in the reference frame and, thereby, the most crucial step is overcome so that the desired theorem is finally obtained. In short, we derive both time decay of the semigroup generated by the linear part and spatial decay of a small steady flow and, by use of them, we prove the asymptotic stability of the steady flow. Moreover, we derive some definite decay rates of the disturbance in various Lebesgue spaces. The decay of the semigroup is described as L_p-L_q estimate, while the decay of the steady flow is understood in terms of Lorentz spaces, especially, weak-L_q spaces.