Study on the Navier-Stokes equations on unbounded domains by way of real analysis

本文研究的是全平面或二维外区域上的Navier-Stokes方程。结果表明，如果存在一个强对称的小的定常外力，则方程有一个在无穷远处迅速衰减的小的定常解。同时还证明了，如果上述定态解足够小，则它在初始扰动下是稳定的，而不受其大小的限制。本文还研究了无限层中的Navier-Stokes方程。它表明，如果该方程被视为在Besov空间，非平凡的解决方案，没有外力存在，如果p是无限的，这些解决方案对应的Poiffille流。

This research is concerned with the Navier-Stokes equations on either the whole plane or two-dimensional exterior domains. It was shown that, if there exists a small stationary external force with strong symmetry, the equation has a small stationary solution decaying rapidly at infinity. It was also shown that, if the stationary solution above is sufficiently small, it is stable under initial perturbation without restriction on the size. The Navier-Stokes equaions in an infinite layer is also studied. It is shown that, if the equation is treated in the Besov spaces, nontrivial solutions with no external forces exist if p is infinite, and that these solutions correspond to the Poiseuille flows.