Some studies on the stationary Navier-Stokes equations

Ph.D. ; In this thesis, we study the stationary solutions to the incompressible Navier-Stokes equations, including both the existence of weak solutions in bounded domains and the asymptotic behavior of solutions in exterior domains. These are some of difficult and challenging problems in this area. ; First, we consider the stationary Navier-Stokes equations in a 3D helically symmetric domain. We prove an existence theorem of helical invariant solutions for the nonhomogeneous boundary value problem. Our proof is based on Leray’s contradiction arguments and Korobkov-Pileckas-Russo approach which developed in [41, 39]. But unlike the previous works which rely strongly on a Bernoulli’s law for solutions to the Euler equations with low regularity, this property fails for helical flows. Indeed, we prove a new formulation of Bernoulli’s law for helical invariant solutions of the Euler equations which is weaker than the ones appeared before. In this way, we can still modify Korobkov-Pileckas-Russo’s approach to work in our case. ; Second, we study the solutions of the stationary Navier-Stokes equations with finite Dirichlet integrals in R3, which can be regarded a simplified case of the exterior problem. When the velocity converges to zero at infinity, currently there is no result on the decay rates of this kind of solutions. We prove some spatial decay properties of axisymmetric solutions and discuss their relations to the Liouville type theorem. Our proof is based on the line integral technique devoloped in [14, 65] and log-type Sobolev inequality used in [8]. We combine these two methods and give a uniform approach to obtain spatial decay rates. ; 在這篇論文中，我們研究不可壓縮Navier-Stokes 方程的穩態解在有界區域的存在性以及在外區域的漸近性態。它們是流體動力學中具有挑戰性的難題。 ; 首先，我們考慮穩態Navier-Stokes 方程在一個螺旋對稱區域中的非齊次邊值問題，證明了相應的螺旋不變弱解的存在性。我們的證明基於Leray 的反證法以及Korobkov-Pileckas-Russo 論文中的方法。但是與他們之前的工作不同的是，歐拉方程的螺旋不變弱解不滿足伯努利定律。我們證明了一個新形式的伯努利定律克服這個困難，我們的定理弱於之前的結果。利用這個定理，我們可以推廣Korobkov 等人的方法證明弱解的存在性。 ; 其次，我們研究穩態Navier-Stokes 方程在三維全空間具有有限Dirichlet ...