Some Studies on the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

本論文致力於研究黏性退化的正壓可壓Navier-Stokes流。我們主要考慮有界區域上黏性退化的可壓Navier-Stokes方程初邊值問題的弱解的存在性，邊界條件取Navier型滑動邊界條件，初值是一般初值。我們採用Li-Xin在[22]中的逼近方程， 考慮密度滿足第二類邊界條件，速度滿足摩擦力等於邊界曲率的Navier型滑動邊界條件。首先，當有界區域具有非負的曲率時，這樣的逼近系統的光滑解滿足下面的先驗估計：能量估計，BD熵和Mellet-Vasseur型估計。然後，我們證明了這個逼近系統確實存在唯一的強解。最後，我們用Mellet-Vasseur的緊性理論[19]說明逼近解的極限函數在某種意義下確實是一個弱解。因此，我們用[22]的證明框架說明了，當黏性壓力張量形式為__ru+(1�_)__divuI，_ 2 [1; 118 )， 在三維 2 [2_�1; 3_�1]\(1; 3)，或者在二維 2 [2_ � 1;1) \ (1;1)，對一般的大初值，摩擦力等於邊界曲率的Navier型滑動邊界條件，可壓Navier-Stokes方程確實存在一個全局弱解。This thesis is devoted to some studies on the barotropic compressible Navier-Stokes ows with degenerate viscosities. We mainly consider the global existence of weak solutions to compressible Navier-Stokes equations with density-dependent viscosities in bounded domains with Navier-slip boundary conditions for general initial data.We use the approximate equations constructed by Li-Xin in [22], and impose the Neumann boundary condition for the density and Navier-slip boundary conditions for u where the friction equals curvature of the boundary. First, we show that the following a priori estimates: the energy estimates, BD-entropy, and the Mellet-Vasseur type estimates, hold for smooth solutions to the approximate system under the condition that the domains have nonnegative curvature. Then we prove that this approximate system indeed has a strong solution. Finally, we use the compactness due to Mellet-Vasseur [19] to show that the limit is indeed a weak solution in some sense. Hence, by following the proof scheme in [22], we show that for2 [1; 118 ); 2 [2�1; 3�1]\(1; 3) when N = 3 or 2 [2when N = 2, there exists a global weak solution to the compressible Navier-Stokes equations with the stress tensor having the form ru+(1�) divuI in bounded domains with Navier-slip boundary conditions for the general initial data.Zhang, Rong.Thesis M.Phil. Chinese University of Hong Kong 2016.Includes bibliographical references (leaves ).Abstracts also in Chinese.Title from PDF title page (viewed on …).Detailed summary in vernacular field only.Detailed summary in vernacular field only