在本計劃裡，我們研讀非線性守恆律解的漸進穩定性。在這個問題裡面要處理的方程式有(1)氣體動力學裡具有黏性項的尤拉方程、(2)解對於空間具有對稱性質的尤拉-泊桑方程，且此方程具有黏性項、 (3)磁流體力學的模型。其中我們關注的主題是討論與時間無關的解的非線性漸進穩定行為，此種解又分為光滑與不連續，我們將針對不同的情況去研究。首先我們用在奇異擾動中的幾何觀念的技巧，及動態系統的手法去證明某些無黏性流體的穩態解，都有一個相對應的具有黏性的穩態解。所以要研讀穩態解的漸進穩定行為就必須從具有黏性項的穩態解著手。在這研究中，我們先討論在尤拉方程中某些獨特的穩態解的線性穩定性，接著推廣先前的方法去得到穩態解的非線性的漸進穩定性。然後推廣此類方法到尤拉-泊桑方程及磁流體力學的模型。 ; In this project we study the asymptotic stability of solutions to nonlinear balance laws. The nonlinear balance laws we deal with here include Euler equations with viscosity and variable area duct in gas dynamic, Euler-Poisson equations with viscosity (radical symmetric case) and magnetohydrodynamics (MHD) model. The main topic we focus here is to establish the nonlinear asymptotic stability of either classical steady states or standing shocks. First, by the technique of geometric singular perturbations, we show that for some profiles of steady states for the invicid flow, there exit corresponding viscous profiles. Then we study the asymptotic stability of those viscous profiles. In this project, we first study the linear stability of some interesting steady states for the Euler equations in transonic flows, followed by the nonlinear asymptotic stability of steady states. Then we try to extend the analysis to this problem to Euler-Poisson equations and MHD model. ; 研究期間 9808 ~ 9907
