在本計畫中我們考慮三個子問題：深水波方程、不可壓縮極限問題與建構一個在兩個具有 Sobolev 正則性的集合上的微分同胚映射問題。關於深水波方程，我們考慮使用漸近展開的方式來研究小初值的水波問題，並設法提出水波方程的三階逼近。關於不可壓縮極限問題，我們考慮帶有科氏力的等熵可壓縮 Navier-Stokes 方程在馬赫數趨近到零時的極限問題。關於建構微分同胚映射的問題，我們希望建構一個在兩個具有 Sobolev 正則性的集合上的微分同胚映射，其中此映射的 Jacobian 和邊界條件皆與時間相關，並找到此映射其對時間微分的估計。 ;In this project we consider three problems: the water wave equation, the incompressible limit problem, and the problem of diffeomorphisms with prescribing Jacobian and boundary data. As for the water wave equation, we make us of the asymptotic expansion to investigate the water wave equation with small initial data. We propose the cubic order model of the water wave equation which is never seen before. As for the incompressible limit problem, we consider the limit of isentropic compressible Navier-Stokes equations as the Mach number approaches zero. As for the diffeomorphism problem, we hope to construct a diffeomorphism between two domains of Sobolev class regularity with prescribed time-dependent Jacobian and boundary conditions, as well as estimates for the time derivatives of the diffeomorphisms.
