在第二個研究子題裡，我們討論帶有表面張力的 Navier-Stokes 方程在自由邊界下的解存在性。為了處理與時間有關的邊界，我們引進 ALE formulation，找到某個良好的微分映射，將原本隨著時間而改變的邊界利用此映射變換後固定，然後經過光滑化、線化性的處理並使用固定點定理，最後證明出此系統的解存在性。;There are two sub-projects in this dissertation: one is the problem of the incompressible limits for rotational compressible MHD flows and another is the well-posedness of Navier-Stokes equations with surface tension in an optimal Sobolev space.
In the first sub-project, we consider the compressible models of magnetohydrodynamic flows which gives rise to a variety of mathematical problems in many areas. We introduce the asymptotic limit for the compressible rotational magnetohydrodynamic flows with the well-prepared initial data such that a rigorous quasi-geostrophic equation with diffusion term governed by the magnetic field from a compressible rotational magnetohydrodynamic flows is derived. After that, we show the two results: the existence of the unique global strong solution of quasi-geostrophic equation with good regularity on the velocity and magnetic field and the derivation of quasi-geostrophic equation with diffusion.
In the second sub-project, we establish the existence of a solution to the 2-dimensional Navier-Stokes equations with surface tension on a moving domain. To deal with the free boundary problem, we adopt the ALE formulation which transform the moving domain into a fixed domain. Next, we show the well-posedness of this systems in an optimal sobolev space and no compatibility conditions are required to guarantee the existence of a solution.