The approach of an analytic semigroup for free boundary problems of viscous compressible fluids

本文考虑了描述不可压缩粘性流体孤立有限体积运动的Navier-Stokes方程的自由边界问题，该问题不考虑表面张力。利用拉格朗日坐标，将自由边界问题转化为固定边界上的拟线性方程组。我们的目的是证明一个局部的时间唯一性存在定理，对于任何初始数据和一个整体的时间唯一性存在定理的一些小的初始数据.为了处理拟线性方程，首先我们证明了线性化问题的解的Lp-Lq最大正则性，这是描述的Neumann问题的Stokes方程.我们考虑这个问题的分析半群的方法。我们的主要问题是利用Weis（'01，Math.Ann.）和Denk-Hieber-Pruss（'03，Mem.AMS）.在线性化问题的Lp-Lq极大正则性结果的基础上，利用压缩映射原理，证明了在外力为零的情况下，对于任意初值和外力，局部时间唯一性存在定理和对于正交于刚性空间的小初值，整体时间唯一性存在定理.

In this research, we consider a free boundary problem for the Navier-Stokes equation which describes the motion of an isolated finite volume of viscous incompressible fluid without taking surface tension into account By using the Lagrange coordinates, free boundary problem is written by the quasi-linear equations on the fixed boundary. Our purpose is to prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data.To treat quasi-linear equations, first we prove the Lp-Lq maximal regularity of solutions to the linearized problem, which is described by the Neumann problem for the Stokes equation. We consider this problem by analytic semigroup approach. Our main issues is to use R-boundedness and operator valued Fourier multiplier theorem which are recently developed by Weis ('01, Math.Ann.) and Denk-Hieber-Pruss ('03, Mem.AMS).Based on the Lp-Lq maximal regularity result of the linearized problem, by using the contraction mapping principle, we proved a local in time unique existence theorem for any initial data and external force and a global in time unique existence theorem for some small initial data which are orthogonal to the rigid space in the case where external force vanishes.