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

本文研究了描述孤立有限体积粘性不可压缩流体在不考虑表面张力的情况下运动的Navier-Stokes方程的自由边界问题，利用拉格朗日坐标系，将自由边界问题写成固定边界上的拟线性方程。我们的目的是证明任何初始数据的局部时间唯一存在定理和一些小初始数据的全局时间唯一存在定理。为了处理拟线性方程，首先证明了Stokes方程的Neumann问题所描述的线性化问题解的Lp-Lq极大正则性。我们用解析半群的方法来考虑这个问题。我们的主要问题是使用r有界性和算子值傅立叶乘数定理，这是最近由Weis （‘01, Math.Ann.）和Denk-Hieber-Pruss （’03, mems . ams .）提出的。在线性化问题的Lp-Lq极大正则性结果的基础上，利用收缩映射原理，证明了在外力消失的情况下，对于任意初始数据和外力均存在的局部时间唯一定理，以及对于正交于刚性空间的少量初始数据在外力消失的情况下的全局时间唯一存在定理。

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.