ON THE TWO-PHASE NAVIER-STOKES EQUATIONS WITH SURFACE TENSION
ON THE TWO-PHASE NAVIER-STOKES EQUATIONS WITH SURFACE TENSION
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DOI：
10.4171/ifb/237
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发表时间：
2009-08
影响因子：
1
通讯作者：
Jan Pruess;G. Simonett
中科院分区：
文献类型：
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作者：
Jan Pruess;G. Simonett
纳维-斯托克斯系统的两相自由边界问题是在初始界面接近半平面的情况下考虑的。通过底层线性问题的 Lp 最大正则性，我们展示了问题的局部适定性，并证明该解（特别是界面）立即成为实解析的。在本文中，我们考虑描述两种粘性不可压缩毛细管牛顿流体运动的自由边界问题。流体被未知的界面分开，必须作为问题的一部分来确定。设 1(0) ⊂ R n+1 (n ≥ 1) 为粘性不可压缩流体 Fluid1 所占据的区域，设 2(0) 为 R n+1 中 1(0) 闭包的补集，对应于第二种不可压缩粘性流体 Fluid2 所占据的区域。我们假设这两种流体是不混溶的。令 0 为边界 1(0)（因此也边界 2(0)）的超曲面，并令 (t) 表示 0 在时间 t 的位置。因此，( t) 是一个尖锐的界面，它将分别占据区域 1(t) 和 2(t) 的流体分开，其中 2(t) := R n+1 \ 1(t)。我们用 ν(t, � ) 表示 ( t) 上的法线场，从 1(t) 指向 2(t)。此外，我们分别用 V (t, � ) 和 κ(t, � ) 表示法向速度和 ( t) 相对于 ν(t, � ) 的平均曲率。这里，当 1(t) 在 x ∈ ( t) 的邻域内凸时，假设曲率 κ(x, t) 为负。对于 i = 1,2 ，流体的运动由以下方程组控制：
The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic. In this paper we consider a free boundary problem that describes the motion of two viscous incompressible capillary Newtonian fluids. The fluids are separated by an interface that is unknown and has to be determined as part of the problem. Let 1(0) ⊂ R n+1 (n ≥ 1) be a region occupied by a viscous incompressible fluid, fluid1, and let 2(0) be the complement of the closure of 1(0) in R n+1 , corre- sponding to the region occupied by a second incompressible viscous fluid, fluid2. We assume that the two fluids are immiscible. Let 0 be the hypersurface that bounds 1(0) (and hence also 2(0)) and let ( t) denote the position of 0 at time t. Thus, ( t) is a sharp interface which separates the fluids occupying the regions 1(t) and 2(t), respectively, where 2(t) := R n+1 \ 1(t). We denote the normal field on ( t), pointing from 1(t) into 2(t), by ν(t, � ). Moreover, we de- note by V (t, � ) and κ(t, � ) the normal velocity and the mean curvature of ( t) with respect to ν(t, � ), respectively. Here the curvature κ(x, t) is assumed to be negative when 1(t) is convex in a neighborhood of x ∈ ( t). The motion of the fluids is governed by the following system of equations for i = 1,2 :