Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit
Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit
复制标题
具有表面张力和零表面张力极限的自由边界可压缩3-D欧拉方程的适定性
DOI：
10.1137/120888697
复制
发表时间：
2013
影响因子：
2
通讯作者：
Coutand D
中科院分区：
文献类型：
--
作者：
Coutand D
证明了沿运动自由边界具有表面张力的三维可压缩欧拉方程是适定的；然后，当表面张力趋于零时，我们确定极限。具体来说，我们考虑等熵动力学，并考虑一个状态方程，模拟液体，由Courant和Friedrichs给出[超音速流动和激波，苹果公司]。数学。《科学》第21期，纽约，1976年]。与势能相关的两个相互竞争的非线性：体的压缩和自由边界的表面积动力学，使分析变得困难。与不可压缩欧拉方程的分析不同，不可压缩欧拉方程的边界规则性控制着内部的规则性，可压缩欧拉方程需要对产生声波的非线性波动方程进行额外的分析。采用一种特殊的抛物线正则化方法，结合消失黏度法，建立了存在性理论。选择人工抛物项是为了在零粘度极限下与欧拉方程渐近一致。有了正表面张力问题的解，我们继续获得与表面张力参数无关的先验估计。这需要选择满足泰勒符号条件的初始数据。通过传递到零表面张力的极限，证明了自由边界上无曲面且无导数损失的可压缩欧拉系统的适定性。
We prove that the three-dimensional compressible Euler equations with surface tension along the moving free-boundary are well-posed; we then establish the limit as surface tension tends to zero. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [Supersonic Flow and Shock Waves, Appl. Math. Sci. 21, Springer-Verlag, New York, 1976] asfor consantsand. The analysis is made difficult by two competing nonlinearities associated with the potential energy:compressionin the bulk andsurface area dynamicson the free-boundary. Unlike the analysis of the incompressible Euler equations, wherein boundary regularity controls regularity in the interior, the compressible Euler equation requires the additional analysis of nonlinear wave equations generating sound waves. An existence theory is developed by a specially chosen parabolic regularization together with the vanishing viscosity method. The artificial parabolic term is chosen so as to be asymptotically consistent with the Euler equations in the limit of zero viscosity. Having solutions for the positive surface tension problem, we proceed to obtain a priori estimates which are independent of the surface tension parameter. This requires choosing initial data which satisfy the Taylor sign condition. By passing to the limit of zero surface tension, we prove the well-posedness of the compressible Euler system without surface on the free-boundary and without derivative loss.
影响因子：
6
作者：
R. Temam
通讯作者：
R. Temam
可压缩粘性流体的表面波
DOI：
--
发表时间：
2003
期刊：
影响因子：
--
作者：
N. Tanaka;A. Tani
通讯作者：
A. Tani
DOI：
--
发表时间：
2017
期刊：
影响因子：
--
作者：
S. Nuanprasert;K. Lee;A. Murid;S. Baba;T. Suzuki;Sakajo Takashi
通讯作者：
Sakajo Takashi