Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations
Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations
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DOI：
10.1007/s00220-011-1334-6
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发表时间：
2011-10
影响因子：
2.4
通讯作者：
Zhenhua Guo;Hai-liang Li;Z. Xin
中科院分区：
文献类型：
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作者：
Zhenhua Guo;Hai-liang Li;Z. Xin
考虑了粘性系数依赖于密度的可压缩Navier-Stokes方程组，其原型是浅水运动的粘性Saint-Venat模型。本文证明了在应力自由边界条件和任意大数据量下，CNS自由边值问题的球对称弱解在时间上是全局存在的，自由边界将流体和真空分开，并以有限的速度传播，作为远离对称中心连续的粒子路径.详细地给出了该解的正则性和拉格朗日结构。特别是，它示出的粒子路径是唯一定义的从任何非真空区域远离对称中心，沿沿着真空状态不会形成在任何有限的时间和初始的解是保存。从任意一个非真空点开始，粒子的路径也是唯一地定义为时间向后的，它要么到达某个初始的非真空点，要么在一个小的中间时间停止并与真空连续连接。此外，自由边界被示出为在时间上以代数速率向外扩展，并且流体密度随着时间的增长在远离对称中心的几乎所有地方衰减并趋于零。这最终导致真空状态的形成几乎无处不在的时间去到无穷大。
The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.