Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations
Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations
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DOI：
10.1007/s00220-008-0495-4
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发表时间：
2008-04
影响因子：
2.4
通讯作者：
Hai-Liang Li;Jing Li;Zhouping Xin
中科院分区：
文献类型：
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作者：
Hai-Liang Li;Jing Li;Zhouping Xin
本文考虑粘性依赖于密度的可压缩流体的Navier-Stokes方程组。这些方程尤其包括最近严格推导出的浅水运动圣维南方程组，这些方程组是从具有运动自由面的不可压缩流的Navier-Stokes方程组[14]中推导出来的。当真空态出现时，这些可压缩系统是退化的。我们研究这类系统的初边值问题的有界空间域或周期域。对弱解和真空态的动力学进行了严格的研究，首先证明了满足有限初始熵的一般大初值下熵弱解在时间上全局存在。其次，对于较规则的初始数据，在较短的时间内，存在唯一的、规则的、具有良好速度场的整体熵弱解，初始真空的界面在这段时间内沿粒子路径沿着传播.然后，证明了对于任何整体熵弱解，任何（可能存在的）真空态必在有限时间内消失。当真空态消失时，速度（即使足够规则且定义良好）在有限时间内爆炸。当真空态消失后，整体熵弱解变为强解，并随时间指数地趋向于非真空平衡态。
The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.