可压Navier-Stokes方程真空状态的动力学行为

Dynamics of vacuum states for the compressible Navier-Stokes equations

本文主要考虑了一维可压Navier-Stokes方程真空状态的动力学行为.对于任意的熵弱解,如果初始状态不存在真空,我们证明了密度函数关于时间和空间变量是连续的且对于任意时间它是处处为正的.同时,我们还得到了含有间断连接的真空状态的整体熵弱解的存在性,结果显示其真空区域以代数速率被压缩,并在有限时间内消失.

The dynamics of vacuum states for 1D compressible Navier-Stokes equations are considered. For any global entropy weak solution, we show that the flow density is continuous on both space and time, and is positive everywhere for all the time, if no vacuum state exists initially （i.e., non-formation of vacuum states happens）. Furthermore, we prove that there is a global weak solution which contains one piece of discontinuous finite vacuum for some time period, meanwhile the vacuum is shown to be compressed at an algebraic rate and then vanishes within finite time.