Global existence and time decay estimate of solutions to the compressible Navier–Stokes–Korteweg system under critical condition
Global existence and time decay estimate of solutions to the compressible Navier–Stokes–Korteweg system under critical condition
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DOI：
10.3233/asy-201600
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发表时间：
2019-05
影响因子：
1.4
通讯作者：
Kobayashi Takayuki;Kazuyuki Tsuda
中科院分区：
文献类型：
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作者：
Kobayashi Takayuki;Kazuyuki Tsuda
在本研究中，我们研究可压缩Navier-Stokes-Korteweg方程组在一个定常状态下整体解的存在性。该系统描述了具有扩散界面的相变的液-汽型两相流。以前的工作假设压力是一个单调的函数的密度变化类似于通常的可压缩Navier-Stokes系统。另一方面，由于相变，压力实际上是非单调函数，并且线性化系统在临界情况下失去对称性，使得在给定的恒定状态下压力的导数为0。我们证明了在小数据的临界情形下，当动量为导数形式时，整体L2解是存在的，并得到了解的抛物型衰减率。这是证明的基础上分解的解决方案的低频部分和高频部分。
In this research, we study the global existence of solutions to the compressible Navier–Stokes–Korteweg system around a constant state. This system describes liquid-vapor type two-phase flow with a phase transition with diffuse interface. Previous works assume that pressure is a monotone function for change of density similarly to the usual compressible Navier–Stokes system. On the other hand, due to phase transition the pressure is in fact non-monotone function, and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. We show that global L 2 solutions are available for the critical case of small data, whose momentum is in its derivative form, and obtain parabolic type decay rate of the solutions. This is proved based on the decomposition of solutions to a low frequency part and a high frequency part.