Global existence and large time behaviour for the pressureless Euler–Naver–Stokes system in ℝ3
Global existence and large time behaviour for the pressureless Euler–Naver–Stokes system in ℝ3
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DOI：
10.1017/prm.2023.16
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发表时间：
2023-02
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影响因子：
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通讯作者：
Shanshan Guo;Guochun Wu;Yinghui Zhang
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文献类型：
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作者：
Shanshan Guo;Guochun Wu;Yinghui Zhang
本文研究了无压Euler方程与等熵可压缩Navier-Stokes方程通过阻力强迫项耦合的两相流模型的整体Cauchy问题。该模型首先由Choi-Kwon [J. Differential Equations，261（1）（2016），pp. 654-711]通过采用Vlasov/可压缩Navier-Stokes方程的流体动力学极限。在初始扰动足够小的假设下，Choi-Kwon [J. Differential Equations，261（1）（2016），pp. 654-711]建立了三维周期域$\mathbb {T}^3 $的整体适定性和大时间行为。然而，到目前为止，三维Cauchy问题的整体适定性和大时间行为仍然没有得到解决。本文通过证明三维Cauchy问题经典解的整体存在性和最优衰减率来解决这个问题。本文的一个重要发现是，为了克服Euler方程中无压力项所带来的困难，我们充分利用Navier-Stokes方程的阻力强迫项和耗散结构来封闭无压力Euler方程中变量的能量估计。
We investigate the global Cauchy problem for a two–phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier–Stokes equations through a drag forcing term. This model was first derived by Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] by taking the hydrodynamic limit of the Vlasov/compressible Navier–Stokes equations. Under the assumption that the initial perturbation is sufficiently small, Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] established the global well–posedness and large time behaviour for the three dimensional periodic domain $\mathbb {T}^3$. However, up to now, the global well–posedness and large time behaviour for the three dimensional Cauchy problem still remain unsolved. In this paper, we resolve this problem by proving the global existence and optimal decay rates of classic solutions for the three dimensional Cauchy problem when the initial data is near its equilibrium. One of key observations here is that to overcome the difficulties arising from the absence of pressure in the Euler equations, we make full use of the drag forcing term and the dissipative structure of the Navier–Stokes equations to closure the energy estimates of the variables for the pressureless Euler equations.