摘要
本文研究由流体动理学(fluid-kinetic)耦合模型通过渐近分析推导出的无压Euler-Navier-Stokes两相流系统的整体适定性及长时间行为.Euler方程中缺少压力项给密度ρ(t,x)的一致估计带来很大挑战.为了克服这个困难,本文采取的办法是假设Navier-Stokes方程的初始扰动(n_(0)-1,v_(0))属于某个负Sobolev空间H^(-s)(R^(3))(s∈(1,3/2));然后利用能量法得到Euler方程中速度∥▽u∥_(H^(3))的时间衰减率(1+t)^(-1+s/2),结合上述估计和特征线方法得到无压Euler方程密度的一致估计;最后,借助对∥▽u∥_(H^(3))的时间衰减估计,本文建立该模型经典解的整体适定性.同时,本文证明解(u,n,v)在L^(2)范数意义下以代数衰减率衰减到平衡状态(0,1,0).
In this paper,we establish the global well-posedness and large-time behavior of the two-phase flow system consisting of the pressureless Euler equations and compressible Navier-Stokes equations coupled through the drag force,which can be derived from the kinetic-fluid model by asymptotic analysis.The absence of the pressure term in the Euler part brings enormous challenges for the uniform estimates of density ρ(t,x).To overcome that difficulty,our strategy is first to obtain the time-decay rate(1+t)^(-1+s/2) of||▽u||_(H^(3)) under the additional assumption that the initial perturbation(n_(0)-1,v_(0))for the viscous compressible flow belongs to some negative Sobolev space H^(-s)(R^(3))(s∈(1,3/2)).Then,combining the above decay estimates and characteristic method yields the uniform bound of the density to the pressureless Euler system.Finally,we establish the global well-posedness of the classical solution for this model with the help of the key time-decay rate for ||▽u||_(H^(3)).As a by-product,we also show that the solution(u,n,v)decays to the equilibrium state(0,1,0)at an algebraic time-decay rate in L^(2)-norm.
作者
郭柏灵
汤厚志
赵斌
Boling Guo;Houzhi Tang;Bin Zhao
出处
《中国科学:数学》
北大核心
2025年第8期1549-1566,共18页
Scientia Sinica:Mathematica
基金
国家自然科学基金(批准号:11731014和11571254)
中国博士后科学基金(批准号:2023M733691和2023M740334)资助项目。
