由拟线性反应扩散方程所驱动的非静态不可压缩Navier-Stokes系统  

Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation

作　　者：曾生达 Stanislaw Migdrski 刘振海[3] Shengda Zeng;Stanislaw Migorski;Zhenhai Liu

机构地区：[1]玉林师范学院数学与统计学院、广西高校复杂系统优化与大数据处理重点实验室,玉林537000 [2]Faculty of Mathematics and Computer Science,Jagiellonian University in Krakow,Krakow 30348,Poland [3]广西民族大学数学与物理学院,南宁530006

出　　处：《中国科学：数学》2022年第3期331-354,共24页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11671101);广西自然科学基金(批准号:2018GXNSFDA138002)资助项目。

摘　　要：本文考虑一类由非单调多值边界条件的非静态不可压缩Navier-Stokes方程和Neumann边界条件的广义拟线性反应扩散方程所构成的新动力系统,该系统可形式化为非线性抛物方程和发展型H半变分不等式所组成的耦合系统.为获得全局弱解存在性,首先,运用后向Euler法与反馈迭代技术引进耦合系统的时间半离散逼近问题;而后,沿用弱-弱上半连续集值算子的满射定理证明逼近问题解的存在性和先验估计;最后,采用极限过程建立系统全局弱解的存在性定理.In this paper,we consider a new class of a dynamic system which couples a nonst at ionary incompressible Navier-Stokes equation involving nonmonotone and multivalued boundary conditions of subdifferential type,and a generalized quasilinear reaction-diffusion equation with the Neumann boundary condition.The variational formulation of the system is provided in the form of an evolutionary hemivariational inequality coupled with the nonlinear parabolic equation.The main result concerning with the existence of global weak solutions to the system is proved.Firstly,the backward Euler technique and the feedback iterative method are used to develop a temporally semi-discrete approximate problem.Next,invoking a surjectivity result for weakly-weakly upper semicontinuous multivalued operators,the existence and a priori estimates for the solution to the semidiscrete approximate problem are established.Finally,using a limiting procedure for solutions to the approximate problem,a theorem on the existence of weak solutions to the system is obtained.

关 键 词：非静态不可压缩Navier-Stokes方程 H半变分不等式 Clarke次梯度 半群 反应扩散方程 半离散逼近系统 满射性 

分 类 号：O175[理学—数学]