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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (5): 807-821.doi: 10.3969/j.issn.1005-3085.2023.05.009

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含非线性阻尼的2D $g$-Navier-Stokes系统解的全局吸引子及渐近光滑效应

王小霞1,  姜金平1,  侯延仁2   

  1. 1. 延安大学数学与计算机科学学院，延安 716000; 
    2. 西安交通大学数学与统计学院，西安 710049
  • 收稿日期:2022-04-10 接受日期:2022-12-29 出版日期:2023-10-15 发布日期:2023-12-15
  • 基金资助:
    国家自然科学基金(11971378)；陕西省自然科学基础研究计划(2018JM1042)；陕西省大学生创新训练计划(S202110719115).

The Global Attractor and Asymptotic Smoothing Effect of the Solution for 2D $g$-Navier-Stokes System with Nonlinear Dampness

WANG Xiaoxia1,  JIANG Jinping1,  HOU Yanren2   

  1. 1. College of Mathematics and Computer, Yan'an University, Yan'an 716000;
    2. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
  • Received:2022-04-10 Accepted:2022-12-29 Online:2023-10-15 Published:2023-12-15
  • Supported by:
    The National Natural Science Foundation of China (11971378); the Natural Science Basic Research Program of Shaanxi Province (2018JM1042); the Innovation Training Plan for College Students in Shaanxi Province (S202110719115).

摘要：

Navier-Stokes方程作为流体动力学研究的基本方程之一，描述了作用于液体任意给定区域的力的动态平衡，反映了粘性流体流动的基本力学规律，在流体力学中具有十分重要的意义。作为Navier-Stokes方程的一种推广，近年来有关$g$-Navier-Stokes方程的研究工作方兴未艾。针对一类含非线性阻尼的自治$g$-Navier-Stokes系统的动力学性质展开研究，借助解的先验估计和能量方程方法，在${\bf R}^2$上证明了解的全局渐近紧性，得到解半群全局吸引子的存在性，并对解的渐近光滑效应进行了分析，进一步推广且改进了近年来已有的部分经典研究成果，丰富了$g$-Navier-Stokes方程的相关研究理论。

关键词: $g$-Navier-Stokes方程, 全局吸引子, 非线性阻尼, 渐近光滑效应

Abstract:

As one of the basic equations in fluid dynamics research, Navier-Stokes equation describes the dynamic balance of the force acting on any given region of liquid, reflecting the basic mechanical law of viscous fluid flow and has very important significance in fluid mechanics. As the generalization of the Navier-Stokes equation, the research on $g$-Navier-Stokes equation is flourishing in recent years. The dynamic properties of a class of autonomous $g$-Navier-Stokes systems with nonlinear dampness are studied. The global asymptotic compactness is proved by the prior estimation of solutions and the energy equation method on ${\bf R}^2$. The existence of global attractors about solvable semigroups is obtained, and the asymptotic smooth effect of solutions is analyzed. Some of the classical research results in recent years are further extended and improved, and the relevant research theories of $g$-Navier-Stokes equations are enriched.

Key words: $g$-Navier-Stokes equation, global attractors, nonlinear dampness, asymptotic sm-oothing effect

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