不可压微极流方程组的稳定性研究
Research on the Stability of Incompressible Micropolar Equations
DOI: 10.12677/pm.2025.155169, PDF,   
作者: 侯 洁：成都理工大学数学科学学院，四川 成都
关键词: 微极流方程组；稳定性；频率分解；Micropolar Equations； Stability； Frequency Decomposition
摘要: 本文研究了不可压微极流方程组的稳定性问题。论文首先介绍了微极流方程组的背景以及它的研究现状，给出了本文的研究内容与结论。其次给出了论文相关的不等式及相关的引理。最后证明了论文的主要定理，通过频率分解方法得到了方程组在平衡态附近扰动后解的稳定性。
Abstract: This paper investigates the stability problem of incompressible micropolar equations. It begins by introducing the background of micropolar equations and their current research status, presenting the research content and conclusions of this study. Subsequently, relevant inequalities and lemmas pertinent to the paper are provided. Finally, the main theorem of the paper is proved, and the stability of solutions to the equations following a perturbation near the equilibrium state is obtained through the method of frequency decomposition.
文章引用：侯洁. 不可压微极流方程组的稳定性研究[J]. 理论数学, 2025, 15(5): 203-211. https://doi.org/10.12677/pm.2025.155169

参考文献

[1] Eringen, A.C. (2002) Electrodynamics of Continua II: Fluids and Complex Media. Springer.
[2] Cowin, S.C. (1968) Polar Fluids. The Physics of Fluids, 11, 1919-1927. [Google Scholar] [CrossRef] 
[3] Eringen, A.C. (1969) Micropolar Fluids with Stretch. International Journal of Engineering Science, 7, 115-127. [Google Scholar] [CrossRef] 
[4] Łukaszewicz, G. (1999) Selected Applications. In: Bellomo, N. and Tezduyar, T.E., Eds., Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 181-235. [Google Scholar] [CrossRef] 
[5] Constantin, P. and Foias, C. (1988) Navier-Stokes Equations. University of Chicago Press. [Google Scholar] [CrossRef] 
[6] Cont, R. and Tankov, P. (2004) Financial Modeling with Jump Processes. Chapman and Hall/CRC Press.
[7] Gill, A.E. (1982) Atmosphere-Ocean Dynamics. Academic Press.
[8] Katz, N.H. and Pavlović, N. (2002) A Cheap Caffarelli-Kohn-Nirenberg Inequality for the Navier—Stokes Equation with Hyper-Dissipation. Geometric and Functional Analysis, 12, 355-379. [Google Scholar] [CrossRef] 
[9] Kato, T. and Ponce, G. (1988) Commutator Estimates and the Euler and Navier-Stokes Equations. Communications on Pure and Applied Mathematics, 41, 891-907. [Google Scholar] [CrossRef] 
[10] Kenig, C.E., Ponce, G. and Vega, L. (1991) Well-Posedness of the Initial Value Problem for the Korteweg-De Vries Equation. Journal of the American Mathematical Society, 4, 323-347. [Google Scholar] [CrossRef] 
[11] Stokes, V.K. (1984) Theories of Fluids with Microstructure. Springer.
[12] Bahouri, H., Chemin, J.-Y. and Danchin, R. (2011) Fourier Analysis and Nonlinear Partial Differential Equations. Springer.
[13] Chen, Q. and Miao, C. (2012) Global Well-Posedness for the Micropolar Fluid System in Critical Besov Spaces. Journal of Differential Equations, 252, 2698-2724. [Google Scholar] [CrossRef] 
[14] Rashidi, M.M., Keimanesh, M., Bég, O.A. and Hung, T.K. (2010) Magnetohydrodynamic Biorheological Transport Phenomena in a Porous Medium: A Simulation of Magnetic Blood Flow Control and Filtration. International Journal for Numerical Methods in Biomedical Engineering, 27, 805-821. [Google Scholar] [CrossRef] 
[15] Raptis, A. (2000) Boundary Layer Flow of a Micropolar Fluid through a Porous Medium. Journal of Porous Media, 3, 95-97. [Google Scholar] [CrossRef] 
[16] Turkyilmazoglu, M. (2014) A Note on Micropolar Fluid Flow and Heat Transfer over a Porous Shrinking Sheet. International Journal of Heat and Mass Transfer, 72, 388-391. [Google Scholar] [CrossRef] 
[17] Turkyilmazoglu, M. (2016) Flow of a Micropolar Fluid Due to a Porous Stretching Sheet and Heat Transfer. International Journal of Non-Linear Mechanics, 83, 59-64. [Google Scholar] [CrossRef] 
[18] Chen, M. (2013) Global Well-Posedness of the 2D Incompressible Micropolar Fluid Flows with Partial Viscosity and Angular Viscosity. Acta Mathematica Scientia, 33, 929-935. [Google Scholar] [CrossRef] 
[19] Dong, B. and Zhang, Z. (2010) Global Regularity of the 2D Micropolar Fluid Flows with Zero Angular Viscosity. Journal of Differential Equations, 249, 200-213. [Google Scholar] [CrossRef] 
[20] Dong, B., Li, J. and Wu, J. (2017) Global Well-Posedness and Large-Time Decay for the 2D Micropolar Equations. Journal of Differential Equations, 262, 3488-3523. [Google Scholar] [CrossRef] 
[21] Shang, H. and Zhao, J. (2017) Global Regularity for 2D Magneto-Micropolar Equations with Only Micro-Rotational Velocity Dissipation and Magnetic Diffusion. Nonlinear Analysis: Theory, Methods & Applications, 150, 194-209. [Google Scholar] [CrossRef] 
[22] Xue, L. (2011) Well-Posedness and Zero Microrotation Viscosity Limit of the 2D Micropolar Fluid Equations. Mathematical Methods in the Applied Sciences, 34, 1760-1777. [Google Scholar] [CrossRef] 
[23] Dong, B. and Chen, Z. (2009) Regularity Criteria of Weak Solutions to the Three-Dimensional Micropolar Flows. Journal of Mathematical Physics, 50, Article 103525. [Google Scholar] [CrossRef] 
[24] Chen, Z. and Price, W.G. (2006) Decay Estimates of Linearized Micropolar Fluid Flows in R3 Space with Applications to L3-Strong Solutions. International Journal of Engineering Science, 44, 859-873. [Google Scholar] [CrossRef] 
[25] Dong, B. and Chen, Z. (2009) Asymptotic Profiles of Solutions to the 2D Viscous Incompressible Micropolar Fluid Flows. Discrete & Continuous Dynamical Systems A, 23, 765-784. [Google Scholar] [CrossRef] 
[26] Liu, H., Sun, C. and Meng, F. (2019) Global Well-Posedness of the 3D Magneto-Micropolar Equations with Damping. Applied Mathematics Letters, 94, 38-43. [Google Scholar] [CrossRef] 
[27] Wang, D., Wu, J. and Ye, Z. (2020) Global Regularity of the Three-Dimensional Fractional Micropolar Equations. Journal of Mathematical Fluid Mechanics, 22, Article No. 28. [Google Scholar] [CrossRef] 
[28] Zhu, W. and Zhao, J. (2018) Existence and Regularizing Rate Estimates of Solutions to the 3-D Generalized Micropolar System in Fourier-Besov Spaces. Mathematical Methods in the Applied Sciences, 41, 1703-1722. [Google Scholar] [CrossRef] 
[29] Ye, H., Wang, Q. and Jia, Y. (2022) Well-Posedness and Large Time Decay for the 3D Micropolar Equations with Only Velocity Dissipation. Nonlinear Analysis, 219, Article 112796. [Google Scholar] [CrossRef] 
[30] Li, H. (2019) Global Regularity for the 3D Micropolar Equations. Applied Mathematics Letters, 92, 70-75. [Google Scholar] [CrossRef] 
[31] Liu, S. and Si, Z. (2020) Global Well-Posedness of the 3D Micropolar Equations with Partial Viscosity and Damping. Applied Mathematics Letters, 109, Article 106543. [Google Scholar] [CrossRef] 

为你推荐