非齐次Dirichlet边界条件下Navier-Stokes-Brinkman-Forchheimer方程的适定性
Well-Posedness of the Navier-Stokes-Brinkman-Forchheimer Equations with Nonhomogeneous Dirichlet Boundary Conditions
摘要: 本文研究了在非齐次Dirichlet边界条件下，具有非自治外力的Navier-Stokes-Brinkman-Forchheimer方程的适定性。通过应用经典的Faedo-Galerkin方法，证明了弱解的存在唯一性。其中运用泛函分析中的紧性定理等内容处理非线性项与惯性项的收敛问题，进而证明了存在性。利用非线性项的增长条件证明了弱解的唯一性。
Abstract: This paper investigates the well-posedness of the Navier-Stokes-Brinkman-Forchheimer equations with nonhomogeneous Dirichlet boundary conditions, where the external force is non-autonomous. By applying the classical Faedo-Galerkin method, we establish the existence and uniqueness of weak solutions. Compactness theorems from functional analysis are employed to handle the convergence of nonlinear and inertial terms, thereby proving existence. The uniqueness of weak solutions is proved by utilizing the growth condition of the nonlinear term.
文章引用：李雨欣. 非齐次Dirichlet边界条件下Navier-Stokes-Brinkman-Forchheimer方程的适定性[J]. 理论数学, 2026, 16(4): 142-154. https://doi.org/10.12677/pm.2026.164099

参考文献

[1] Rajagopal, K.R. (2007) On a Hierarchy of Approximate Models for Flows of Incompressible Fluids through Porous Solids. Mathematical Models and Methods in Applied Sciences, 17, 215-252. [Google Scholar] [CrossRef] 
[2] Straughan, B. (2008) Stability and Wave Motion in Porous Media. Springer.
[3] Aulisa, E., Bloshanskaya, L., Hoang, L. and Ibragimov, A. (2009) Analysis of Generalized Forchheimer Flows of Compressible Fluids in Porous Media. Journal of Mathematical Physics, 50, Article ID: 103102. [Google Scholar] [CrossRef] 
[4] Brinkman, H.C. (1949) A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles. Flow, Turbulence and Combustion, 1, 27-34. [Google Scholar] [CrossRef] 
[5] Giorgi, T. (1997) Derivation of the Forchheimer Law via Matched Asymptotic Expansions. Transport in Porous Media, 29, 191-206. [Google Scholar] [CrossRef] 
[6] Brinkman, H.C. (1949) On the Permeability of Media Consisting of Closely Packed Porous Particles. Flow, Turbulence and Combustion, 1, 81-86. [Google Scholar] [CrossRef] 
[7] Shenoy, A.V. (1994) Non-Newtonian Fluid Heat Transfer in Porous Media. In: Advances in Heat Transfer, Elsevier, 101-190. [Google Scholar] [CrossRef] 
[8] Gordeev, R.G. (1976) The Existence of a Periodic Solution in a Tide Dynamics Problem. Journal of Soviet Mathematics, 6, 1-4. [Google Scholar] [CrossRef] 
[9] Nield, D.A. (1991) The Limitations of the Brinkman-Forchheimer Equation in Modeling Flow in a Saturated Porous Medium and at an Interface. International Journal of Heat and Fluid Flow, 12, 269-272. [Google Scholar] [CrossRef] 
[10] Nield, D.A. (1994) Modelling High Speed Flow of a Compressible Fluid in a Saturated Porous Medium. Transport in Porous Media, 14, 85-88. [Google Scholar] [CrossRef] 
[11] Hajduk, K.W. and Robinson, J.C. (2017) Energy Equality for the 3D Critical Convective Brinkman-Forchheimer Equations. Journal of Differential Equations, 263, 7141-7161. [Google Scholar] [CrossRef] 
[12] Kalantarov, V.K. and Zelik, S. (2012) Smooth Attractors for the Brinkman-Forchheimer Equations with Fast Growing Nonlinearities. Communications on Pure and Applied Analysis, 11, 2037-2054. [Google Scholar] [CrossRef] 
[13] Çelebi, A.O., Kalantarov, V.K. and U[gtilde]urlu, D. (2005) Continuous Dependence for the Convective Brinkman-Forchheimer Equations. Applicable Analysis, 84, 877-888. [Google Scholar] [CrossRef] 
[14] Cai, X. and Jiu, Q. (2008) Weak and Strong Solutions for the Incompressible Navier-Stokes Equations with Damping. Journal of Mathematical Analysis and Applications, 343, 799-809. [Google Scholar] [CrossRef] 
[15] Zhang, Z., Wu, X. and Lu, M. (2011) On the Uniqueness of Strong Solution to the Incompressible Navier-Stokes Equations with Damping. Journal of Mathematical Analysis and Applications, 377, 414-419. [Google Scholar] [CrossRef] 
[16] Kalantarov, V. and Zelik, S. (2021) Asymptotic Regularity and Attractors for Slightly Compressible Brinkman-Forchheimer Equations. Applied Mathematics & Optimization, 84, 3137-3171. [Google Scholar] [CrossRef] 
[17] Stone, D. and Zelik, S. (2023) The Non-Autonomous Navier-Stokes-Brinkman-Forchheimer Equation with Dirichlet Boundary Conditions: Dissipativity, Regularity, and Attractors. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 155, 352-393. [Google Scholar] [CrossRef] 
[18] Robinson, J. (2001) Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press.

为你推荐