不可压缩对流Brinkman-Forchheimer方程弱解的研究

Study on the Weak Solutions to the Incompressible Convective Brinkman-Forchheimer Equations

摘要: 本文研究当系数满足$32{\mu }^2\beta \alpha \ge 1$ 时，不可压缩对流Brinkman-Forchheimer方程的全局解的存在性、连续性以及唯一性。通过古典的能量方法、Friedrich方法证明了全局解的存在性，同时运用Sobolev紧嵌入定理，结合不等式估计证明了在限制条件下解的连续性和唯一性。该结果改进了已有文献中对阻尼指数$r$ 和参数$\alpha ,\beta$ 的限制条件，揭示了Brinkman-Forchheimer方程模型的物理现象，为多孔介质流体动力学的数学理论提供了重要基础。

Abstract: This paper study the existence, uniqueness, and continuity of global solutions for the incompressible convective Brinkman-Forchheimer equations under the condition that the coefficients satisfy $32{\mu }^2\beta \alpha \ge 1$ . The existence of global solutions is established via classical energy methods and the Friedrichs method. Furthermore, by employing the Sobolev compact embedding theorem combined with key inequality estimates, the uniqueness and continuity of solutions are proved under the specified constraint. These results improve upon the restrictions on the damping exponent $r$ and the parameters $\alpha ,\beta$ found in existing literature. They also contribute to revealing the physical phenomena inherent in the Brinkman-Forchheimer model and provide a significant foundation for the mathematical theory of fluid dynamics in porous media.