非零普朗特数下达西-贝纳德对流问题的中心流形约化方法
A center manifold reduction approach to the Darcy-Bénard convection problem with non-zero Prandtl number
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中文总结 AI 辅助
针对非零普朗特数的达西-贝纳德对流问题，本文开发适配其特性的中心流形约化方法，证明中心流形的存在性与局部吸引性，推导显式近似并通过数值模拟验证结果。
中文摘要 AI 辅助
我们研究矩形域内二维达西-贝纳德对流（DBC）的分岔，该模型是多孔介质热对流的经典模型，在地球物理学与工程领域有应用。其动量方程缺乏平流和粘性耗散，仅由线性达西阻尼项正则化，因此线性化算子生成的半群既非解析也非紧，非线性项也不满足利普希茨条件，导致该系统超出标准中心流形定理的适用范围。为克服这些障碍，我们开发了适用于DBC系统的中心流形约化方法。主要结果是构造性证明了中心流形函数h的存在性，以及中心流形的局部吸引性——小解向其指数收敛，且不依赖于全线性半群的解析性和非线性项的利普希茨性质。我们利用系统的部分耗散结构规避困难：温度方程由拉普拉斯算子控制，其生成解析半群，提供了所需的光滑性，以补偿速度场的正则性缺失和全局利普希茨界的缺失。通过精心设计的不等式，我们既建立了中心流形函数h的构造，也证明了附近解的指数收敛性。在两种场景下推导了中心流形的显式近似——单简单特征值和两个不同特征值，得到常微分方程的约化系统，对该系统的分析可确定分岔类型。我们还给出了数值模拟以验证理论结果。
英文摘要
We study the bifurcation of two-dimensional Darcy-Bénard convection (DBC) in a rectangular domain, a canonical model for thermal convection in porous media with applications in geophysics and engineering. The momentum equation lacks advection and viscous dissipation, being regularized solely by a linear Darcy damping term. As a result, the linearized operator generates a semigroup that is neither analytic nor compact and the nonlinear term fails to be Lipschitz. The system is thus placed outside the scope of the standard center-manifold theorem. To overcome these obstructions, we develop a center-manifold reduction adapted to DBC system. Our main result is a constructive proof of the existence of the center manifold function h and local attractivity of the center manifold-the exponential convergence of small solutions toward it-without relying on analyticity of the full linear semigroup and Lipschitz nonlinearity. We circumvent these difficulties by exploiting the partially dissipative structure: the temperature equation is governed by the Laplacian, which generates an analytic semigroup and provides the smoothing needed to compensate for the lack of regularity in the velocity and the absence of global Lipschitz bounds. Through carefully designed inequalities, we establish both the construction of the center manifold function h and the exponential convergence of nearby solutions. Explicit approximations for the center manifold are derived in two scenarios-one simple eigenvalue and two distinct eigenvalues-yielding reduced systems of ordinary differential equations whose analysis determines the bifurcation type. Numerical simulations are presented to corroborate the theoretical results.
