A center manifold reduction approach to the Darcy-B\'{e}nard convection problem with non-zero Prandtl number

Abstract

We study the bifurcation of two-dimensional Darcy-B\'{e}nard 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.