Gevrey Regularity for the Attractor of a Partially Dissipative Model of Bénard Convection in a Porous Medium

Convective flow though a porous medium can be modeled by Darcyʹs law—a linear, weakly damped momentum equation—coupled with an advection–diffusion equation for the energy. The solution semigroup for this system is not smoothing, and the solution of the momentum equation does not gain regularity with respect to its initial value in finite time. However, it is known that the semigroup is asymptotically smoothing, so that the system possesses a finite dimensional global attractor as well as exponential attractors. We show that the global attractor is contained in a special Gevrey class of regularity and, in particular, is real analytic. The key idea is the use of a Fourier splitting method to approximate every orbit asymptotically in time by a Gevrey-regular function