M. Oliver and E.S. Titi,
Gevrey regularity for the attractor of a partially dissipative model of Bénard convection in a porous medium,
J. Differential Equations 163 (2000), 292-311.

Abstract:

Convective flow though a porous medium can be modeled by Darcy's law - a linear, weakly dissipative momentum equation - coupled with an advection-diffusion equation for the energy. The solution semigroup for this system is not compact, and thus the solution does not gain regularity with respect to its initial value in finite times. However, it is known that the semigroup is asymptotically compact, 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 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.
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