Global gevrey regularity for the Benard convection in a porous medium withzero Darcy-Prandtl number

In this paper, we prove the existence and uniqueness for the three-dimensio nal Benard convection model in a porous medium with zero Darcy-Prandtl numb er using the Galerkin procedure. In addition, we show that the solutions to this problem are analytic in time with values in a Gevrey class regularity . We also prove that the solution of the standard Galerkin method converges exponentially fast, in the wave number, to the exact solution. This gives an analytical justification to the two-dimensional computational results of Graham, Steen, and Titi [J. Nonlin. Sci. 3 (1993), 153-167]. MSC numbers. 76R05, 76S05, 35K55.