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The Benard Problem for General Fluid Dynamical Equations and Remarks on the Boussinesq Approximation PAUL C, FIFE Communicated by P. R. GanaBeptan 1, Introduction. The object of this paper is to give a rigorous mathematical treatment of Bénard convection, using eneral fiuid dynamical equations. situation has been the object of extensive experimental investigation, as well as of analytical study within the framework of the ‘“Boussinesqg approximation” to the compressible Navier-Stokcs Equations. Those analytical investigations closest in spirit to the present one are found in papers by Velte [8], Judovich [3], Rabinowitz [5] and Fife and Joseph [2]. ‘The present paper proceeds from the general dynamical equations of a com- pressible fluid with an equation of state of the form p= poll + alT — 7.) + ACP). Here p is the density and T the temperature (p, and Ty, are a reference density and temperature, respectively), and h is a function of T which is arbitrary except that it is assumed “small enough” in a certain sense. The requirement that p depend only on 7’ is the most restrictive assumption on our treatment. The stress tensor, heat conduction coefficient, and internal energy are allowed to depend on the temperature and pressure; furthermore the stress tensor is allowed to be nonlinear as a function of the deformation tensor (this gocs beyond the Navier-Stokes Equations, in which linearity is assumed). Restrictions on the temperature and pressure dependence of these quantities are given in Section 2, but are typified by the fact that the heat conduction coefficient is supposed to Supported in part by NSF Grant No. GP-11660 at the University of Arizona. ‘The author thanks the reviewer for holpful comments. 308 Indiana University Mathematics Journal, Vol. 20, No, 4 (1970), 
