Maximum Principles for the Primitive Equations of the Atmosphere

Introduction. The primitive equations of the atmosphere and of the ocean have been studied, from the mathematical viewpoint, in Lions, Temam, and Wang ([6], [7]; see also [8], [9]) for the coupling of the atmosphere and the ocean. Concerning the equations of temperature, the classical methods used for the maximum principle apply to the temperature equation for the ocean but they seemingly do not apply to the temperature equation for the atmosphere. This does not appear to be due to a mathematical technicality but rather to a difference of structure of the equations, the water being incompressible and the air being compressible. On physical grounds, it was suggested to us to consider a modified temperature equation for the atmosphere, namely the equation for the potential temperature θ, whose definition is recalled below. We have been able, in this way, to derive the desired estimates. The aim of this article is to introduce the potential temperature equation and to derive L estimates for θ (which provide, afterwards, positivity and L estimates for the classical temperature T ). The article is organized as follows: in Section 1, we recall the primitive equations of the atmosphere (PEs) and introduce the potential temperature and the corresponding equations. In Section 2, we provide the weak formulation of the PEs in a limited domain (with suitable, physically reasonable boundary conditions), and we establish the existence of weak solutions. Then, maximum principles are established in Section 3 by a combination of the truncation (Stampacchia) method for the positivity and classical methods for the L bound. In Section 4, we describe the similar results for the whole atmosphere and present the changes in the proofs which are necessary in this case.