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Maximum principles for the primitive equations of the atmosphere. (English) Zbl 1030.86004

From the introduction: The primitive equations of the atmosphere and of the ocean have been studied, from the mathematical viewpoint [in J. L. Lions, R. Témam and S. Wang, Nonlinearity 5, 237-288 (1992; Zbl 0746.76019), Nonlinearity 5, 1007-1053 (1992; Zbl 0766.35039), Comput. Mech. Adv. 1, 5-54, 55-119 (1993; Zbl 0805.76011 and Zbl 0805.76052), J. Math. Pures Appl. (9) 74, 105-163 (1995; Zbl 0866.76025)] 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 byt 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 $\theta$. 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 ${L}^{\infty }$ estimates for $\theta$ (which provide, afterwards, positivity and ${L}^{\infty }$ 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 ${L}^{\infty }$ 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.

##### MSC:
 86A10 Meteorology and atmospheric physics 35B50 Maximum principles (PDE) 76B60 Atmospheric waves 76U05 Rotating fluids
##### Keywords:
existence of weak solutions; maximum principles