Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation

Abstract

Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation RAIRO-Modélisation mathématique et analyse numérique, tome 26, n o 7 (1992), p. 855-865. <http://www.numdam.org/item?id=M2AN_1992__26_7_855_0> © AFCET, 1992, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

On the strong solutions of the primitive equations in 2D domains

“…In these works, compactness method is used to obtain the velocity u in a space with the restriction ∇ · u = 0 and the pressure is recovered, in the latter part of the argument, by a specific De Rham's lemma on the surface. In domains without sidewalls, the existence of a weak solution is obtained as a consequence of a limit process applied to the Navier-Stokes equations with anysotropic viscosity when the ratio depth over horizontal diameter (of the domain) tends to zero, see Besson-Laydi [5] for the stationary case and Azerad-Guillén [6] for the evolution case. Finally, the existence of a weak solution in domains without sidewalls can be proved by internal approximation arguments: a mixed (velocity-pressure) variational formulation of the stationary problem is approximated by a conform Finite Element method in Chacón-Guillén [7] and a semi-discretization in time of the evolution problem is proved that converges to continuous problem in Guillén-Redondo [8,9].…”

Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

“…It is then natural to perform the geometrical scaling z = z ε /ε (see [AG01,BL92]), obtaining the adimensional domain (figure 1):…”

“…Equations (11)-(13) can be obtained as limit of the anisotropic Navier-Stokes equations (5)-(7) when ε tends to zero. This fact is justified on rigorous mathematical grounds in [BL92,AG01]. Many works about existence and regularity results of (11)-(13) are based on replacing this problem by the next (equivalent) reduced problem: find u : Ω × (0, T ) → R 2 , the horizontal velocity and p s : S × (0, T ) → R an (artificial) surface pressure, satisfying…”

“…But when ε → 0, this system tends to the hydrostatic Stokes problem (1)-(3) (see [AG01] for the nonlinear unsteady case and [BL92] for the steady case). According to Theorem 6, when a finite element approximation is made for the limit problem (1)-(3), its stability requires a choice of finite element spaces verifying not only the condition (IS) P h , which is related to (IS) h , but also the constraint (IS) V h , which is a new restriction, not required in the standard Stokes problem.…”

Analysis of the hydrostatic approximation in oceanography with compression term

“…one in which the horizontal dimension is very large compared with the vertical one, the hydrostatic approximation is the basic model. This model is obtained through a limit process from the anisotropic stationary Navier-Stokes equations and it is studied as such by Besson and Laydi in [2] and by Bresch and Simon in [3]. Several codes have also been developed to solve this problem [15].…”