Partial differential equations
On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations
Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 639-644.

In this work, we obtain the hydrostatic approximation by taking the small aspect ratio limit to the Navier–Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in general large scale motions meaning that the vertical scale is significantly smaller than horizontal. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier–Stokes equations to the compressible Primitive Equations. This is the first work to use relative entropy inequality for proving hydrostatic approximation and derive the compressible Primitive Equations.

Dans ce travail, nous obtenons l’approximation hydrostatique en prenant la limite du petit rapport d’aspect des équations de Navier–Stokes. Le rapport d’aspect (le rapport de la profondeur à la largeur horizontale) est une contrainte géométrique dans les mouvements géophysiques signifiant que l’échelle verticale est nettement plus petite que l’horizontale. Nous utilisons l’inégalité d’entropie relative pour prouver rigoureusement la limite des équations de Navier–Stokes compressibles aux équations primitives compressibles. C’est le premier travail qui utilise l’inégalité d’entropie relative pour prouver l’approximation hydrostatique et dériver les équations primitives compressibles.

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Accepted:
Published online:
DOI: 10.5802/crmath.186
Gao, Hongjun 1; Nečasová, Šárka 2; Tang, Tong 3

1 School of Mathematics, Southeast University, Nanjing 211189, P.R. China
2 Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic
3 School of Mathematical Science, Yangzhou University, Yangzhou 225002, P.R. China
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Gao, Hongjun; Nečasová, Šárka; Tang, Tong. On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 639-644. doi : 10.5802/crmath.186. http://www.numdam.org/articles/10.5802/crmath.186/

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