logo CRAS
Comptes Rendus. Mathématique
Équations aux dérivées partielles
On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations
[Sur l’approximation hydrostatique des équations de Navier–Stokes anisotropes compressibles]
Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 639-644.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/crmath.186
Hongjun Gao 1 ; Šárka Nečasová 2 ; Tong Tang 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
@article{CRMATH_2021__359_6_639_0,
     author = {Hongjun Gao and \v{S}\'arka Ne\v{c}asov\'a and Tong Tang},
     title = {On the hydrostatic approximation of compressible anisotropic {Navier{\textendash}Stokes} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {639--644},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {6},
     year = {2021},
     doi = {10.5802/crmath.186},
     language = {en},
}
TY  - JOUR
AU  - Hongjun Gao
AU  - Šárka Nečasová
AU  - Tong Tang
TI  - On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 639
EP  - 644
VL  - 359
IS  - 6
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.186
DO  - 10.5802/crmath.186
LA  - en
ID  - CRMATH_2021__359_6_639_0
ER  - 
%0 Journal Article
%A Hongjun Gao
%A Šárka Nečasová
%A Tong Tang
%T On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations
%J Comptes Rendus. Mathématique
%D 2021
%P 639-644
%V 359
%N 6
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.186
%R 10.5802/crmath.186
%G en
%F CRMATH_2021__359_6_639_0
Hongjun Gao; Šárka Nečasová; Tong Tang. On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations. Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 639-644. doi : 10.5802/crmath.186. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.186/

[1] Pascal Azérad; Francisco Guillén Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., Volume 33 (2001), pp. 847-859 | Article | MR 1884725 | Zbl 0999.35072

[2] Didier Bresch; Pierre-Emmanuel Jabin Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. Math., Volume 188 (2018) no. 2, pp. 577-684 | Article | MR 3862947 | Zbl 1405.35133

[3] Chongsheng Cao; Edriss S. Titi Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math., Volume 166 (2007) no. 1, pp. 245-267 | MR 2342696 | Zbl 1151.35074

[4] Bernard Ducomet; Šárka Nečasová; Milan Pokorný; M. Angeles Rodríguez-Bellido Derivation of the Navier-Stokes-Poisson system with radiation for an accretion disk, J. Math. Fluid Mech., Volume 20 (2018) no. 2, pp. 697-719 | Article | MR 3808590 | Zbl 1448.76123

[5] Eduard Feireisl; Bum Ja Jin; Antonín Novotný Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., Volume 14 (2012) no. 4, pp. 717-730 | Article | MR 2992037 | Zbl 1256.35054

[6] Ken Furukawa; Yoshikazu Giga; Matthias Hieber; Amru Hussein; Takahito Kashiwabara; Marc Wrona Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations (2018) (https://arxiv.org/abs/1808.02410) | Zbl 1452.35150

[7] Hongjun Gao; Šárka Nečasová; Tong Tang On the hydrostatic approximation of compressible anisotropic Navier–Stokes equations (2020) (preprint Institute of Mathematics)

[8] Jinkai Li; Edriss S. Titi The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., Volume 124 (2019), pp. 30-58 | MR 3926040 | Zbl 1412.35224

[9] Jacques Louis Lions; Roger Temam; Shouhong Wang New formulations of the primitive equations of atmosphere and applications, Nonlinearity, Volume 5 (1992) no. 2, pp. 237-288 | Article | MR 1158375 | Zbl 0746.76019

[10] Xin Liu; Edriss S. Titi Local well-posedness of strong solutions to the three-dimensional compressible Primitive Equations (2018) (https://arxiv.org/abs/1806.09868v1) | Zbl 07364847

[11] David Maltese; Antonín Novotný Compressible Navier-Stokes equations on thin domains, J. Math. Fluid Mech., Volume 16 (2014) no. 3, pp. 571-594 | Article | MR 3247369 | Zbl 1308.35170

[12] Joseph Pedlosky Geophysical Fluid Dynamics, Springer, 1979 | Zbl 0429.76001

Cité par Sources :