On the two-dimensional compressible isentropic Navier-Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 6, p. 1091-1109

We analyze the compressible isentropic Navier-Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma ={c}_{p}/{c}_{v}=2$. These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in Lions (1998). The second part is devoted to the numerical analysis and mainly deals with problems of geophysical fluids. We compare the simulations obtained with this compressible isentropic Navier-Stokes model and those obtained with a shallow water model (Di Martino et al., 1999). At first, the computations are executed on a simplified domain in order to validate the method by comparison with existing numerical results and then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analytical example presented by Weigant (1995) which shows that even if the data are rather smooth, we cannot have bounds on $\rho$ in ${L}^{p}$ for $p$ large if $\gamma <2$ when $N=2$.

DOI : https://doi.org/10.1051/m2an:2003007
Classification:  35Q30
Keywords: Navier-Stokes, compressible, shallow water, time-discretisation, Galerkin
@article{M2AN_2002__36_6_1091_0,
author = {Giacomoni, Catherine and Orenga, Pierre},
title = {On the two-dimensional compressible isentropic Navier-Stokes equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {36},
number = {6},
year = {2002},
pages = {1091-1109},
doi = {10.1051/m2an:2003007},
zbl = {1048.35056},
language = {en},
url = {http://www.numdam.org/item/M2AN_2002__36_6_1091_0}
}

Giacomoni, Catherine; Orenga, Pierre. On the two-dimensional compressible isentropic Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 6, pp. 1091-1109. doi : 10.1051/m2an:2003007. http://www.numdam.org/item/M2AN_2002__36_6_1091_0/

[1] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compensated-compactness and Hardy spaces. J. Math. Pures Appl. 72 (1993) 247-286. | Zbl 0864.42009

[2] R.J. Diperna and P.L. Lions, On the cauchy problem for boltzman equations: global existence and weak stability. C.R. Acad. Sci. Paris Sér. I Math. 306 (1988) 343-346. | Zbl 0662.35016

[3] R.J. Diperna and P.L. Lions, Ordinary differential equations, transport theory and sobolev spaces. Invent. Math. 98 (1989) 511-547. | Zbl 0696.34049

[4] V. Girault and P.A. Raviart, Finite Elements Methods of the Navier-Stokes Equations. Springer-Verlag (1986). | MR 851383 | Zbl 0585.65077

[5] P.L. Lions, Mathematical Topics in Fluid Mechanics, Incompressible models. Vol. 1, Oxford Science Publications (1996). | MR 1422251 | Zbl 0866.76002

[6] P.L. Lions, Mathematical Topics in Fluid Mechanics, Compressible models. Vol. 2, Oxford Science Publications (1998). | MR 1637634 | Zbl 0908.76004

[7] B. Di Martino, F.J. Chatelon and P. Orenga, The nonlinear Galerkin's method applied to the shallow water equations. Math. Models Methods Appl. Sci. 9 (1999) 825-854. | Zbl 0957.76062

[8] P. Orenga, Analyse de quelques problèmes d'océanographie physique. Ph.D. thesis, Université de Corse, Corte (1992).

[9] P. Orenga, Construction d'une base spéciale pour la résolution de quelques problèmes non linéaires d'océanographie physique en dimension deux, in Nonlinear partial differential equations and their applications, D. Cioranescu and J.L. Lions, Vol. 13. Longman, Pitman Res. Notes Math. Ser. 391 (1998) 234-258. | Zbl 0954.35137

[10] V.A. Solonnikov, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 56 (1976) 128-142. English translation in J. Soviet Math. 14 (1980) 1120-1133.

[11] V.A. Weigant, An exemple of non-existence globally in time of a solution of the Navier-Stokes equations for a compressible viscous barotropic fluid. Russian Acad. Sci. Doklady Mathematics 50 (1995) 397-399. | Zbl 0877.35092

[12] E. Zeidler, Fixed-point theorems, in Nonlinear Functional Analysis and its Applications, Vol. 1, Springer-Verlag (1986). | MR 816732 | Zbl 1063.54504