# On the two-dimensional compressible isentropic Navier–Stokes equations

Catherine Giacomoni; Pierre Orenga

- Volume: 36, Issue: 6, page 1091-1109
- ISSN: 0764-583X

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topGiacomoni, Catherine, and Orenga, Pierre. "On the two-dimensional compressible isentropic Navier–Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.6 (2002): 1091-1109. <http://eudml.org/doc/245203>.

@article{Giacomoni2002,

abstract = {We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma = \displaystyle \{\{c_\{p\}\}/\{c_\{v\}\}\}=2$. 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 $N=2$.},

author = {Giacomoni, Catherine, Orenga, Pierre},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Navier–Stokes; compressible; shallow water; time-discretisation; Galerkin; time-discretized problem; Galerkin method; finite element Modulef software; square domain; shallow water model; Navier-Stokes equations},

language = {eng},

number = {6},

pages = {1091-1109},

publisher = {EDP-Sciences},

title = {On the two-dimensional compressible isentropic Navier–Stokes equations},

url = {http://eudml.org/doc/245203},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Giacomoni, Catherine

AU - Orenga, Pierre

TI - On the two-dimensional compressible isentropic Navier–Stokes equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 6

SP - 1091

EP - 1109

AB - We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma = \displaystyle {{c_{p}}/{c_{v}}}=2$. 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 $N=2$.

LA - eng

KW - Navier–Stokes; compressible; shallow water; time-discretisation; Galerkin; time-discretized problem; Galerkin method; finite element Modulef software; square domain; shallow water model; Navier-Stokes equations

UR - http://eudml.org/doc/245203

ER -

## References

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- [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. Zbl0954.35137
- [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. Zbl0877.35092
- [12] E. Zeidler, Fixed-point theorems, in Nonlinear Functional Analysis and its Applications, Vol. 1, Springer-Verlag (1986). Zbl1063.54504MR816732

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