An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 3, pp. 477-486.

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low Mach number limit for standard compressible flows given in P.-L. Lions’ book that means with constant viscosity coefficients.

DOI : 10.1051/m2an:2005026
Classification : 35Q30
Mots clés : compressible flows, Navier-Stokes equations, low Mach (Froude) number limit shallow-water equations, lake equations, nonconstant density
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     title = {An example of low {Mach} {(Froude)} number effects for compressible flows with nonconstant density (height) limit},
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Bresch, Didier; Gisclon, Marguerite; Lin, Chi-Kun. An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 3, pp. 477-486. doi : 10.1051/m2an:2005026. http://www.numdam.org/articles/10.1051/m2an:2005026/

[1] T. Alazard, Incompressible limit of the non-isentropic Euler equations with solid wall boundary conditions. Submitted (2004). | Zbl

[2] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm. Math. Phys. 238 (2003) 211-223. | Zbl

[3] D. Bresch, B. Desjardins and D. Gérard-Varet, Rotating fluids in a cylinder. Discrete Contin. Dynam. Systems Ser. A 11 (2004) 47-82. | Zbl

[4] D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems. Comm. Partial Differential Equations 28 (2003) 1009-1037. | Zbl

[5] D. Bresch, B. Desjardins, E. Grenier and C.-K. Lin, Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case. Stud. Appl. Math. 109 (2002) 125-148. | Zbl

[6] R. Danchin, Fluides légèrement compressibles et limite incompressible. Séminaire École Polytechnique (France), Exposé No. III (2000). | Numdam | MR | Zbl

[7] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78 (1999) 461-471. | Zbl

[8] I. Gallagher, Résultats récents sur la limite incompressible. Séminaire Bourbaki (France), No. 926 (2003). | Numdam | MR

[9] J.F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar Shallow water; Numerical results. Discrete Contin. Dynam. Systems Ser. B 1 (2001) 89-102. | Zbl

[10] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. 76 (1997) 477-498. | Zbl

[11] C.D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493-1515. | Zbl

[12] C.D. Levermore, M. Oliver and E.S. Titi, Global well-posedness for a models of shallow water in a basin with a varying bottom. Indiana Univ. Math. J. 45 (1996) 479-510. | Zbl

[13] P.-L. Lions, Mathematical topics in fluid dynamics, Vol. 2, Compressible models. Oxford Science Publication, Oxford (1998). | MR | Zbl

[14] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluids. J. Math. Pures Appl. 77 (1998) 585-627. | Zbl

[15] G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations. Arch. Rational Mech. Anal. 158 (2001) 61-90. | Zbl

[16] G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, in Séminaire Équations aux Dérivées Partielles, École Polytechnique (2001). | MR

[17] M. Oliver, Justification of the shallow water limit for a rigid lid with bottom topography. Theor. Comp. Fluid Dyn. 9 (1997) 311-324. | Zbl

[18] J. Pedlosky, Geophysical fluid dynamics. Berlin Heidelberg-New York, Springer-Verlag (1987). | Zbl

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