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January  2004, 11(1): 47-82. doi: 10.3934/dcds.2004.11.47

Rotating fluids in a cylinder

D. Bresch 1, , B. Desjardins 2, and  D. Gérard-Varet 3,

1. 

LMC - IMAG, UMR 5523 (CNRS-UJF-INPG), B.P. 53, 38041 Grenoble Cedex 9, France
2. 

CEA/DIF, B.P. 12, 91680 Bruyères le Châtel, France
3. 

UMPA (CNRS UMR 128), E.N.S. Lyon, 46 allée d'Italie, 69364 Lyon cedex 07, France

Received  February 2003 Revised  November 2003 Published  April 2004

We study various singularly perturbed models related to rotating flows in a cylinder. At first we consider the three dimensional incompressible Navier--Stokes equations with turbulent viscosity, in the low Rossby limit. We prove a strong convergence result for ill prepared data, under a geometrical assumption on the cylinder section and a genericity condition on the singular operator.
In a second section, we discuss the compressible Navier--Stokes equations with anisotropic viscosity tensor in the combined low Mach and low Rossby number limit. In the case of well prepared initial data, we prove that global weak solutions with Dirichlet boundary conditions converge to the solution of a two--dimensional quasi-geostrophic model taking into account the compressibility. In the case of ill prepared data, we only show that we can hope a strong convergence result under the same kind of assumptions as in the incompressible case.
Keywords: geophysical flows, incompressible and compressible Navier–Stokes equations, rapidly rotating fluids, quasi-geostrophic equations, Singular perturbations, propagation of waves, low Mach number, free surface..
Mathematics Subject Classification: Primary: 76D05, 35Q35, 35Q10, 35Q30, 76U0.
Citation: D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 47-82. doi: 10.3934/dcds.2004.11.47
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