Revista Matemática Iberoamericana


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Volume 35, Issue 6, 2019, pp. 1763–1807
DOI: 10.4171/rmi/1101

Published online: 2019-08-02

Asymptotics of fast rotating density-dependent incompressible fluids in two space dimensions

Francesco Fanelli[1] and Isabelle Gallagher[2]

(1) Université Claude Bernard – Lyon 1, Villeurbanne, France
(2) École Normale Supérieure, Paris, France

In the present paper we study the fast rotation limit for viscous incompressible fluids with variable density, whose motion is influenced by the Coriolis force. We restrict our analysis to two dimensional flows. In the case when the initial density is a small perturbation of a constant state, we recover in the limit the convergence to the homogeneous incompressible Navier–Stokes equations (up to an additional term, due to density fluctuations). For general non-homogeneous fluids, the limit equations are instead linear, and the limit dynamics is described in terms of the vorticity and the density oscillation function: we lack enough regularity on the latter to prove convergence on the momentum equation itself. The proof of both results relies on a compensated compactness argument, which enables one to treat also the possible presence of vacuum.

Keywords: Incompressible fluids, Navier–Stokes equations, variable density, vacuum, Coriolis force, singular perturbation problem, low Rossby number

Fanelli Francesco, Gallagher Isabelle: Asymptotics of fast rotating density-dependent incompressible fluids in two space dimensions. Rev. Mat. Iberoam. 35 (2019), 1763-1807. doi: 10.4171/rmi/1101