Highly rotating viscous compressible fluids in presence of capillarity effects

Francesco Fanelli (1, 2, 3)

1 UCBL - Université Claude Bernard Lyon 1
2 ICJ - Institut Camille Jordan
3 EDPA - Équations aux dérivées partielles, analyse

We study here a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain $\R^2\times\,]0,1[\,$ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter $\veps\ra0$, we perform the incompressible and high rotation limits simultaneously; moreover, we consider both the constant and vanishing capillarity regimes. In this last case, the limit problem is identified as a $2$-D incompressible Navier-Stokes equation in the variables orthogonal to the rotation axis; if the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure, due to the presence of an additional surface tension term. In the vanishing capillarity regime, various rates at which the capillarity coefficient goes to $0$ are considered: in general, this produces an anisotropic scaling in the system. The proof of the results is based on suitable applications of the RAGE theorem, combined with microlocal symmetrization arguments.

Mots clés: Navier-Stokes-Korteweg system; rotating fluids; singular perturbation problem; low Mach number; low Rossby number; vanishing capillarity.

Domaines: Equations aux dérivées partielles [math.AP]

Francesco Fanelli. Highly rotating viscous compressible fluids in presence of capillarity effects. Mathematische Annalen, 2016, ⟨10.1007/s00208-015-1358-x⟩. ⟨hal-01300655⟩