Highly rotating viscous compressible fluids in presence of capillarity effects

Abstract

We study here a singular limit problem for a Navier–Stokes–Korteweg system with Coriolis force, in the domain \(\mathbb {R}^2\times \,]0,1[\,\) and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter \(\varepsilon \rightarrow 0\), 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.