Francesco Fanelli

Highly rotating viscous compressible fluids in presence of capillarity effects

We study here a singular limit problem for a Navier–Stokes–Korteweg system with Coriolis force, in the domain

A Singular Limit Problem for Rotating Capillary Fluids with Variable Rotation Axis

\mathbb {R}^2\times \,]0,1[\,

The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients

R2×]0,1[ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter

Some local questions for hyperbolic systems with non-regular time dependent coefficients

\varepsilon \rightarrow 0

A Few Remarks on Hyperbolic Systems with Zygmund in Time Coefficients

ε→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.

A Note on Viscous Capillary Fluids in Fast Rotation

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension N ≥ 1. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.