In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coecients in any space dimension N ≥ 1. We will suppose the coecients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradierential 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.

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.2010 Mathematics Subject Classification: 35Q35 (primary); 35B25, 76U05, 35Q86, 35B40, 76M45 (secondary).

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the L ∞ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension d ≥ 2 and for general pressure laws.2010 Mathematics Subject Classification: 35Q35 (primary); 35B65, 76N10, 35B30, 35A02 (secondary).

We study here a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain R 2 × ]0, 1[ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter 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. 2010 Mathematics Subject Classification: 35Q35 (primary); 35B25, 35B40, 35P25, 76U05, 47A40, 28A33, 47A55 (secondary).Notice that the previous scaling corresponds to supposing both the Mach number and the Rossby number to be proportional to ε (see e.g. paper [20], or Chapter 4 of book [13]). We are interested in studying the asymptotic behavior of weak solutions to the previous system, in the regime of small ε, namely for ε → 0. In particular, this means that we are performing the incompressible limit, the high rotation limit and, when α > 0, the vanishing capillarity limit simultaneously.Many are the mathematical contributions to the study of the effects of fast rotation on fluid dynamics, under different assumptions (about e.g. incompressibility of the fluid, about the domain and the boundary conditions. . . ). We refer e.g. to book [9] and the references therein for an extensive analysis of this problem for incompressible viscous fluids. In the context of compressible fluids there are, to our knowledge, few works, dealing with different models: among others that we are going to present more in detail below, we quote here [5] and [17] as important contributions.In the recent paper [12], Feireisl, Gallagher and Novotný studied the incompressible and high rotation limits together, for the 3-D compressible barotropic Navier-Stokes system with Coriolis force, in the general instance of ill-prepared initial data. Their asymptotic result relies on the spectral analysis of the singular perturbation operator: by use of the celebrated RAGE theorem (see e.g. books [30] and [10]), they were able to prove some dispersion properties due to fast rotation, from which they deduced strong convergence of the velocity fields, and this allowed them to pass to the limit in the weak formulation of the system. In paper [11] by Feireisl, Gallagher, Gérard-Varet and Novotný, the effect of the centrifugal force was added to the previous system. Notice that this term scales as 1/ε 2 ; hence, they got interested in both the isotropic a...