Highly rotating viscous compressible fluids in presence of capillarity effects

In this paper we study 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 to be proportional to a small parameter \veps going to $0$, we perform the incompressible and high rotation limits simultaneously. Moreover, we consider both the constant capillarity 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. Various rates at which the capillarity coefficient can vanish are also considered: in most cases this will produce an anisotropic scaling in the system, for which a different analysis is needed. The proof of the results is based on suitable applications of the RAGE theorem.Comment: Version 2 includes a corrigendum, which fixes errors contained in the proofs to Theorems 6.5 and 6.

Weak observability estimates for 1-D wave equations with rough coefficients

In this paper we prove observability estimates for 1-dimensional wave equations with non-Lipschitz coefficients. For coefficients in the Zygmund class we prove a "classical" observability estimate, which extends the well-known observability results in the energy space for $BV$ regularity. When the coefficients are instead log-Lipschitz or log-Zygmund, we prove observability estimates "with loss of derivatives": in order to estimate the total energy of the solutions, we need measurements on some higher order Sobolev norms at the boundary. This last result represents the intermediate step between the Lipschitz (or Zygmund) case, when observability estimates hold in the energy space, and the H\"older one, when they fail at any finite order (as proved in \cite{Castro-Z}) due to an infinite loss of derivatives. We also establish a sharp relation between the modulus of continuity of the coefficients and the loss of derivatives in the observability estimates. In particular, we will show that under any condition which is weaker than the log-Lipschitz one (not only H\"older, for instance), observability estimates fail in general, while in the intermediate instance between the Lipschitz and the log-Lipschitz ones they can hold only admitting a loss of a finite number of derivatives. This classification has an exact counterpart when considering also the second variation of the coefficients.Comment: submitte

The well-posedness issue in endpoint spaces for an inviscid low-Mach number limit system

The present paper is devoted to the well-posedness issue for a low-Mach number limit system with heat conduction but no viscosity. We will work in the framework of general Besov spaces $B^s_{p,r}(\R^d)$, $d\geq 2$, which can be embedded into the class of Lipschitz functions. Firstly, we consider the case of $p\in[2,4]$, with no further restrictions on the initial data. Then we tackle the case of any $p\in\,]1,\infty]$, but requiring also a finite energy assumption. The extreme value $p=\infty$ can be treated due to a new a priori estimate for parabolic equations. At last we also briefly consider the case of any $p\in ]1,\infty[$ but with smallness condition on initial inhomogeneity. A continuation criterion and a lower bound for the lifespan of the solution are proved as well. In particular in dimension 2, the lower bound goes to infinity as the initial density tends to a constant.Comment: This work was superseded by arXiv:1403.0960 and arXiv:1403.096

Conservation of geometric structures for non-homogeneous inviscid incompressible fluids

We obtain a result about propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension $N\geq2$. In particular, we investigate conservation of striated and conormal regularity, which is a natural way of generalizing the 2-D structure of vortex patches. The results we get are only local in time, even in the dimension N=2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N=2 or 3, we investigate also propagation of H\"older regularity in the interior of a bounded domain