672 research outputs found
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 and for general
ill-prepared initial data. Taking the Mach and the Rossby numbers to be
proportional to a small parameter \veps going to , 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 -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 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 , , which can be
embedded into the class of Lipschitz functions.
Firstly, we consider the case of , with no further restrictions on
the initial data. Then we tackle the case of any , but
requiring also a finite energy assumption. The extreme value can be
treated due to a new a priori estimate for parabolic equations. At last we also
briefly consider the case of any 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 . 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
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