949 research outputs found
Blow up Analysis for Anomalous Granular Gases
We investigate in this article the long-time behaviour of the solutions to
the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for
hard spheres. This model describes a diluted gas composed of hard spheres under
statistical description, that dissipates energy during collisions. We assume
that the gas is "anomalous", in the sense that energy dissipation increases
when temperature decreases. This allows the gas to cool down in finite time. We
study existence and uniqueness of blow up profiles for this model, together
with the trend to equilibrium and the cooling law associated, generalizing the
classical Haff's Law for granular gases. To this end, we investigate the
asymptotic behaviour of the inelastic Boltzmann equation with and without drift
term by introducing new strongly "nonlinear" self-similar variables.Comment: 20
Experimental Study of the Intrinsic and Extrinsic Transport Properties of Graphite and Multigraphene Samples
This work deals with the intrinsic and extrinsic properties of the graphene
layers inside the graphite structure, in particular the influence of defects
and interfaces. We discuss the evidence for ballistic transport found in
mesoscopic graphite samples and the possibility to obtain the intrinsic carrier
density of graphite, without the need of free parameters or arbitrary
assumptions. The influence of internal interfaces on the transport properties
of bulk graphite is described in detail. We show that in specially prepared
multigraphene samples the transport properties show clear signs for the
existence of granular superconductivity within the graphite interfaces. We
argue that the superconducting-insulator or metal-insulator transition (MIT)
reported in the literature for bulk graphite is not intrinsic of the graphite
structure but it is due to the influence of these interfaces. Current-Voltage
characteristics curves reveal Josephson-like behavior at the interfaces with
superconducting critical temperatures above 150K.Comment: 26 pages, 15 figures. To be published in "Graphene, Book 2" by
Intech, Open Access Publisher 2011, ISBN: 979-953-307-180-
Recent development in kinetic theory of granular materials: analysis and numerical methods
33 pagesOver the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods
Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,
This paper studies the transport of a mass in by a
flow field . We focus on kernels for
for which the smooth densities are known to develop
singularities in finite time. For this range This paper studies the transport
of a mass in by a flow field . We
focus on kernels for for which the
smooth densities are known to develop singularities in finite time. For this
range we prove the existence for all time of radially symmetric measure
solutions that are monotone decreasing as a function of the radius, thus
allowing for continuation of the solution past the blowup time. The monotone
constraint on the data is consistent with the typical blowup profiles observed
in recent numerical studies of these singularities. We prove monotonicity is
preserved for all time, even after blowup, in contrast to the case
where radially symmetric solutions are known to lose monotonicity. In the case
of the Newtonian potential (), under the assumption of radial
symmetry the equation can be transformed into the inviscid Burgers equation on
a half line. This enables us to prove preservation of monotonicity using the
classical theory of conservation laws. In the case and at
the critical exponent we exhibit initial data in for which the
solution immediately develops a Dirac mass singularity. This extends recent
work on the local ill-posedness of solutions at the critical exponent.Comment: 30 page
Hydrodynamics of granular particles on a line
We investigate a lattice model representing a granular gas in a thin channel.
We deduce the hydrodynamic description for the model from the microscopic
dynamics in the large system limit, including the lowest finite-size
corrections. The main prediction from hydrodynamics, when finite-size
corrections are neglected, is the existence of a steady "uniform longitudinal
flow" (ULF), with the granular temperature and the velocity gradient both
uniform and directly related. Extensive numerical simulations of the system
show that such a state can be observed in the bulk of a finite-size system by
attaching two thermostats with the same temperature at its boundaries. The
relation between the ULF state and the shocks appearing in the late stage of a
cooling gas of inelastic hard rods is discussed.Comment: 12 pages, 6 figures, to be published on Physical Review E (in press
Recent development in kinetic theory of granular materials: analysis and numerical methods
33 pagesOver the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions
This paper is devoted to the analysis of non-negative solutions for a
generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system
with and porous medium-like non-linear diffusion. Here, the non-linear
diffusion is chosen in such a way that its scaling and the one of the Poisson
term coincide. We exhibit that the qualitative behaviour of solutions is
decided by the initial mass of the system. Actually, there is a sharp critical
mass such that if solutions exist globally in time,
whereas there are blowing-up solutions otherwise. We also show the existence of
self-similar solutions for . While characterising the eventual
infinite time blowing-up profile for , we observe that the long time
asymptotics are much more complicated than in the classical Patlak-Keller-Segel
system in dimension two
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