949 research outputs found

    Blow up Analysis for Anomalous Granular Gases

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    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

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    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

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    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,

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    This paper studies the transport of a mass ÎŒ\mu in ℜd,d≄2,\real^d, d \geq 2, by a flow field v=−∇K∗Όv= -\nabla K*\mu. We focus on kernels K=∣x∣α/αK=|x|^\alpha/ \alpha for 2−d≀α<22-d\leq \alpha<2 for which the smooth densities are known to develop singularities in finite time. For this range This paper studies the transport of a mass ÎŒ\mu in ℜd,d≄2,\real^d, d \geq 2, by a flow field v=−∇K∗Όv= -\nabla K*\mu. We focus on kernels K=∣x∣α/αK=|x|^\alpha/ \alpha for 2−d≀α<22-d\leq \alpha<2 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 α>2\alpha >2 where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential (α=2−d\alpha=2-d), 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 2−d<α<22 -d < \alpha < 2 and at the critical exponent pp we exhibit initial data in LpL^p 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

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    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

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    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

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    This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with d≄3d\ge3 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 McM_c such that if M∈(0,Mc]M \in (0,M_c] solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for M∈(0,Mc)M \in (0,M_c). While characterising the eventual infinite time blowing-up profile for M=McM=M_c, 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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