4,652 research outputs found
Free cooling and high-energy tails of granular gases with variable restitution coefficient
We prove the so-called generalized Haff's law yielding the optimal algebraic
cooling rate of the temperature of a granular gas described by the homogeneous
Boltzmann equation for inelastic interactions with non constant restitution
coefficient. Our analysis is carried through a careful study of the infinite
system of moments of the solution to the Boltzmann equation for granular gases
and precise Lp estimates in the selfsimilar variables. In the process, we
generalize several results on the Boltzmann collision operator obtained
recently for homogeneous granular gases with constant restitution coefficient
to a broader class of physical restitution coefficients that depend on the
collision impact velocity. This generalization leads to the so-called
L1-exponential tails theorem. for this model
Analysis of a chemotaxis system modeling ant foraging
In this paper we analyze a system of PDEs recently introduced in [P. Amorim,
{\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail
formation}], in order to describe the dynamics of ant foraging. The system is
made of convection-diffusion-reaction equations, and the coupling is driven by
chemotaxis mechanisms. We establish the well-posedness for the model, and
investigate the regularity issue for a large class of integrable data. Our main
focus is on the (physically relevant) two-dimensional case with boundary
conditions, where we prove that the solutions remain bounded for all times. The
proof involves a series of fine \emph{a priori} estimates in Lebesgue spaces.Comment: 39 page
Propagation of and Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann Equation
We consider the -dimensional space homogeneous Boltzmann equation for
elastic collisions for variable hard potentials with Grad (angular) cutoff. We
prove sharp moment inequalities, the propagation of -Maxwellian weighted
estimates, and consequently, the propagation -Maxwellian weighted
estimates to all derivatives of the initial value problem associated to the
afore mentioned problem.
More specifically, we extend to all derivatives of the initial value problem
associated to this class of Boltzmann equations corresponding sharp moment
(Povzner) inequalities and time propagation of -Maxwellian weighted
estimates as originally developed A.V. Bobylev in the case of hard spheres in 3
dimensions; an improved sharp moments inequalities to a larger class of angular
cross sections and -exponential bounds in the case of stationary states to
Boltzmann equations for inelastic interaction problems with `heating' sources,
by A.V. Bobylev, I.M. Gamba and V.Panferov, where high energy tail decay rates
depend on the inelasticity coefficient and the the type of `heating' source;
and more recently, extended to variable hard potentials with angular cutoff by
I.M. Gamba, V. Panferov and C. Villani in the elastic case collision case and
so -Maxwellian weighted estimated were shown to propagate if initial
states have such property. In addition, we also extend to all derivatives the
propagation of -Maxwellian weighted estimates to solutions of the
initial value problem to the Boltzmann equations for elastic collisions for
variable hard potentials with Grad (angular) cutoff.Comment: 24 page
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