2,567 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
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
Klein-Gordon equation from Maxwell-Lorentz dynamics
We consider Maxwell-Lorentz dynamics: that is to say, Newton's law under the
action of a Lorentz's force which obeys the Maxwell equations. A natural class
of solutions are those given by the Lagrangian submanifolds of the phase space
when it is endowed with the symplectic structure modified by the
electromagnetic field. We have found that the existence of this type of
solution leads us directly to the Klein-Gordon equation as a compatibility
condition. Therefore, surprisingly, quite natural assumptions on the classical
theory involve a quantum condition without any process of limit. This result
could be a partial response to the inquiries of Dirac.Comment: 8 pages; some misprints corrected; added some coments in the abstract
and also explaining the relationship with the work of P.A.M. Dirac; also
added 2 reference
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