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\centerline{\large \bf Kinetic Models of Granular Flows}


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\centerline{G. Toscani}
\centerline{\it Dipartimento di Matematica, Universit\`a di Pavia, Italy}
\centerline{e-mail {\tt toscani@dimat.unipv.it}}


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%   Text of the abstract

We review recent results on one-dimensional kinetic models of the
Boltzmann equation with dissipative collisions and variable
coefficient of restitution. In particular, we investigate the
behavior of the Boltzmann equation in the so-called quasi elastic
limit  for a wide range of the rate function. By this limit procedure
we obtain a class of nonlinear equations classified as nonlinear
friction equations.
We analyze the long-time asymptotics of these nonlinear friction
equations, which split naturally into two classes, which are
characterized by the behavior in time of their similarity solutions
(homogeneous cooling state). If the similarity solution relaxes
towards equilibrium in infinite time,  we show convergence in the
Vasershtein metric of any solution with the same mass and mean
velocity towards the similarity solution by computing explicit
relaxation rates.
Finally we introduce numerical schemes for these dissipative models.
We study the numerical passage of the Boltzmann
equation with singular kernel to nonlinear friction equations in
the  quasi elastic limit. To this aim we introduce a
Fourier spectral method for the Boltzmann equation
and show that the kernel modes that define the spectral method
have the correct quasi elastic limit providing a consistent
spectral method for the limiting nonlinear friction equation.

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