We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t→∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial datum has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L^1-norm, as well as various Sobolev norms.

Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria / Bisi, Marzia; Carrillo, J. A.; Toscani, G.. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 118:(2005), pp. 301-331. [10.1007/s10955-004-8785-5]

Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria

BISI, Marzia;
2005-01-01

Abstract

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t→∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial datum has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L^1-norm, as well as various Sobolev norms.
2005
Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria / Bisi, Marzia; Carrillo, J. A.; Toscani, G.. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 118:(2005), pp. 301-331. [10.1007/s10955-004-8785-5]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/1444088
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