Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

Abstract

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in $L^1(\R \times \R,\d \x \otimes \d\v)$ associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.

19 pages, to appear in Journal of Statistical Physics

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Cited by in corpus (8)

• Entropy dissipation estimates for the linear Boltzmann operator
• Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particles Thermal Bath
• Spectral analysis of transport equations with bounce-back boundary conditions
• The Derivation of the Linear Boltzmann Equation from a Rayleigh Gas Particle Model
• Kinetic description of a Rayleigh Gas with annihilation
• Derivation of a Non-autonomous Linear Boltzmann Equation from a Heterogeneous Rayleigh Gas
• Long Range Particle Dynamics and the Linear Boltzmann Equation
• Uniqueness in the weakly inelastic regime of the equilibrium state of the inelastic Boltzmann equation driven by a particle bath