Strong Convergence towards homogeneous cooling states for dissipative Maxwell models

Abstract

We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogenous cooling state present in the literature implies the strong convergence in Sobolev norms and in the $L^1$ norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the $L^1$ distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.

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References in corpus (4)

• On the Boltzmann equation for diffusively excited granular media
• Scaling Solutions of Inelastic Boltzmann Equations with Over-populated High Energy Tails
• Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem
• Tanaka Theorem for Inelastic Maxwell Models

Cited by in corpus (4)

• Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particles Thermal Bath
• Granular gas of inelastic and rough Maxwell particles
• An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux
• Kullback--Leibler Divergence of a Freely Cooling Granular Gas