Existence of self-similar profile for a kinetic annihilation model

Abstract

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard-spheres such that, whenever two particles meet, they either annihilate with probability $\alpha \in (0,1)$ or they undergo an elastic collision with probability $1-\alpha$. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for $\alpha$ smaller than some explicit threshold value ${\alpha }_{\ast }$, a self-similar solution exists.

This new version supersedes and replaces the previous one. We found a mistake in the previous (and published) version of the manuscript and explained how to fix it in "Erratum to "Existence of self-similar profile for a kinetic annihilation model" [J. Differential Equations 254 (7) (2013) 3023-3080]. J. Differential Equations 257 (2014), no. 8, 3071-3074." This version provides a complete and corrected version of the previous manuscript

References in corpus (8)

• Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions
• Regularity theory for the spatially homogeneous Boltzmann equation with cut-off
• Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres
• Ballistic Annihilation
• Kinetics and scaling in ballistic annihilation
• About $L^p$ estimates for the spatially homogeneous Boltzmann equation
• Hydrodynamics of probabilistic ballistic annihilation
• Maxwell and very hard particle models for probabilistic ballistic annihilation: hydrodynamic description