Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions

Abstract

We study the high-energy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hard-sphere Boltzmann equations, which imply that the velocity distribution functions $f(v)$ behave in a certain sense as $C\exp (-r|v{|}^s)$ for $|v|$ large. The values of $s$, which we call {\em the orders of tails}, range from $s=1$ to $s=2$, depending on the model of external forcing. The method we use is based on the moment inequalities and careful estimating of constants in the integral form of the Povzner-type inequalities.

22 pages

References in corpus (13)

• The Granular Phase Diagram
• Multiscaling in Infinite Dimensional Collision Processes
• Computer simulation of uniformly heated granular fluids
• On the Boltzmann equation for diffusively excited granular media
• Scaling Solutions of Inelastic Boltzmann Equations with Over-populated High Energy Tails
• Influence of correlations on the velocity statistics of scalar granular gases
• Transport Coefficients for Granular Media from Molecular Dynamics Simulations
• The Inelastic Maxwell Model
• Velocity Tails for Inelastic Maxwell Models
• Velocity Distributions and Correlations in Homogeneously Heated Granular Media
• Nontrivial Velocity Distributions in Inelastic Gases
• Scale Invariance and Lack of Self-Averaging in Fragmentation
• Velocity Distributions in Homogeneously Cooling and Heated Granular Fluids

Cited by in corpus (50)

• Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials
• On the Boltzmann equation for diffusively excited granular media
• Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem
• Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres
• Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior
• Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails
• On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates
• Factorization for non-symmetric operators and exponential H-theorem
• Self-similarity and power-like tails in nonconservative kinetic models
• Convolution inequalities for the Boltzmann collision operator
• The Boltzmann equation for driven systems of inelastic soft spheres
• The Boltzmann equation for a multi-species mixture close to global equilibrium
• Decay estimates for large velocities in the Boltzmann equation without cutoff
• On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity
• Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy
• A new approach to the creation and propagation of exponential moments in the Boltzmann equation
• Celebrating Cercignani's conjecture for the Boltzmann equation
• Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media
• Uniform shear flow in dissipative gases. Computer simulations of inelastic hard spheres and (frictional) elastic hard spheres
• Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform
• The dissipative linear Boltzmann equation for hard spheres
• On existence and uniqueness to homogeneous Boltzmann flows of monatomic gas mixtures
• Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off
• Boltzmann Model for viscoelastic particles: asymptotic behavior, pointwise lower bounds and regularity
• The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation system for bosons at very low temperature
• Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation
• Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particles Thermal Bath
• Stability of global equilibrium for the multi-species Boltzmann equation in $L^{\infty }$ settings
• Producing suprathermal tails in the stationary velocity distribution
• On exponential moments of the homogeneous Boltzmann equation for hard potentials without cutoff
• On the Cauchy problem for Boltzmann equation modelling a polyatomic gas
• A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media
• Blow up Analysis for Anomalous Granular Gases
• Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting
• Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation
• Moment estimates and well-posedness of the binary-ternary Boltzmann equation
• Moment inequalities and high-energy tails for electron distribution function of the Boltzmann transport equation in semiconductors
• Nonlinear approximation theory for the homogeneous Boltzmann equation
• Coagulation equations for non-spherical clusters
• High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations
• Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data
• Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules
• Existence of self-similar profile for a kinetic annihilation model
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate
• Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation
• Two proofs of Haff's law for dissipative gases: the use of entropy and the weakly inelastic regime
• Uniqueness in the weakly inelastic regime of the equilibrium state of the inelastic Boltzmann equation driven by a particle bath
• Spectral - Lagrangian methods for Collisional Models of Non - Equilibrium Statistical States
• Propagation of $L^1$ and $L^{\infty }$ Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann Equation
• Exponential Convergence to the Maxwell Distribution For Spatially Inhomogenous Boltzmann Equations