On the Boltzmann equation for diffusively excited granular media

Abstract

We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative $L^2$ function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjecturedby previous authors, and we prove pointwise lower estimates for the solutions.

40 pages, 1 figure

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Cited by in corpus (32)

• Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions
• 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
• Convolution inequalities for the Boltzmann collision operator
• The Boltzmann equation for driven systems of inelastic soft spheres
• The rich behavior of the Boltzmann equation for dissipative gases
• Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy
• Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media
• Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform
• Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case
• Strong Convergence towards homogeneous cooling states for dissipative Maxwell models
• On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation
• Boltzmann Model for viscoelastic particles: asymptotic behavior, pointwise lower bounds and regularity
• Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particles Thermal Bath
• Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction
• Blow up Analysis for Anomalous Granular Gases
• On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials
• Universality of Cluster Dynamics
• The fragmentation equation with size diffusion: Well-posedness and long-term behavior
• Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting
• Measure Valued Solution to the Spatially Homogeneous Boltzmann Equation with Inelastic Long-Range Interactions
• Moment inequalities and high-energy tails for electron distribution function of the Boltzmann transport equation in semiconductors
• Convergence and error estimates for the Lagrangian based Conservative Spectral method for Boltzmann Equations
• On the Krein-Rutman theorem and beyond
• Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions
• On the self-similar behaviour of coagulation systems with injection
• High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations
• Uniqueness in the weakly inelastic regime of the equilibrium state of the inelastic Boltzmann equation driven by a particle bath
• Mass-conserving self-similar solutions to coagulation-fragmentation equations
• 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