# Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

@article{Bisi2008EquilibriumST, title={Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath}, author={Marzia Bisi and Jos{\'e} A. Carrillo and Bertrand Lods}, journal={Journal of Statistical Physics}, year={2008}, volume={133}, pages={841-870} }

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions to the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.

## 14 Citations

### Uniqueness in the Weakly Inelastic Regime of the Equilibrium State to the Boltzmann Equation Driven by a Particle Bath

- MathematicsSIAM J. Math. Anal.
- 2011

It is proved that $F_\alpha$ is bounded from above and from below by two explicit universal (i.e., independent ...) properties, and an accurate spectral analysis of the associated linearized collision operator in the elastic limit is given.

### Uniqueness in the weakly inelastic regime of the equilibrium state of the inelastic Boltzmann equation driven by a particle bath

- Mathematics
- 2011

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient $\alpha \in (0,1)$) under the thermalization induced by a host medium with a…

### One-Dimensional Dissipative Boltzmann Equation: Measure Solutions, Cooling Rate, and Self-Similar Profile

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- 2018

A dynamical fixed point theorem is applied on a suitable stable set, for the model dynamics, of Borel measures, of borel measures to prove the existence of a nontrivial self-similar profile after appropriate scaling of the Boltzmann equation.

### Some alternative methods for hydrodynamic closures to dissipative kinetic models

- Engineering
- 2009

Abstract
Two different strategies for deriving hydrodynamic equations for dissipative kinetic models are presented and discussed. The homogeneity scaling approach, not very well-known in the physical…

### Inelastic Boltzmann equation driven by a particle thermal bath

- MathematicsKinetic & Related Models
- 2021

<p style='text-indent:20px;'>We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient <inline-formula><tex-math…

### Direction-dependent turning leads to anisotropic diffusion and persistence

- Biology
- 2021

It is shown that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment, and generate enhanced diffusion in structured domains.

### Kinetic models with non-local sensing determining cell polarization and speed according to independent cues

- BiologyJournal of mathematical biology
- 2019

How the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics is analyzed by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation.

## References

SHOWING 1-10 OF 48 REFERENCES

### Dissipative Linear Boltzmann Equation for Hard Spheres

- Mathematics, Physics
- 2003

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard-spheres collision kernel describing inelastic interactions of a gas…

### Thermalization of a particle by dissipative collisions

- Physics
- 1998

One considers the stationary state of a test particle immersed in a homogeneous fluid in equilibrium at temperature T, undergoing dissipative collisions with the fluid particles. It is shown that the…

### On the Boltzmann Equation for Diffusively Excited Granular Media

- Mathematics
- 2004

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…

### On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions

- Mathematics, Physics
- 2000

We investigate a Boltzmann equation for inelastic scattering in which the relative velocity in the collision frequency is approximated by the thermal speed. The inelasticity is given by a velocity…

### Cooling Process for Inelastic Boltzmann Equations for Hard Spheres, Part I: The Cauchy Problem

- Mathematics
- 2006

We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution…

### Moment Equations for a Granular Material in a Thermal Bath

- Mathematics, Physics
- 2002

We compute the moment equations for a granular material under the simplifying assumption of pseudo-Maxwellian particles approximating dissipative hard spheres. We obtain the general moment equations…