On the Boltzmann Equation for Diffusively Excited Granular Media

By Irene M. Gamba, V. Panferov, Cédric Villani. 2003

143 citations recorded by the source corpus.

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 L2(N) 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 conjectured by previous authors, and we prove pointwise lower estimates for the solutions.

Subfields: Applied Mathematics, Computational Mechanics, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/e92db4ca5276bf0671c7c81908ae6b657f085b7e

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Citations in the corpus

Optimal local well-posedness theory for the kinetic wave equation (2017, result): “This should of course be contrasted with the cases of the classical Boltzmann equation [1, 4, 21, 22] and the quantum Boltzmann equation for bosons at very low temperature [2] (this is also the weak turbulence kinetic equation for anharmonic crystal lattices), for which the decay of the solution is ”
Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations (2007, methodology): “where the last term gives a positive contribution, see [63].”
Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions (2003, background): “The first one is the pure diffusion thermal bath,(36,33,19) in which case”
General relative entropy inequality: an illustration on growth models (2005, methodology): “In the case of example 3 the model is mass conservative and it is possible to prove existence of a stationary solution with the help of the Schauder Theorem (see for instance [18, 16] for details), in other words 0 is the first eigenvalue.”