American Institute of Mathematical Sciences

Advanced
September  2010, 3(3): 445-456. doi: 10.3934/krm.2010.3.445

The uniformly heated inelastic Boltzmann equation in Fourier space

Ralf Kirsch 1, and  Sergej Rjasanow 2,

1. 

Department of Flow and Material Simulation, Fraunhofer ITWM, Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany
2. 

Department of Mathematics, Saarland University, P.O. Box 15 11 50, D-66041 Saarbrücken, Germany

Received  January 2010 Revised  May 2010 Published  July 2010

In this article, we present an alternative formulation of the Boltzmann equation for diffusively driven granular media. The equation is considered with minimal a priori assumptions, i.e. in weak form in the sense of tempered distributions. Using shifted test functions and the Fourier transform, it is seen that the transformed problem contains only a threefold integral. For constant restitution coefficients and the variable hard spheres model, explicit expressions of the integral kernel in the transformed collision operator are obtained. The version of the equation derived here is a true extension of the elastic case. Some well-known results for Maxwell molecules with inelastic interactions are recovered.
Keywords: Granular materials, homogeneous Boltzmann equation..
Mathematics Subject Classification: 82C40, 76P0.
Citation: Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[1]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[2]

K. A. Ariyawansa, Leonid Berlyand, Alexander Panchenko. A network model of geometrically constrained deformations of granular materials. Networks & Heterogeneous Media, 2008, 3 (1) : 125-148. doi: 10.3934/nhm.2008.3.125
[3]

Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159
[4]

Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic & Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299
[5]

Léo Glangetas, Mohamed Najeme. Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 407-427. doi: 10.3934/krm.2013.6.407
[6]

Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic & Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453
[7]

Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187
[8]

Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic & Related Models, 2015, 8 (2) : 309-333. doi: 10.3934/krm.2015.8.309
[9]

Yoshinori Morimoto. A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinetic & Related Models, 2012, 5 (3) : 551-561. doi: 10.3934/krm.2012.5.551
[10]

Haolei Wang, Lei Zhang. Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions. Electronic Research Archive, 2020, 28 (1) : 405-421. doi: 10.3934/era.2020023
[11]

Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic & Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020
[12]

Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic & Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036
[13]

Qun Lin, Antoinette Tordesillas. Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions. Journal of Industrial & Management Optimization, 2014, 10 (1) : 337-362. doi: 10.3934/jimo.2014.10.337
[14]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[15]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145
[16]

Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014
[17]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[18]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223
[19]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[20]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences