# American Institute of Mathematical Sciences

December  2008, 1(4): 521-555. doi: 10.3934/krm.2008.1.521

## On some properties of linear and linearized Boltzmann collision operators for hard spheres

 1 Department of Mathematics, Karlstad University, SE-65188 Karlstad, Sweden, Sweden

Received  August 2008 Revised  September 2008 Published  October 2008

The linear and the linearized Boltzmann collision operators for hard spheres are studied by a method based on reduction of integral equations to differential equations. We use this approach (in combination with numerical methods) to study the eigenvalues of the operators. We also use the differential equations to investigate large energy asymptotics of solutions to linear integral equations related to the Chapman-Enskog expansion.
Citation: A. V. Bobylev, E. Mossberg. On some properties of linear and linearized Boltzmann collision operators for hard spheres. Kinetic & Related Models, 2008, 1 (4) : 521-555. doi: 10.3934/krm.2008.1.521
 [1] Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic & Related Models, 2014, 7 (4) : 755-778. doi: 10.3934/krm.2014.7.755 [2] 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 [3] Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068 [4] Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459 [5] Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009 [6] Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic & Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008 [7] Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 [8] 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 [9] Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 [10] Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic & Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020 [11] Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034 [12] Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039 [13] Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205 [14] Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic & Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499 [15] El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401 [16] Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237 [17] Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic & Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551 [18] Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 [19] Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579 [20] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673

2018 Impact Factor: 1.38