American Institute of Mathematical Sciences

Advanced
June  2010, 3(2): 335-351. doi: 10.3934/krm.2010.3.335

Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels

C. David Levermore 1, and  Weiran Sun 2,

1. 

Department of Mathematics & Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, United States
2. 

Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Received  September 2009 Revised  November 2009 Published  May 2010

We prove an $L^p$ compactness result for the gain parts of the linearized Boltzmann collision operator associated with weakly cutoff collision kernels that derive from a power-law intermolecular potential. We replace the Grad cutoff assumption previously made by Caflisch [1], Golse and Poupaud [7], and Guo [11] with a weaker local integrability assumption. This class includes all classical kernels to which the DiPerna-Lions theory applies that derive from a repulsive inverse-power intermolecular potential. In particular, our approach allows the treatment of both hard and soft potential cases.
Keywords: Compactness, Fredholm property, linearized Boltzmann operator..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: C. David Levermore, Weiran Sun. Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels. Kinetic & Related Models, 2010, 3 (2) : 335-351. doi: 10.3934/krm.2010.3.335
[1]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315
[2]

Ricardo J. Alonso, Irene M. Gamba. Gain of integrability for the Boltzmann collisional operator. Kinetic & Related Models, 2011, 4 (1) : 41-51. doi: 10.3934/krm.2011.4.41
[3]

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
[4]

Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493
[5]

Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinetic & Related Models, 2013, 6 (3) : 625-648. doi: 10.3934/krm.2013.6.625
[6]

Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[7]

Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
[8]

Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441
[9]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[10]

Radjesvarane Alexandre, Lingbing He. Integral estimates for a linear singular operator linked with Boltzmann operators part II: High singularities $1\le\nu<2$. Kinetic & Related Models, 2008, 1 (4) : 491-513. doi: 10.3934/krm.2008.1.491
[11]

Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic & Related Models, 2012, 5 (4) : 769-786. doi: 10.3934/krm.2012.5.769
[12]

Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019
[13]

Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036
[14]

Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715
[15]

Lorenzo Brasco, Marco Squassina, Yang Yang. Global compactness results for nonlocal problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 391-424. doi: 10.3934/dcdss.2018022
[16]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473
[17]

Carlangelo Liverani. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1203-1215. doi: 10.3934/dcds.2005.13.1203
[18]

Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139
[19]

M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495
[20]

Yong Zhou, V. Vijayakumar, R. Murugesu. Controllability for fractional evolution inclusions without compactness. Evolution Equations & Control Theory, 2015, 4 (4) : 507-524. doi: 10.3934/eect.2015.4.507

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences