2009, 24(1): 187-212. doi: 10.3934/dcds.2009.24.187

Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff

1. 

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501

2. 

17-26 Iwasaki, Hodogaya, Yokohama 240-0015

3. 

Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France

4. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  February 2007 Revised  December 2007 Published  January 2009

Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.
Citation: 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
[1]

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

[2]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic & Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919

[3]

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

[4]

Nadia Lekrine, Chao-Jiang Xu. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation. Kinetic & Related Models, 2009, 2 (4) : 647-666. doi: 10.3934/krm.2009.2.647

[5]

Renjun Duan, Shuangqian Liu, Tong Yang, Huijiang Zhao. Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. Kinetic & Related Models, 2013, 6 (1) : 159-204. doi: 10.3934/krm.2013.6.159

[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]

Lingbing He, Yulong Zhou. High order approximation for the Boltzmann equation without angular cutoff. Kinetic & Related Models, 2018, 11 (3) : 547-596. doi: 10.3934/krm.2018024

[8]

Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1

[9]

Alexander Arbieto, Carlos Matheus. On the periodic Schrödinger-Debye equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 699-713. doi: 10.3934/cpaa.2008.7.699

[10]

Shengfan Zhou, Jinwu Huang, Xiaoying Han. Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices. Communications on Pure & Applied Analysis, 2010, 9 (1) : 193-210. doi: 10.3934/cpaa.2010.9.193

[11]

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

[12]

Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059

[13]

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

[14]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395

[15]

A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35

[16]

Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic & Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383

[17]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399

[18]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305

[19]

Niclas Bernhoff. On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinetic & Related Models, 2010, 3 (2) : 195-222. doi: 10.3934/krm.2010.3.195

[20]

Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic & Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (39)

[Back to Top]