American Institute of Mathematical Sciences

Advanced
December  2009, 2(4): 647-666. doi: 10.3934/krm.2009.2.647

Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation

Nadia Lekrine 1, and  Chao-Jiang Xu 1,

1. 

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

Received  April 2009 Revised  August 2009 Published  October 2009

In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier.
Keywords: Non-cutoff Kac's equation, Boltzmann equation, Gevrey regularizing effect, Cauchy problem, Fourier analysis..
Mathematics Subject Classification: 35A05, 35B65,35D10, 42A38, 60H07, 82B4.
Citation: 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
[1]

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
[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]

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
[4]

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
[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]

Milana Pavić-Čolić, Maja Tasković. Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules. Kinetic & Related Models, 2018, 11 (3) : 597-613. doi: 10.3934/krm.2018025
[7]

Hao Tang, Zhengrong Liu. On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2229-2256. doi: 10.3934/dcds.2016.36.2229
[8]

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
[9]

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
[10]

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, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187
[11]

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
[12]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731
[13]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012
[14]

Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic & Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479
[15]

Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic & Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333
[16]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149
[17]

Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic & Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015
[18]

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
[19]

Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[20]

Laurent Desvillettes, Clément Mouhot, Cédric Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. Kinetic & Related Models, 2011, 4 (1) : 277-294. doi: 10.3934/krm.2011.4.277

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy