American Institute of Mathematical Sciences

Advanced
July  2019, 18(4): 1769-1782. doi: 10.3934/cpaa.2019083

Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces

Koya Nishimura

Independent scholar

Received  June 2018 Revised  June 2018 Published  January 2019

We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $ L^r_v L^\infty_x $ spaces where $ r \in (1,\infty] $ by using the excess conservation laws and entropy inequality introduced in [5].

Keywords: Boltzmann, hard potentials, Maxwellian, kinetic theory, global existence.
Mathematics Subject Classification: Primary: 35Q20, 35A01; Secondary: 82B40.
Citation: Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083
References:
[1]

R. Duan, F. Huang, Y. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large Amplitude initial data, Arch. Rational Mech. Anal., 225 (2017), 375-424.  doi: 10.1007/s00205-017-1107-2.  Google Scholar

[2]

Robert T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[3]

Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[4]

Yan Guo, Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[5]

Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148.  doi: 10.1090/S0033-569X-09-01180-4.  Google Scholar

[6]

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

[7]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2 \cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310.  doi: 10.1142/S0219530506000784.  Google Scholar

show all references

References:
[1]

R. Duan, F. Huang, Y. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large Amplitude initial data, Arch. Rational Mech. Anal., 225 (2017), 375-424.  doi: 10.1007/s00205-017-1107-2.  Google Scholar

[2]

Robert T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[3]

Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[4]

Yan Guo, Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[5]

Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148.  doi: 10.1090/S0033-569X-09-01180-4.  Google Scholar

[6]

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

[7]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2 \cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310.  doi: 10.1142/S0219530506000784.  Google Scholar

[1]

Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135
[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]

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
[5]

Xinkuan Chai. The Boltzmann equation near Maxwellian in the whole space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435
[6]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385
[7]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
[8]

Alexander V. Bobylev, Irene M. Gamba. Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules. Kinetic & Related Models, 2017, 10 (3) : 573-585. doi: 10.3934/krm.2017023
[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]

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901
[11]

Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169
[12]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345
[13]

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
[14]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
[15]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621
[16]

Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405
[17]

Marc Briant. Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. Kinetic & Related Models, 2015, 8 (2) : 281-308. doi: 10.3934/krm.2015.8.281
[18]

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
[19]

Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039
[20]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (77)
  • HTML views (173)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences