-
Previous Article
Existence and regularity of solutions for an evolution model of perfectly plastic plates
- CPAA Home
- This Issue
-
Next Article
Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition
Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces
Independent scholar |
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 [
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. |
| [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. |
| [3] |
Yan Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
| [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. |
| [5] |
Yan Guo,
Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148.
doi: 10.1090/S0033-569X-09-01180-4. |
| [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. |
| [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. |
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. |
| [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. |
| [3] |
Yan Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
| [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. |
| [5] |
Yan Guo,
Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148.
doi: 10.1090/S0033-569X-09-01180-4. |
| [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. |
| [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. |
| [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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435 |
| [6] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
| [7] |
Alexander V. Bobylev, Irene M. Gamba. Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules. Kinetic and Related Models, 2017, 10 (3) : 573-585. doi: 10.3934/krm.2017023 |
| [8] |
Yoshinori Morimoto. A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinetic and Related Models, 2012, 5 (3) : 551-561. doi: 10.3934/krm.2012.5.551 |
| [9] |
Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385 |
| [10] |
Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303 |
| [11] |
Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic and Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169 |
| [12] |
Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic and Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901 |
| [13] |
Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic and Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345 |
| [14] |
A. V. Bobylev, E. Mossberg. On some properties of linear and linearized Boltzmann collision operators for hard spheres. Kinetic and Related Models, 2008, 1 (4) : 521-555. doi: 10.3934/krm.2008.1.521 |
| [15] |
Lican Kang, Yanming Lai, Yanyan Liu, Yuan Luo, Jing Zhang. High-dimensional linear regression with hard thresholding regularization: Theory and algorithm. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022034 |
| [16] |
Marc Briant. Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. Kinetic and Related Models, 2015, 8 (2) : 281-308. doi: 10.3934/krm.2015.8.281 |
| [17] |
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 and Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036 |
| [18] |
Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic and Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405 |
| [19] |
Fei Meng, Fang Liu. On the inelastic Boltzmann equation for soft potentials with diffusion. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5197-5217. doi: 10.3934/cpaa.2020233 |
| [20] |
Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]