- Previous Article
- DCDS Home
- This Issue
-
Next Article
The semirelativistic Choquard equation with a local nonlinear term
The Boltzmann equation with frictional force for very soft potentials in the whole space
1. | School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China |
2. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
3. | Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China |
We develop a general energy method for proving the optimal time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space, for both hard potentials and soft potentials. With the help of this method, we establish the global existence and temporal convergence rates of solution near a given global Maxwellian to the Cauchy problem on the Boltzmann equation with frictional force for very soft potentials i.e. $ -3<\gamma<-2 $.
References:
[1] |
R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang,
Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.
doi: 10.1007/s00205-010-0290-1. |
[2] |
R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang,
Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.
doi: 10.1007/s00220-011-1242-9. |
[3] |
R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang,
The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[4] |
R. J. Duan, Y. J. Lei, T. Yang and H. J. Zhao,
The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials, Comm. Math. Phys., 351 (2017), 95-153.
doi: 10.1007/s00220-017-2844-7. |
[5] |
R. J. Duan and S. Q. Liu,
The Vlasov-Poisson-Boltzmann System without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45.
doi: 10.1007/s00220-013-1807-x. |
[6] |
R. J. Duan, S. Q. Liu, T. Yang and H. J. Zhao,
Stabilty of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials, Kinetic and Related Models, 6 (2013), 159-204.
doi: 10.3934/krm.2013.6.159. |
[7] |
R. J. Duan and R. M. Strain,
Optimal time decay of the Vlasov-Poisson-Boltzmann system in ${\mathbb{R}^3}$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[8] |
R. J. Duan and R. M. Strain,
Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64 (2011), 1497-1546.
doi: 10.1002/cpa.20381. |
[9] |
R. J. Duan, S. Ukai, T. Yang and H. J. Zhao,
Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Comm. Math. Phys., 277 (2008), 189-236.
doi: 10.1007/s00220-007-0366-4. |
[10] |
R. J. Duan and T. Yang,
Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41 (2010), 2353-2387.
doi: 10.1137/090745775. |
[11] |
R. J. Duan, T. Yang and H. J. Zhao,
The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
[12] |
R. J. Duan, T. Yang and H. J. Zhao,
The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Methods Models Appl. Sci., 23 (2013), 979-1028.
doi: 10.1142/s0218202513500012. |
[13] |
Y. Z. Fan and Y. J. Lei,
Global solutions and time decay of the non-cutoff Vlasov-Maxwell-Boltzmann system in the whole space, J. Stat. Phys., 161 (2015), 1059-1097.
doi: 10.1007/s10955-015-1380-0. |
[14] |
R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[15] |
H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ., in Rarefied Gas Dynamics Vol. Ⅰ, Academic Press, New York, 1963.
![]() |
[16] |
P. T. Gressman and R. M. Strain,
Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.
doi: 10.1090/s0894-0347-2011-00697-8. |
[17] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[18] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[19] |
Y. Guo,
The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[20] |
Y. Guo,
The Vlasov-Poisson-Laudau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.
doi: 10.1090/S0894-0347-2011-00722-4. |
[21] |
Y. Guo and Y. J. Wang,
Decay of dissipative equation and negative sobolev spaces, Comm.Partial Differential Equations, 37 (2012), 2165-2208.
doi: 10.1080/03605302.2012.696296. |
[22] |
L. Hsiao and T. P. Liu,
Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605.
doi: 10.1007/BF02099268. |
[23] |
Y. J. Lei and L. Wan,
The Boltzmann equation with frictional force for soft potentials in the whole space, J. Differential Equations, 258 (2015), 3491-3534.
doi: 10.1016/j.jde.2015.01.021. |
[24] |
S. Q. Liu and H. X. Liu,
Optimal convergence rate of the Landau equation with frictional force, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 1781-1804.
|
[25] |
T. P. Liu, T. Yang and S. H. Yu,
Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192.
doi: 10.1016/j.physd.2003.07.011. |
[26] |
R. M. Strain and Y. Guo,
Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[27] |
S. Ukai, T. Yang and H. J. Zhao,
Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3 (2005), 157-193.
doi: 10.1142/S0219530505000522. |
[28] |
S. W. Vong,
The Boltzmann equation with frictional force, J. Differential Equations, 222 (2006), 95-136.
doi: 10.1016/j.jde.2005.07.007. |
[29] |
Y. Wang and Z. H. Jiang,
Optimal time decay of the Boltzmann equation with frictional force, J. Math. Anal. Appl., 374 (2011), 499-515.
|
[30] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao,
The Vlasov-Posson-Boltzmann system with angular cutoff for soft potential, J. Differential Equations, 255 (2013), 1196-1232.
doi: 10.1016/j.jde.2013.05.005. |
[31] |
T. Yang and H. J. Yu,
Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 1035-1062.
doi: 10.1016/S0252-9602(09)60085-0. |
show all references
References:
[1] |
R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang,
Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.
doi: 10.1007/s00205-010-0290-1. |
[2] |
R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang,
Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.
doi: 10.1007/s00220-011-1242-9. |
[3] |
R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang,
The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[4] |
R. J. Duan, Y. J. Lei, T. Yang and H. J. Zhao,
The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials, Comm. Math. Phys., 351 (2017), 95-153.
doi: 10.1007/s00220-017-2844-7. |
[5] |
R. J. Duan and S. Q. Liu,
The Vlasov-Poisson-Boltzmann System without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45.
doi: 10.1007/s00220-013-1807-x. |
[6] |
R. J. Duan, S. Q. Liu, T. Yang and H. J. Zhao,
Stabilty of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials, Kinetic and Related Models, 6 (2013), 159-204.
doi: 10.3934/krm.2013.6.159. |
[7] |
R. J. Duan and R. M. Strain,
Optimal time decay of the Vlasov-Poisson-Boltzmann system in ${\mathbb{R}^3}$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[8] |
R. J. Duan and R. M. Strain,
Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64 (2011), 1497-1546.
doi: 10.1002/cpa.20381. |
[9] |
R. J. Duan, S. Ukai, T. Yang and H. J. Zhao,
Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Comm. Math. Phys., 277 (2008), 189-236.
doi: 10.1007/s00220-007-0366-4. |
[10] |
R. J. Duan and T. Yang,
Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41 (2010), 2353-2387.
doi: 10.1137/090745775. |
[11] |
R. J. Duan, T. Yang and H. J. Zhao,
The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
[12] |
R. J. Duan, T. Yang and H. J. Zhao,
The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Methods Models Appl. Sci., 23 (2013), 979-1028.
doi: 10.1142/s0218202513500012. |
[13] |
Y. Z. Fan and Y. J. Lei,
Global solutions and time decay of the non-cutoff Vlasov-Maxwell-Boltzmann system in the whole space, J. Stat. Phys., 161 (2015), 1059-1097.
doi: 10.1007/s10955-015-1380-0. |
[14] |
R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[15] |
H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ., in Rarefied Gas Dynamics Vol. Ⅰ, Academic Press, New York, 1963.
![]() |
[16] |
P. T. Gressman and R. M. Strain,
Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.
doi: 10.1090/s0894-0347-2011-00697-8. |
[17] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[18] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[19] |
Y. Guo,
The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[20] |
Y. Guo,
The Vlasov-Poisson-Laudau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.
doi: 10.1090/S0894-0347-2011-00722-4. |
[21] |
Y. Guo and Y. J. Wang,
Decay of dissipative equation and negative sobolev spaces, Comm.Partial Differential Equations, 37 (2012), 2165-2208.
doi: 10.1080/03605302.2012.696296. |
[22] |
L. Hsiao and T. P. Liu,
Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605.
doi: 10.1007/BF02099268. |
[23] |
Y. J. Lei and L. Wan,
The Boltzmann equation with frictional force for soft potentials in the whole space, J. Differential Equations, 258 (2015), 3491-3534.
doi: 10.1016/j.jde.2015.01.021. |
[24] |
S. Q. Liu and H. X. Liu,
Optimal convergence rate of the Landau equation with frictional force, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 1781-1804.
|
[25] |
T. P. Liu, T. Yang and S. H. Yu,
Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192.
doi: 10.1016/j.physd.2003.07.011. |
[26] |
R. M. Strain and Y. Guo,
Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[27] |
S. Ukai, T. Yang and H. J. Zhao,
Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3 (2005), 157-193.
doi: 10.1142/S0219530505000522. |
[28] |
S. W. Vong,
The Boltzmann equation with frictional force, J. Differential Equations, 222 (2006), 95-136.
doi: 10.1016/j.jde.2005.07.007. |
[29] |
Y. Wang and Z. H. Jiang,
Optimal time decay of the Boltzmann equation with frictional force, J. Math. Anal. Appl., 374 (2011), 499-515.
|
[30] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao,
The Vlasov-Posson-Boltzmann system with angular cutoff for soft potential, J. Differential Equations, 255 (2013), 1196-1232.
doi: 10.1016/j.jde.2013.05.005. |
[31] |
T. Yang and H. J. Yu,
Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 1035-1062.
doi: 10.1016/S0252-9602(09)60085-0. |
[1] |
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 |
[2] |
Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150 |
[3] |
Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021002 |
[4] |
Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 |
[5] |
Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495 |
[6] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[7] |
Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020180 |
[8] |
Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 |
[9] |
Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287 |
[10] |
Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 |
[11] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[12] |
François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221 |
[13] |
Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020032 |
[14] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[15] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
[16] |
Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 |
[17] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[18] |
Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268 |
[19] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
[20] |
Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]