American Institute of Mathematical Sciences

Advanced
July  2019, 39(7): 4303-4329. doi: 10.3934/dcds.2019174

The Boltzmann equation with frictional force for very soft potentials in the whole space

Yingzhe Fan 1, and  Yuanjie Lei 2,3,,

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

* Corresponding author: Yuanjie Lei

Received  November 2016 Published  April 2019

Fund Project: The corresponding author is supported by NSFC grant No.11601169 and 11871335.

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 $.

Keywords: Boltzmann equation with frictional force, soft potentials, energy method, optimal time decay rates, global existence.
Mathematics Subject Classification: 82B40, 35A01, 35Q83.
Citation: Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

[17]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[18]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15] H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ., in Rarefied Gas Dynamics Vol. Ⅰ, Academic Press, New York, 1963.   Google Scholar
[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.  Google Scholar

[17]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[18]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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