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

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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 and Continuous Dynamical Systems, 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.
[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.

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