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June  2020, 19(6): 3113-3136. doi: 10.3934/cpaa.2020135

Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off

Lvqiao Liu and  Hao Wang ,

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

* Corresponding author

Received  June 2019 Revised  December 2019 Published  March 2020

In this article we study the large-time behavior of perturbative classical solutions to the Fokker-Planck-Boltzmann equation for non-cutoff hard potentials. When the initial data is a small pertubation of an equilibrium state, global existence and temporal decay estimates of classical solutions are established.

Keywords: Fokker-Planck-Boltzmann equation, energy method, non cut-off, time decay.
Mathematics Subject Classification: Primary: 35Q84, 35Q20; Secondary: 82C31.
Citation: 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
References:
[1]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Commun. Partial Differ. Equ., 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.  Google Scholar

[2]

M. Bisi, J. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equilibria, J. Statist. Phys., 118 (2005), 301-331.  doi: 10.1007/s10955-004-8785-5.  Google Scholar

[3]

R. E. Caflisch, The Boltzmann equation with a soft potential-II. Nonlinear, spatially-periodic, Commun. Math. Phys., 74 (1980), 97-109.   Google Scholar

[4]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[5]

R. J. DiPerna and P. L. Lions, On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23.   Google Scholar

[6]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[7]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[8]

R. 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

[9]

R. 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

[10]

R. Duan, T. Yang and H. Zhao, The Vlasov–Poisson–Boltzmann system in the whole space: The hard potential case, J. Differ. Equ., 252 (2012), 6356-6386.  doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[11]

R. Duan, T. Yang and H. Zhao, The Vlasov–Poisson–Boltzmann system for soft potentials, Math. Models Meth. Appl. Sci., 23 (2013), 979-1028.  doi: 10.1142/S0218202513500012.  Google Scholar

[12]

R. Duan and S. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, Commun. Math. Phys., 324 (2013), 1-45.  doi: 10.1007/s00220-013-1807-x.  Google Scholar

[13]

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

[14]

P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384.  doi: 10.1016/j.aim.2011.05.005.  Google Scholar

[15]

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

[16]

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

[17]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9.  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]

K. Hamdache, Estimations uniformes des solutions de l'equation de Boltzmann par les methodes de viscosité artificielle et de diffusion de Fokker-Planck, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 187–190.  Google Scholar

[20]

H. L. Li and A. Matsumura, Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian, Arch. Ration. Mech. Anal., 189 (2008), 1-44.  doi: 10.1007/s00205-007-0057-5.  Google Scholar

[21]

T. P. Liu, T. Yang and S. H. Yu, Energy method for Boltzmann equation, Physica D, 188 (2004), 178-192.  doi: 10.1016/j.physd.2003.07.011.  Google Scholar

[22]

T. P. Liu and S. H. Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[23]

R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models, 5 (2012), 583-613.  doi: 10.3934/krm.2012.5.583.  Google Scholar

[24]

R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Commun. Math. Phys., 300 (2010), 529-597.  doi: 10.1007/s00220-010-1129-1.  Google Scholar

[25]

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

[26]

L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation, Kinet. Relat. Models, 7 (2014), 169-194.  doi: 10.3934/krm.2014.7.169.  Google Scholar

[27]

T. Yang, H. Yu and and H. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rat. Mech. Anal., 182 (2006), 415-470.  doi: 10.1007/s00205-006-0009-5.  Google Scholar

[28]

T. Yang and H. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Commun. Math. Phys., 268 (2006), 569-605.  doi: 10.1007/s00220-006-0103-4.  Google Scholar

show all references

References:
[1]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Commun. Partial Differ. Equ., 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.  Google Scholar

[2]

M. Bisi, J. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equilibria, J. Statist. Phys., 118 (2005), 301-331.  doi: 10.1007/s10955-004-8785-5.  Google Scholar

[3]

R. E. Caflisch, The Boltzmann equation with a soft potential-II. Nonlinear, spatially-periodic, Commun. Math. Phys., 74 (1980), 97-109.   Google Scholar

[4]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[5]

R. J. DiPerna and P. L. Lions, On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23.   Google Scholar

[6]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[7]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[8]

R. 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

[9]

R. 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

[10]

R. Duan, T. Yang and H. Zhao, The Vlasov–Poisson–Boltzmann system in the whole space: The hard potential case, J. Differ. Equ., 252 (2012), 6356-6386.  doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[11]

R. Duan, T. Yang and H. Zhao, The Vlasov–Poisson–Boltzmann system for soft potentials, Math. Models Meth. Appl. Sci., 23 (2013), 979-1028.  doi: 10.1142/S0218202513500012.  Google Scholar

[12]

R. Duan and S. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, Commun. Math. Phys., 324 (2013), 1-45.  doi: 10.1007/s00220-013-1807-x.  Google Scholar

[13]

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

[14]

P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384.  doi: 10.1016/j.aim.2011.05.005.  Google Scholar

[15]

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

[16]

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

[17]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9.  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]

K. Hamdache, Estimations uniformes des solutions de l'equation de Boltzmann par les methodes de viscosité artificielle et de diffusion de Fokker-Planck, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 187–190.  Google Scholar

[20]

H. L. Li and A. Matsumura, Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian, Arch. Ration. Mech. Anal., 189 (2008), 1-44.  doi: 10.1007/s00205-007-0057-5.  Google Scholar

[21]

T. P. Liu, T. Yang and S. H. Yu, Energy method for Boltzmann equation, Physica D, 188 (2004), 178-192.  doi: 10.1016/j.physd.2003.07.011.  Google Scholar

[22]

T. P. Liu and S. H. Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[23]

R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models, 5 (2012), 583-613.  doi: 10.3934/krm.2012.5.583.  Google Scholar

[24]

R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Commun. Math. Phys., 300 (2010), 529-597.  doi: 10.1007/s00220-010-1129-1.  Google Scholar

[25]

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

[26]

L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation, Kinet. Relat. Models, 7 (2014), 169-194.  doi: 10.3934/krm.2014.7.169.  Google Scholar

[27]

T. Yang, H. Yu and and H. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rat. Mech. Anal., 182 (2006), 415-470.  doi: 10.1007/s00205-006-0009-5.  Google Scholar

[28]

T. Yang and H. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Commun. Math. Phys., 268 (2006), 569-605.  doi: 10.1007/s00220-006-0103-4.  Google Scholar

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