2014, 7(1): 169-194. doi: 10.3934/krm.2014.7.169

Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation

1. 

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

Received  June 2013 Revised  October 2013 Published  December 2013

The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.
Citation: Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169
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,, Comm. Partial Differential Equations, 26 (2001), 43. doi: 10.1081/PDE-100002246.

[2]

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

[3]

C. Cercignani, The Boltzmann Equation and Its Applications,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4612-1039-9.

[4]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994).

[5]

R. J. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation,, Comm. Math. Phys., 120 (1988), 1. doi: 10.1007/BF01223204.

[6]

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

[7]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$,, Arch. Ration. Mech. Anal., 199 (2011), 291. doi: 10.1007/s00205-010-0318-6.

[8]

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

[9]

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

[10]

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

[11]

F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation,, Arch. Rational Mech. Anal., 103 (1988), 81. doi: 10.1007/BF00292921.

[12]

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

[13]

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

[14]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081. doi: 10.1512/iumj.2004.53.2574.

[15]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation,, Comm. Pure Appl. Math., 59 (2006), 626. doi: 10.1002/cpa.20121.

[16]

K. Hamdache, Estimations uniformes des solutions de l'équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck,, (French) [Uniform estimates for solutions of the perturbed Boltzmann equation by artificial viscosity or Fokker-Planck diffusion], 302 (1986), 187.

[17]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,, Ph.D thesis, (1983).

[18]

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

[19]

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

[20]

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

[21]

S. K. Loyalka, Rarefied gas dynamic problems in environmental sciences,, in Proceedings 15th International Symposium on Rarefied Gas Dynamics (eds. V. Boffi and C. Cercignani), (1986).

[22]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Partial Differential Equations, 31 (2006), 1321. doi: 10.1080/03605300600635004.

[23]

W. A. Strauss, Decay and asymptotics for $u_{t t} - \Delta u=F(u)$,, J. Functional Analysis, 2 (1968), 409. doi: 10.1016/0022-1236(68)90004-9.

[24]

R. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287. doi: 10.1007/s00205-007-0067-3.

[25]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0.

[26]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

[27]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179. doi: 10.3792/pja/1195519027.

[28]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions,, Anal. Appl. (Singap.), 4 (2006), 263. doi: 10.1142/S0219530506000784.

[29]

M.-Y. Zhong and H.-L. Li, Long time behavior of the Fokker-Planck-Boltzmann equation with soft potential,, Quart. Appl. Math., 70 (2012), 721. doi: 10.1090/S0033-569X-2012-01269-3.

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,, Comm. Partial Differential Equations, 26 (2001), 43. doi: 10.1081/PDE-100002246.

[2]

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

[3]

C. Cercignani, The Boltzmann Equation and Its Applications,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4612-1039-9.

[4]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994).

[5]

R. J. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation,, Comm. Math. Phys., 120 (1988), 1. doi: 10.1007/BF01223204.

[6]

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

[7]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$,, Arch. Ration. Mech. Anal., 199 (2011), 291. doi: 10.1007/s00205-010-0318-6.

[8]

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

[9]

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

[10]

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

[11]

F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation,, Arch. Rational Mech. Anal., 103 (1988), 81. doi: 10.1007/BF00292921.

[12]

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

[13]

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

[14]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081. doi: 10.1512/iumj.2004.53.2574.

[15]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation,, Comm. Pure Appl. Math., 59 (2006), 626. doi: 10.1002/cpa.20121.

[16]

K. Hamdache, Estimations uniformes des solutions de l'équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck,, (French) [Uniform estimates for solutions of the perturbed Boltzmann equation by artificial viscosity or Fokker-Planck diffusion], 302 (1986), 187.

[17]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,, Ph.D thesis, (1983).

[18]

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

[19]

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

[20]

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

[21]

S. K. Loyalka, Rarefied gas dynamic problems in environmental sciences,, in Proceedings 15th International Symposium on Rarefied Gas Dynamics (eds. V. Boffi and C. Cercignani), (1986).

[22]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Partial Differential Equations, 31 (2006), 1321. doi: 10.1080/03605300600635004.

[23]

W. A. Strauss, Decay and asymptotics for $u_{t t} - \Delta u=F(u)$,, J. Functional Analysis, 2 (1968), 409. doi: 10.1016/0022-1236(68)90004-9.

[24]

R. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287. doi: 10.1007/s00205-007-0067-3.

[25]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0.

[26]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

[27]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179. doi: 10.3792/pja/1195519027.

[28]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions,, Anal. Appl. (Singap.), 4 (2006), 263. doi: 10.1142/S0219530506000784.

[29]

M.-Y. Zhong and H.-L. Li, Long time behavior of the Fokker-Planck-Boltzmann equation with soft potential,, Quart. Appl. Math., 70 (2012), 721. doi: 10.1090/S0033-569X-2012-01269-3.

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