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Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity
Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China |
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.
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. |
[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. |
[3] |
R. E. Caflisch,
The Boltzmann equation with a soft potential-II. Nonlinear, spatially-periodic, Commun. Math. Phys., 74 (1980), 97-109.
|
[4] |
C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[5] |
R. J. DiPerna and P. L. Lions,
On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Commun. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[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. |
[18] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
R. E. Caflisch,
The Boltzmann equation with a soft potential-II. Nonlinear, spatially-periodic, Commun. Math. Phys., 74 (1980), 97-109.
|
[4] |
C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[5] |
R. J. DiPerna and P. L. Lions,
On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Commun. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[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. |
[18] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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