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September  2012, 5(3): 583-613. doi: 10.3934/krm.2012.5.583

Optimal time decay of the non cut-off Boltzmann equation in the whole space

Robert M. Strain 1,

1. 

University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104, United States

Received  March 2012 Revised  April 2012 Published  August 2012

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space $\mathbb{R}^n _x$ with $n≥3$ .We use the existence theory of global in time nearby Maxwellian solutions from [12,11].It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption [26,1]. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\frac{N}{2}+\frac{N}{2r}})$ in the $L^2_v$$(L^r_x)$-norm for any $2\leq r\leq \infty$.
Keywords: fractional derivatives, Boltzmann equation, Kinetic Theory, soft potentials, long-range interaction, non cut-off, time decay, convergence rates., hard potentials.
Mathematics Subject Classification: 76P05, 82C40, 35F20, 26A3.
Citation: Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic and Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583
References:
[1]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0.

[2]

Russel E. Caflisch, The Boltzmann equation with a soft potential. I, II, Comm. Math. Phys., 74 (1980), 71-95, 97-109. doi: 10.1007/BF01197579.

[3]

Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case, Arch. Ration. Mech. Anal., 203 (2012), 343-377. doi: 10.1007/s00205-011-0482-3.

[4]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.

[5]

R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189. doi: 10.1088/0951-7715/24/8/003.

[6]

R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\varepsilon$（$H^N_x$）, J. Differential Equations, 244 (2008), 3204-3234. doi: 10.1016/j.jde.2007.11.006.

[7]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$, Arch. Rational Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6.

[8]

R. 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.

[9]

R. Duan, S. Ukai, T. Yang and H. 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. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.

[11]

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.

[12]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci. U. S. A., 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107.

[13]

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

[14]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.

[15]

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

[16]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.

[17]

Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.

[18]

C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl. (9), 87 (2007), 515-535.

[19]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567. doi: 10.1007/s00220-006-0109-y.

[20]

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

[21]

Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.

[22]

Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.

[23]

R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $mathbbR^3_x$, Kinetic and Related Models, 5 (2012), 383-415.

[24]

M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997.

[25]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.

[26]

Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 477-519 (57-99). doi: 10.2977/prims/1195183569.

[27]

C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics," Vol. I, North-Holland, Amsterdam, (2002), 71-305.

[28]

C. Villani, "Hypocoercivity," Mem. Amer. Math. Soc., 202 (2009), iv+141.

show all references

References:
[1]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0.

[2]

Russel E. Caflisch, The Boltzmann equation with a soft potential. I, II, Comm. Math. Phys., 74 (1980), 71-95, 97-109. doi: 10.1007/BF01197579.

[3]

Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case, Arch. Ration. Mech. Anal., 203 (2012), 343-377. doi: 10.1007/s00205-011-0482-3.

[4]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.

[5]

R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189. doi: 10.1088/0951-7715/24/8/003.

[6]

R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\varepsilon$（$H^N_x$）, J. Differential Equations, 244 (2008), 3204-3234. doi: 10.1016/j.jde.2007.11.006.

[7]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$, Arch. Rational Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6.

[8]

R. 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.

[9]

R. Duan, S. Ukai, T. Yang and H. 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. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.

[11]

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.

[12]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci. U. S. A., 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107.

[13]

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

[14]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.

[15]

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

[16]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.

[17]

Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.

[18]

C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl. (9), 87 (2007), 515-535.

[19]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567. doi: 10.1007/s00220-006-0109-y.

[20]

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

[21]

Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.

[22]

Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.

[23]

R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $mathbbR^3_x$, Kinetic and Related Models, 5 (2012), 383-415.

[24]

M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997.

[25]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.

[26]

Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 477-519 (57-99). doi: 10.2977/prims/1195183569.

[27]

C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics," Vol. I, North-Holland, Amsterdam, (2002), 71-305.

[28]

C. Villani, "Hypocoercivity," Mem. Amer. Math. Soc., 202 (2009), iv+141.

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