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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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. Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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. Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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