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

Robert M. Strain

doi:10.3934/krm.2012.5.583

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2012, Volume 5, Issue 3: 583-613. Doi: 10.3934/krm.2012.5.583

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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

Received: February 29, 2012

Revised: March 31, 2012

Published: August 2012

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Abstract

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 $2\leq r\leq \infty$.

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Mathematics Subject Classification: 76P05, 82C40, 35F20, 26A33.

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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.