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September  2017, 10(3): 573-585. doi: 10.3934/krm.2017023

Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules

1. 

Keldysh Institute for Applied Mathematics RAS, Miusskaya Sq. 4, 125047 Moscow, RF, Russia

2. 

Department of Mathematics and ICES, The University of Texas at Austin, 1 University Station C0200, Texas 78712, USA

Received  July 2016 Revised  August 2016 Published  December 2016

We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desired local estimate. We remark that propagation of such upper Maxwellian bounds were obtained by Gamba, Panferov and Villani for the case of hard spheres and hard potentials with angular cut-off. That manuscript introduced the main ideas and tools needed to prove such local estimates on the basis of similar integral estimates. The case of pseudo-Maxwell molecules needs, however, a special consideration performed in the present paper.

Citation: Alexander V. Bobylev, Irene M. Gamba. Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules. Kinetic & Related Models, 2017, 10 (3) : 573-585. doi: 10.3934/krm.2017023
References:
[1]

R. Alonso and I. M. Gamba, $ L^1$-$L^\infty$ Maxwellian bounds for the derivatives of the solution of the homogeneous Boltzmann equation, Journal de Mathematiques Pures et Appliquees(9), 89 (2008), 575-595.  doi: 10.1016/j.matpur.2008.02.006.  Google Scholar

[2]

L. Arkeryd, $ L^∞$ estimates for the space homogeneous Boltzmann equation, J. Statist. Phys., 31 (1983), 347-361.  doi: 10.1007/BF01011586.  Google Scholar

[3]

A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwellian molecules(Russian), Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044.   Google Scholar

[4]

A. V. Bobylev, On expansion of the Boltzmann integral into Landau series, Sov. Phys. Dokl., 207 (1976), 40-742.   Google Scholar

[5]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Mathematical physics reviews, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, 7 (1988), 111-233.  Google Scholar

[6]

A. V. Bobylev, Moment inequalities for the Boltzmann equations and application to spatially homogeneous problems, J. Statist. Phys., 88 (1997), 1183-1214.  doi: 10.1007/BF02732431.  Google Scholar

[7]

A. V. Bobylev and V. V. Vedeniapin, The maximum principle for discrete models of the Boltzmann equation and the relation of integrals of direct and inverse collisions of the Boltzmann equation, (in Russian), Doklady AN SSSR, 233 (1977), 519-522. English translation in Soviet Math. Dokl. Vol., 18 (1977), 413-417.  Google Scholar

[8]

I. M. GambaV. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal, 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9.  Google Scholar

[9] L. D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford-New York-Toronto, Ont, 1976.   Google Scholar
[10]

B. Wennberg, Stability and Exponential Convergence for the Boltzmann Equation, PhD thesis, Department of Mathematics, Chalmers/Gothenburg University, 1993. Google Scholar

[11]

B. Wennberg, Stability and exponential convergence for the Boltzmann equation, Arch. Rational Mech. Anal., 130 (1995), 103-144.  doi: 10.1007/BF00375152.  Google Scholar

show all references

References:
[1]

R. Alonso and I. M. Gamba, $ L^1$-$L^\infty$ Maxwellian bounds for the derivatives of the solution of the homogeneous Boltzmann equation, Journal de Mathematiques Pures et Appliquees(9), 89 (2008), 575-595.  doi: 10.1016/j.matpur.2008.02.006.  Google Scholar

[2]

L. Arkeryd, $ L^∞$ estimates for the space homogeneous Boltzmann equation, J. Statist. Phys., 31 (1983), 347-361.  doi: 10.1007/BF01011586.  Google Scholar

[3]

A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwellian molecules(Russian), Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044.   Google Scholar

[4]

A. V. Bobylev, On expansion of the Boltzmann integral into Landau series, Sov. Phys. Dokl., 207 (1976), 40-742.   Google Scholar

[5]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Mathematical physics reviews, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, 7 (1988), 111-233.  Google Scholar

[6]

A. V. Bobylev, Moment inequalities for the Boltzmann equations and application to spatially homogeneous problems, J. Statist. Phys., 88 (1997), 1183-1214.  doi: 10.1007/BF02732431.  Google Scholar

[7]

A. V. Bobylev and V. V. Vedeniapin, The maximum principle for discrete models of the Boltzmann equation and the relation of integrals of direct and inverse collisions of the Boltzmann equation, (in Russian), Doklady AN SSSR, 233 (1977), 519-522. English translation in Soviet Math. Dokl. Vol., 18 (1977), 413-417.  Google Scholar

[8]

I. M. GambaV. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal, 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9.  Google Scholar

[9] L. D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford-New York-Toronto, Ont, 1976.   Google Scholar
[10]

B. Wennberg, Stability and Exponential Convergence for the Boltzmann Equation, PhD thesis, Department of Mathematics, Chalmers/Gothenburg University, 1993. Google Scholar

[11]

B. Wennberg, Stability and exponential convergence for the Boltzmann equation, Arch. Rational Mech. Anal., 130 (1995), 103-144.  doi: 10.1007/BF00375152.  Google Scholar

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