September  2012, 5(3): 441-458. doi: 10.3934/krm.2012.5.441

A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem

1. 

Department of Mathematics, College of Natural Sciences, Chung-Ang University, 84 Heukseok-Ro, Dongjak-Gu, Seoul 156-756, South Korea

Received  November 2011 Revised  February 2012 Published  August 2012

We present a Fourier transform formula of quadratic-form type for the collision operator with a Maxwellian kernel under the momentum transfer condition. As an application, we extend the work of Toscani and Villani on the uniform stability of the Cauchy problem for the associated Boltzmann equation to any physically relevant Maxwellian molecules in the long-range interactions with a minimal requirement for the initial data.
Citation: Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic and Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441
References:
[1]

R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations, I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., 15 (2005), 907-920. doi: 10.1142/S0218202505000613.

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interations, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.

[3]

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

[4]

J. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), 75-198.

[5]

L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation, Summer School on Methods and Models in Kinetic Theory, Riv. Mat. Univ. Parma (7), 2 (2003), 1-99.

[6]

L. Desvillettes and M. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Rational Mech. Anal., 193 (2009), 227-253. doi: 10.1007/s00205-009-0233-x.

[7]

N. Fournier and G. Héléne, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys., 131 (2008), 749-781. doi: 10.1007/s10955-008-9511-5.

[8]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grasing collisions, J. Stat. Phys., 89 (1997), 752-776. doi: 10.1007/BF02765543.

[9]

A. Pulvirenti and G. Toscani, The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation, Ann. Mat. Pura Appl. (IV), 171 (1996), 181-204. doi: 10.1007/BF01759387.

[10]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Stat. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.

[11]

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

[12]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.

show all references

References:
[1]

R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations, I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., 15 (2005), 907-920. doi: 10.1142/S0218202505000613.

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interations, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.

[3]

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

[4]

J. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), 75-198.

[5]

L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation, Summer School on Methods and Models in Kinetic Theory, Riv. Mat. Univ. Parma (7), 2 (2003), 1-99.

[6]

L. Desvillettes and M. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Rational Mech. Anal., 193 (2009), 227-253. doi: 10.1007/s00205-009-0233-x.

[7]

N. Fournier and G. Héléne, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys., 131 (2008), 749-781. doi: 10.1007/s10955-008-9511-5.

[8]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grasing collisions, J. Stat. Phys., 89 (1997), 752-776. doi: 10.1007/BF02765543.

[9]

A. Pulvirenti and G. Toscani, The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation, Ann. Mat. Pura Appl. (IV), 171 (1996), 181-204. doi: 10.1007/BF01759387.

[10]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Stat. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.

[11]

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

[12]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.

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