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June  2015, 8(2): 309-333. doi: 10.3934/krm.2015.8.309

On the homogeneous Boltzmann equation with soft-potential collision kernels

1. 

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

Received  August 2014 Revised  January 2015 Published  March 2015

We consider the well-posedness problem for the space-homogeneous Boltzmann equation with soft-potential collision kernels. By revisiting the classical Fourier inequalities and fractional integrals, we deduce a set of bilinear estimates for the collision operator on the space of integrable functions possessing certain degree of smoothness and we apply them to prove the local-in-time existence of a solution to the Boltzmann equation in both integral form and the original one. Uniqueness and stability of solutions are also established.
Citation: Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic & Related Models, 2015, 8 (2) : 309-333. doi: 10.3934/krm.2015.8.309
References:
[1]

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

[2]

L. Arkeryd, On the Boltzmann equation, Part I: Existence, Part II: The full initial value problem,, Arch. Rational Mech. Anal., 45 (1972), 1. Google Scholar

[3]

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,, Arch. Rational Mech. Anal., 77 (1981), 11. doi: 10.1007/BF00280403. Google Scholar

[4]

F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel,, Rev. Mat. Iberoamericana, 14 (1998), 47. doi: 10.4171/RMI/233. Google Scholar

[5]

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

[6]

E. Carlen, M. Carvalho and X. Lu, On strong convergence to equilibrium for the Boltzmann equation with soft potentials,, J. Stat. Phys., 135 (2009), 681. doi: 10.1007/s10955-009-9741-1. Google Scholar

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer-Verlag, (1994). doi: 10.1007/978-1-4419-8524-8. Google Scholar

[8]

Y.-K. Cho and H. Yun, On the gain of regularity for the positive part of Boltzmann collision operator associated with soft potentials,, Kinetic and Related Models, 5 (2012), 769. doi: 10.3934/krm.2012.5.769. Google Scholar

[9]

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

[10]

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. doi: 10.1007/s10955-008-9511-5. Google Scholar

[11]

L. Grafakos, On multilinear fractional integrals,, Studia Math., 102 (1992), 49. Google Scholar

[12]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions,, J. Stat. Phys., 89 (1997), 751. doi: 10.1007/BF02765543. Google Scholar

[13]

C. Kenig and E. M. Stein, Multilinear estimates and fractional integration,, Math. Research Letters, 6 (1999), 1. doi: 10.4310/MRL.1999.v6.n1.a1. Google Scholar

[14]

P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II,, J. Math. Kyoto Univ., 34 (1994), 391. Google Scholar

[15]

X. Lu and Y. Zhang, On nonnegativity of solutions of the Boltzmann equation,, Transport Theor. Stat., 30 (2001), 641. doi: 10.1081/TT-100107420. Google Scholar

[16]

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation,, Ann. Inst. Henri Poincaré, 16 (1999), 467. doi: 10.1016/S0294-1449(99)80025-0. Google Scholar

[17]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off,, Arch. Rational Mech. Anal., 173 (2004), 169. doi: 10.1007/s00205-004-0316-7. Google Scholar

[18]

K. T. Smith, Primer of Modern Analysis,, Springer-Verlag, (1983). Google Scholar

[19]

E. M. Stein, Singular Integrals and Differentiabilty Properties of Functions,, Princeton Univ. Press, (1970). Google Scholar

[20]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Univ. Press, (1993). Google Scholar

[21]

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

[22]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

show all references

References:
[1]

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

[2]

L. Arkeryd, On the Boltzmann equation, Part I: Existence, Part II: The full initial value problem,, Arch. Rational Mech. Anal., 45 (1972), 1. Google Scholar

[3]

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,, Arch. Rational Mech. Anal., 77 (1981), 11. doi: 10.1007/BF00280403. Google Scholar

[4]

F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel,, Rev. Mat. Iberoamericana, 14 (1998), 47. doi: 10.4171/RMI/233. Google Scholar

[5]

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

[6]

E. Carlen, M. Carvalho and X. Lu, On strong convergence to equilibrium for the Boltzmann equation with soft potentials,, J. Stat. Phys., 135 (2009), 681. doi: 10.1007/s10955-009-9741-1. Google Scholar

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer-Verlag, (1994). doi: 10.1007/978-1-4419-8524-8. Google Scholar

[8]

Y.-K. Cho and H. Yun, On the gain of regularity for the positive part of Boltzmann collision operator associated with soft potentials,, Kinetic and Related Models, 5 (2012), 769. doi: 10.3934/krm.2012.5.769. Google Scholar

[9]

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

[10]

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. doi: 10.1007/s10955-008-9511-5. Google Scholar

[11]

L. Grafakos, On multilinear fractional integrals,, Studia Math., 102 (1992), 49. Google Scholar

[12]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions,, J. Stat. Phys., 89 (1997), 751. doi: 10.1007/BF02765543. Google Scholar

[13]

C. Kenig and E. M. Stein, Multilinear estimates and fractional integration,, Math. Research Letters, 6 (1999), 1. doi: 10.4310/MRL.1999.v6.n1.a1. Google Scholar

[14]

P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II,, J. Math. Kyoto Univ., 34 (1994), 391. Google Scholar

[15]

X. Lu and Y. Zhang, On nonnegativity of solutions of the Boltzmann equation,, Transport Theor. Stat., 30 (2001), 641. doi: 10.1081/TT-100107420. Google Scholar

[16]

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation,, Ann. Inst. Henri Poincaré, 16 (1999), 467. doi: 10.1016/S0294-1449(99)80025-0. Google Scholar

[17]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off,, Arch. Rational Mech. Anal., 173 (2004), 169. doi: 10.1007/s00205-004-0316-7. Google Scholar

[18]

K. T. Smith, Primer of Modern Analysis,, Springer-Verlag, (1983). Google Scholar

[19]

E. M. Stein, Singular Integrals and Differentiabilty Properties of Functions,, Princeton Univ. Press, (1970). Google Scholar

[20]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Univ. Press, (1993). Google Scholar

[21]

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

[22]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

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