On the homogeneous Boltzmann equation with soft-potential collision kernels

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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

Yong-Kum Cho ^1,

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

Abstract
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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.

Keywords: Bilinear mapping principle, Hausdorff-Young type inequality, soft potential., cutoff, collision, fractional integral, Boltzmann equation, Sobolev embedding, regularity.

Mathematics Subject Classification: 35Q82, 47G20, 76P05, 82B4.

Citation: Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic and 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), 327-355. doi: 10.1007/s002050000083.
[2]	L. Arkeryd, On the Boltzmann equation, Part I: Existence, Part II: The full initial value problem, Arch. Rational Mech. Anal., 45 (1972), 1-16.
[3]	L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403.
[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), 47-61. doi: 10.4171/RMI/233.
[5]	A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044.
[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), 681-736. doi: 10.1007/s10955-009-9741-1.
[7]	C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.
[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), 769-786. doi: 10.3934/krm.2012.5.769.
[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), 227-253. doi: 10.1007/s00205-009-0233-x.
[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), 749-781. doi: 10.1007/s10955-008-9511-5.
[11]	L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49-56.
[12]	T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543.
[13]	C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Research Letters, 6 (1999), 1-15. doi: 10.4310/MRL.1999.v6.n1.a1.
[14]	P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461.
[15]	X. Lu and Y. Zhang, On nonnegativity of solutions of the Boltzmann equation, Transport Theor. Stat., 30 (2001), 641-657. doi: 10.1081/TT-100107420.
[16]	S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. Henri Poincaré, 16 (1999), 467-501. doi: 10.1016/S0294-1449(99)80025-0.
[17]	C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rational Mech. Anal., 173 (2004), 169-212. doi: 10.1007/s00205-004-0316-7.
[18]	K. T. Smith, Primer of Modern Analysis, Springer-Verlag, New York, 1983.
[19]	E. M. Stein, Singular Integrals and Differentiabilty Properties of Functions, Princeton Univ. Press, 1970.
[20]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.
[21]	C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.
[22]	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. doi: 10.1016/S1874-5792(02)80004-0.

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References:

[1]	R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.
[2]	L. Arkeryd, On the Boltzmann equation, Part I: Existence, Part II: The full initial value problem, Arch. Rational Mech. Anal., 45 (1972), 1-16.
[3]	L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403.
[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), 47-61. doi: 10.4171/RMI/233.
[5]	A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044.
[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), 681-736. doi: 10.1007/s10955-009-9741-1.
[7]	C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.
[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), 769-786. doi: 10.3934/krm.2012.5.769.
[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), 227-253. doi: 10.1007/s00205-009-0233-x.
[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), 749-781. doi: 10.1007/s10955-008-9511-5.
[11]	L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49-56.
[12]	T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543.
[13]	C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Research Letters, 6 (1999), 1-15. doi: 10.4310/MRL.1999.v6.n1.a1.
[14]	P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461.
[15]	X. Lu and Y. Zhang, On nonnegativity of solutions of the Boltzmann equation, Transport Theor. Stat., 30 (2001), 641-657. doi: 10.1081/TT-100107420.
[16]	S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. Henri Poincaré, 16 (1999), 467-501. doi: 10.1016/S0294-1449(99)80025-0.
[17]	C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rational Mech. Anal., 173 (2004), 169-212. doi: 10.1007/s00205-004-0316-7.
[18]	K. T. Smith, Primer of Modern Analysis, Springer-Verlag, New York, 1983.
[19]	E. M. Stein, Singular Integrals and Differentiabilty Properties of Functions, Princeton Univ. Press, 1970.
[20]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.
[21]	C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.
[22]	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. doi: 10.1016/S1874-5792(02)80004-0.

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