On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials

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December 2012, 5(4): 769-786. doi: 10.3934/krm.2012.5.769

On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials

Yong-Kum Cho ^1, and Hera Yun ^1,

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

Received February 2012 Revised May 2012 Published November 2012

Abstract
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As for the positive part of Boltzmann's collision operator associated with the collision kernel of soft-potential type, we evaluate its Fourier transform explicitly and prove a set of bilinear estimates for $L^p$ and Sobolev regularity.

Keywords: Hausdorff-Young type inequality, cutoff, Fourier transform, Sobolev space, soft-potential., regularity, collision, bilinear fractional integral, Boltzmann equation.

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

Citation: Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic and Related Models, 2012, 5 (4) : 769-786. doi: 10.3934/krm.2012.5.769

References:

[1]	R. Alonso, E. Carneiro and I. Gamba, Convolution inequalities for the Boltzmann collision operator, Commun. Math. Phys., 298 (2010), 293-322. doi: 10.1007/s00220-010-1065-0.
[2]	R. Alonso and I. Gamba, A revision on classical solutions to the Cauchy Boltzmann problem for soft potentials, J. Stat. Phys., 143 (2011), 740-746. doi: 10.1007/s10955-011-0205-z.
[3]	G. Andrews, R. Askey and R. Roy, "Special Functions," Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press. 71 1999.
[4]	A. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044.
[5]	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.
[6]	L. Desvillettes, About the use of the Fourier transform for the Boltzmannequation, Summer School on Methods and Models in Kinetic Theory, Riv. Mat. Univ. Parma (7), 2 (2003), 1-99.
[7]	L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49-56.
[8]	T. Gustafsson, $L^p$ estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal., 92 (1986), 23-57. doi: 10.1007/BF00250731.
[9]	C. Kenig and E. Stein, Multilinear estimates and fractional integration, Math. Research Letters, 6 (1999), 1-15.
[10]	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.
[11]	X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation, J. Math. Anal. Appl., 228 (1998), 409-435. doi: 10.1006/jmaa.1998.6141.
[12]	E. Lieb and M. Loss, "Analysis," Grad. Stud. Math., Amer. Math. Soc., 14, 1996.
[13]	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.
[14]	E. Stein, "Singular Integrals and Differentiabilty Properties of Functions," Princeton Univ. Press, 1970.
[15]	B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations, 19 (1994), 2057-2074.
[16]	B. Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302.

show all references

References:

[1]	R. Alonso, E. Carneiro and I. Gamba, Convolution inequalities for the Boltzmann collision operator, Commun. Math. Phys., 298 (2010), 293-322. doi: 10.1007/s00220-010-1065-0.
[2]	R. Alonso and I. Gamba, A revision on classical solutions to the Cauchy Boltzmann problem for soft potentials, J. Stat. Phys., 143 (2011), 740-746. doi: 10.1007/s10955-011-0205-z.
[3]	G. Andrews, R. Askey and R. Roy, "Special Functions," Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press. 71 1999.
[4]	A. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044.
[5]	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.
[6]	L. Desvillettes, About the use of the Fourier transform for the Boltzmannequation, Summer School on Methods and Models in Kinetic Theory, Riv. Mat. Univ. Parma (7), 2 (2003), 1-99.
[7]	L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49-56.
[8]	T. Gustafsson, $L^p$ estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal., 92 (1986), 23-57. doi: 10.1007/BF00250731.
[9]	C. Kenig and E. Stein, Multilinear estimates and fractional integration, Math. Research Letters, 6 (1999), 1-15.
[10]	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.
[11]	X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation, J. Math. Anal. Appl., 228 (1998), 409-435. doi: 10.1006/jmaa.1998.6141.
[12]	E. Lieb and M. Loss, "Analysis," Grad. Stud. Math., Amer. Math. Soc., 14, 1996.
[13]	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.
[14]	E. Stein, "Singular Integrals and Differentiabilty Properties of Functions," Princeton Univ. Press, 1970.
[15]	B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations, 19 (1994), 2057-2074.
[16]	B. Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302.

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