# American Institute of Mathematical Sciences

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

 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

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.
Citation: Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic & 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. 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. doi: 10.1007/s10955-011-0205-z. [3] G. Andrews, R. Askey and R. Roy, "Special Functions,", Encyclopedia of Mathematics and Its Applications, 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. [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. 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, 2 (2003), 1. [7] L. Grafakos, On multilinear fractional integrals,, Studia Math., 102 (1992), 49. [8] T. Gustafsson, $L^p$ estimates for the nonlinear spatially homogeneous Boltzmann equation,, Arch. Rational Mech. Anal., 92 (1986), 23. doi: 10.1007/BF00250731. [9] C. Kenig and E. Stein, Multilinear estimates and fractional integration,, Math. Research Letters, 6 (1999), 1. [10] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II,, J. Math. Kyoto Univ., 34 (1994), 391. [11] X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation,, J. Math. Anal. Appl., 228 (1998), 409. doi: 10.1006/jmaa.1998.6141. [12] E. Lieb and M. Loss, "Analysis,", Grad. Stud. Math., 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. 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. [16] B. Wennberg, The geometry of binary collisions and generalized Radon transforms,, Arch. Rational Mech. Anal., 139 (1997), 291.

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##### References:
 [1] R. Alonso, E. Carneiro and I. Gamba, Convolution inequalities for the Boltzmann collision operator,, Commun. Math. Phys., 298 (2010), 293. 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. doi: 10.1007/s10955-011-0205-z. [3] G. Andrews, R. Askey and R. Roy, "Special Functions,", Encyclopedia of Mathematics and Its Applications, 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. [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. 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, 2 (2003), 1. [7] L. Grafakos, On multilinear fractional integrals,, Studia Math., 102 (1992), 49. [8] T. Gustafsson, $L^p$ estimates for the nonlinear spatially homogeneous Boltzmann equation,, Arch. Rational Mech. Anal., 92 (1986), 23. doi: 10.1007/BF00250731. [9] C. Kenig and E. Stein, Multilinear estimates and fractional integration,, Math. Research Letters, 6 (1999), 1. [10] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II,, J. Math. Kyoto Univ., 34 (1994), 391. [11] X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation,, J. Math. Anal. Appl., 228 (1998), 409. doi: 10.1006/jmaa.1998.6141. [12] E. Lieb and M. Loss, "Analysis,", Grad. Stud. Math., 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. 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. [16] B. Wennberg, The geometry of binary collisions and generalized Radon transforms,, Arch. Rational Mech. Anal., 139 (1997), 291.
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