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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 |
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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