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Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
1. | Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l'Université, BP.12, 76801 Saint Etienne du Rouvray |
2. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
3. | Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray |
References:
[1] |
G. E. Andrews, R. Askey and R. Roy, Special Functions,, Cambridge University Press, (1999).
doi: 10.1017/CBO9781107325937. |
[2] |
A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, Soviet Sci. Rev. Sect. C: Math. Phys., 7 (1988), 111.
|
[3] |
C. Cercignani, The Boltzmann Equation and Its Applications,, Applied Mathematical Sciences, (1988).
doi: 10.1007/978-1-4612-1039-9. |
[4] |
L. Desvillettes, G. Furioli and E. Terraneo, Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules,, Trans. Amer. Math. Soc., 361 (2009), 1731.
doi: 10.1090/S0002-9947-08-04574-1. |
[5] |
L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff,, Comm. Partial Differential Equations, 29 (2004), 133.
doi: 10.1081/PDE-120028847. |
[6] |
E. Dolera, On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules,, Boll. Unione Mat. Ital.(9), 4 (2011), 47.
|
[7] |
L. Glangetas and M. Najeme, Analytical regularizing effect for the radial homogeneous Boltzmann equation,, Kinet. Relat. Models, 6 (2013), 407.
doi: 10.3934/krm.2013.6.407. |
[8] |
T. Gramchev , S. Pilipovié and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces,, New Developments in Pseudo-Differential Operators, 189 (2009), 15.
doi: 10.1007/978-3-7643-8969-7_2. |
[9] |
M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory,, UK: Research Studies Press, (1985).
|
[10] |
N. Lekrine and C.-J. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac equation,, Kinet. Relat. Models, 2 (2009), 647.
doi: 10.3934/krm.2009.2.647. |
[11] |
N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Spectral and phase space analysis of the linearized non-cutoff Kac collision operator,, J. Math. Pures Appl., 100 (2013), 832.
doi: 10.1016/j.matpur.2013.03.005. |
[12] |
N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators,, Kinet. Relat. Models, 6 (2013), 625.
doi: 10.3934/krm.2013.6.625. |
[13] |
N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff,, Journal of Differential Equations, 256 (2014), 797.
doi: 10.1016/j.jde.2013.10.001. |
[14] |
H.-G. Li, Cauchy problem for linearized non-cutoff Boltzmann equation with distribution initial datum,, Acta Mathematica Scientia, 35 (2015), 459.
doi: 10.1016/S0252-9602(15)60015-7. |
[15] |
Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff,, J. Pseudo-Differ. Oper., 1 (2010), 139.
doi: 10.1007/s11868-010-0008-z. |
[16] |
Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,, Discrete Contin. Dyn. Syst., 24 (2009), 187.
doi: 10.3934/dcds.2009.24.187. |
[17] |
C. Müller, Analysis of Spherical Symmetries in Euclidean Spaces,, Springer, (1998).
doi: 10.1007/978-1-4612-0581-4. |
[18] |
W. Rudin, Principles of Mathematical Analysis,, McGraw-Hill Book Co., (1964).
|
[19] |
G. Sansone, Orthogonal Functions,, Reprinted by Dover Publications 1991, (1991).
|
[20] |
J. C. Slater, Quantum Theory of Atomic Structure,, Vol. 1, (1960).
doi: 10.1063/1.3057555. |
[21] |
S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutff,, Japan. J. Appl. Math., 1 (1984), 141.
doi: 10.1007/BF03167864. |
[22] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71.
doi: 10.1016/S1874-5792(02)80004-0. |
[23] |
C. S. Wang Chang and G. E. Uhlenbeck, On The Propagation of Sound in Monoatomic Gases,, Univ. of Michigan Press, (1970). Google Scholar |
[24] |
G. N. Watson, A Treatise On The Theory of Bessel Functions,, Cambridge university press, (1995).
|
[25] |
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis,, Cambridge university press, (1996).
doi: 10.1017/CBO9780511608759. |
[26] |
T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff,, J. Differential Equations, 253 (2012), 1172.
doi: 10.1016/j.jde.2012.04.023. |
show all references
References:
[1] |
G. E. Andrews, R. Askey and R. Roy, Special Functions,, Cambridge University Press, (1999).
doi: 10.1017/CBO9781107325937. |
[2] |
A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, Soviet Sci. Rev. Sect. C: Math. Phys., 7 (1988), 111.
|
[3] |
C. Cercignani, The Boltzmann Equation and Its Applications,, Applied Mathematical Sciences, (1988).
doi: 10.1007/978-1-4612-1039-9. |
[4] |
L. Desvillettes, G. Furioli and E. Terraneo, Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules,, Trans. Amer. Math. Soc., 361 (2009), 1731.
doi: 10.1090/S0002-9947-08-04574-1. |
[5] |
L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff,, Comm. Partial Differential Equations, 29 (2004), 133.
doi: 10.1081/PDE-120028847. |
[6] |
E. Dolera, On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules,, Boll. Unione Mat. Ital.(9), 4 (2011), 47.
|
[7] |
L. Glangetas and M. Najeme, Analytical regularizing effect for the radial homogeneous Boltzmann equation,, Kinet. Relat. Models, 6 (2013), 407.
doi: 10.3934/krm.2013.6.407. |
[8] |
T. Gramchev , S. Pilipovié and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces,, New Developments in Pseudo-Differential Operators, 189 (2009), 15.
doi: 10.1007/978-3-7643-8969-7_2. |
[9] |
M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory,, UK: Research Studies Press, (1985).
|
[10] |
N. Lekrine and C.-J. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac equation,, Kinet. Relat. Models, 2 (2009), 647.
doi: 10.3934/krm.2009.2.647. |
[11] |
N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Spectral and phase space analysis of the linearized non-cutoff Kac collision operator,, J. Math. Pures Appl., 100 (2013), 832.
doi: 10.1016/j.matpur.2013.03.005. |
[12] |
N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators,, Kinet. Relat. Models, 6 (2013), 625.
doi: 10.3934/krm.2013.6.625. |
[13] |
N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff,, Journal of Differential Equations, 256 (2014), 797.
doi: 10.1016/j.jde.2013.10.001. |
[14] |
H.-G. Li, Cauchy problem for linearized non-cutoff Boltzmann equation with distribution initial datum,, Acta Mathematica Scientia, 35 (2015), 459.
doi: 10.1016/S0252-9602(15)60015-7. |
[15] |
Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff,, J. Pseudo-Differ. Oper., 1 (2010), 139.
doi: 10.1007/s11868-010-0008-z. |
[16] |
Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,, Discrete Contin. Dyn. Syst., 24 (2009), 187.
doi: 10.3934/dcds.2009.24.187. |
[17] |
C. Müller, Analysis of Spherical Symmetries in Euclidean Spaces,, Springer, (1998).
doi: 10.1007/978-1-4612-0581-4. |
[18] |
W. Rudin, Principles of Mathematical Analysis,, McGraw-Hill Book Co., (1964).
|
[19] |
G. Sansone, Orthogonal Functions,, Reprinted by Dover Publications 1991, (1991).
|
[20] |
J. C. Slater, Quantum Theory of Atomic Structure,, Vol. 1, (1960).
doi: 10.1063/1.3057555. |
[21] |
S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutff,, Japan. J. Appl. Math., 1 (1984), 141.
doi: 10.1007/BF03167864. |
[22] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71.
doi: 10.1016/S1874-5792(02)80004-0. |
[23] |
C. S. Wang Chang and G. E. Uhlenbeck, On The Propagation of Sound in Monoatomic Gases,, Univ. of Michigan Press, (1970). Google Scholar |
[24] |
G. N. Watson, A Treatise On The Theory of Bessel Functions,, Cambridge university press, (1995).
|
[25] |
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis,, Cambridge university press, (1996).
doi: 10.1017/CBO9780511608759. |
[26] |
T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff,, J. Differential Equations, 253 (2012), 1172.
doi: 10.1016/j.jde.2012.04.023. |
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