# American Institute of Mathematical Sciences

June  2016, 9(2): 299-371. doi: 10.3934/krm.2016.9.299

## 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

Received  December 2014 Revised  October 2015 Published  March 2016

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov's regularizing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of the fractional exponent is exactly the same as the singular index of the non-cutoff collisional kernel of the Boltzmann equation. Therefore, we get the sharp regularity of solutions in the Gevrey class and also the sharp decay of solutions with an exponential weight. We also give a method to construct the solution of the Boltzmann equation by solving an infinite system of ordinary differential equations. The key tool is the spectral decomposition of linear and non-linear Boltzmann operators.
Citation: Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299
##### References:
 [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, New York, 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), 111-233. [3] C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 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), 1731-1747. 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), 133-155. 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), 47-68. [7] L. Glangetas and M. Najeme, Analytical regularizing effect for the radial homogeneous Boltzmann equation, Kinet. Relat. Models, 6 (2013), 407-427. 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, Birkhäuser Basel, 189 (2009), 15-31. 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), 647-666. 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), 832-867. 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), 625-648. 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), 797-831. 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), 459-476. 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), 139-159. 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., A special issue on Boltzmann Equations and Applications, 24 (2009), 187-212. 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., New York, 1964. [19] G. Sansone, Orthogonal Functions, Reprinted by Dover Publications 1991, Pure and Applied Mathematics, Vol. IX, Interscience Publishers, New York, 1959. [20] J. C. Slater, Quantum Theory of Atomic Structure, Vol. 1, New York, McGraw-Hill, 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), 141-156. doi: 10.1007/BF03167864. [22] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305. 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, Ann Arbor, Michigan, Reprinted in 1970 in "Studies in Statistical Mechanics," Vol. V. Edited by J. L. Lebowitz and E. Montroll, North-Holland, 1952. [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, Cambridge, 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), 1172-1190. doi: 10.1016/j.jde.2012.04.023.

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##### References:
 [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, New York, 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), 111-233. [3] C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 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), 1731-1747. 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), 133-155. 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), 47-68. [7] L. Glangetas and M. Najeme, Analytical regularizing effect for the radial homogeneous Boltzmann equation, Kinet. Relat. Models, 6 (2013), 407-427. 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, Birkhäuser Basel, 189 (2009), 15-31. 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), 647-666. 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), 832-867. 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), 625-648. 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), 797-831. 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), 459-476. 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), 139-159. 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., A special issue on Boltzmann Equations and Applications, 24 (2009), 187-212. 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., New York, 1964. [19] G. Sansone, Orthogonal Functions, Reprinted by Dover Publications 1991, Pure and Applied Mathematics, Vol. IX, Interscience Publishers, New York, 1959. [20] J. C. Slater, Quantum Theory of Atomic Structure, Vol. 1, New York, McGraw-Hill, 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), 141-156. doi: 10.1007/BF03167864. [22] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305. 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, Ann Arbor, Michigan, Reprinted in 1970 in "Studies in Statistical Mechanics," Vol. V. Edited by J. L. Lebowitz and E. Montroll, North-Holland, 1952. [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, Cambridge, 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), 1172-1190. doi: 10.1016/j.jde.2012.04.023.
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