SHUBIN REGULARITY FOR THE RADIALLY SYMMETRIC SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH DEBYE-YUKAWA POTENTIAL

  • Léo GLANGETAS ,
  • Haoguang LI
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  • 1. Université de Rouen, CNRS UMR 6085, Mathématiques 76801 Saint-Etienne du Rouvray, France;
    2. School of Mathematics and Statistics, South-central University for Nationalities, Wuhan 430074, China

Received date: 2018-08-30

  Revised date: 2019-05-09

  Online published: 2019-12-30

Supported by

The research of the second author was supported by the Natural Science Foundation of China (11701578).

Abstract

In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.

Cite this article

Léo GLANGETAS , Haoguang LI . SHUBIN REGULARITY FOR THE RADIALLY SYMMETRIC SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH DEBYE-YUKAWA POTENTIAL[J]. Acta mathematica scientia, Series B, 2019 , 39(6) : 1487 -1507 . DOI: 10.1007/s10473-019-0602-y

References

[1] Barbaroux J -M, Hundertmark D, Ried T, Vugalter S. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinet Relat Models, 2017, 10:901-924
[2] Bobylev A V. The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Soviet Sci Rev Sect C Math Phys, 1988, 7:111-233
[3] Cercignani C. The Boltzmann Equation and its Applications. Applied Mathematical Sciences, Vol 67. New York:Springer-Verlag, 1988
[4] Chen H, Li W -X, Xu C -J. Analytic smoothness effect of solutions for spatially homogeneous Landau equation. J Differential Equations, 2009, 248:77-94
[5] Chen Y, Desvillettes L, He L. Smoothing effects for classical solutions of the full Landau equation. Arch Ration Mech Anal, 2009, 193:21-55
[6] Desvillettes L, Furioli G, Terraneo E. Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules. Trans Amer Math Soc, 2009, 361:1731-1747
[7] Desvillettes L, Wennberg B. Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm Partial Differ Equ, 2004, 29:133-155
[8] Dolera E. On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules. Boll Unione Mat Ital, 2011, 4:47-68
[9] Glangetas L, Najeme M. Analytical regularizing effect for the radial homogeneous Boltzmann equation. Kinet Relat Models, 2013, 6:407-427
[10] Glangetas L, Li H -G. Sharp regularity and Cauchy problem of the spatially homogeneous Boltzmann equation with Debye-Yukawa potential. J Math Anal Appl, 2016, 444:1438-1461
[11] Glangetas L, Li H -G, Xu C -J. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinet Relat Models, 2016, 9:299-371
[12] Gramchev T, Pilipović S, Rodino L. Classes of degenerate elliptic operators in Gelfand-Shilov spaces//New Developments in Pseudo-Differential Operators. Basel:Birkhäuser, 2009:15-31
[13] Jones M N. Spherical Harmonics and Tensors for Classical Field Theory. UK:Research Studies Press, 1985
[14] Lekrine N, Xu C -J. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac equation. Kinet Relat Models, 2009, 2:647-666
[15] Lerner N, Morimoto Y, Pravda-Starov K, Xu C -J. Spectral and phase space analysis of the linearized non-cutoff Kac collision operator. J Math Pures Appl, 2013, 100:832-867
[16] Lerner N, Morimoto Y, Pravda-Starov K, Xu C -J. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinet Relat Models, 2013, 6:625-648
[17] Lerner N, Morimoto Y, Pravda-Starov K, Xu C -J. Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff. J Differential Equations, 2014, 256:797-831
[18] Li H -G. Cauchy problem for linearized non-cutoff Boltzmann equation with distribution initial datum. Acta Mathematica Scientia, 2015, 35B:459-476
[19] Liu S, Ma X. Regularizing effects for the classical solutions to the Landau equation in the whole space. J Math Anal Appl, 2014, 417:123-143
[20] Morimoto Y. Hypoellipticity for infinitely degenerate elliptic operators. Osaka J Math, 1987, 24:13-35
[21] Morimoto Y. A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinet Relat Models, 2012, 5:551-561
[22] Morimoto Y, Pravda-Starov K, Xu C -J. A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation. Kinet Relat Models, 2013, 6; 715-727
[23] Morimoto Y, Ukai S. Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff. J Pseudo-Differ Oper Appl, 2010, 1:139-159
[24] Morimoto Y, Ukai S, Xu C -J, Yang T. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Disc Contin Dyn Syst, 2009, 24:187-212
[25] Morimoto Y, Xu C -J. Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators. Astérisque, 2003, 284:245-264
[26] Morimoto Y, Xu C -J. Nonlinear hypoellipticity of infinite type. Funkcial Ekvac, 2007, 50:33-65
[27] Slater J C. Quantum Theory of Atomic Structure. Vol 1. New York:McGraw-Hill, 1960
[28] Sansone G. Orthogonal Functions. Pure and Applied Mathematics, Vol IX. New York:Interscience Publishers, 1959, Reprinted by Dover Publications, 1991
[29] Shubin M. Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics. Berlin:Springer-Verlag, 1987
[30] Ukai S. Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan J Appl Math, 1984, 1:141-156
[31] Villani C. A review of mathematical topics in collisional kinetic theory. Handbook of Mathematical Fluid Dynamics, 2002, 1:71-305
[32] Wang-Chang C S, Uhlenbeck G E. On the propagation of sound in monoatomic gases. Univ of Michigan Press. Ann Arbor, Michigan. Reprinted in Studies in Statistical Mechanics. Vol V. North-Holland, 1970:43-75
[33] Zhang T -F, Yin Z. Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff. J Differential Equations, 2012, 253(4):1172-1190
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