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

doi:10.1007/s10473-019-0602-y

Abstract

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.

Key words: Boltzmann equation, shubin regularity, spectral decomposition, Debye-Yukawa potential

CLC Number:

35Q20

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.

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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