CAUCHY PROBLEM FOR LINEARIZED NON-CUTOFF BOLTZMANN EQUATION WITH DISTRIBUTION INITIAL DATUM

doi:10.1016/S0252-9602(15)60015-7

Abstract

Abstract:

In this article, we study the Cauchy problem for the linearized spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. By using the spectral decomposition, we solve the Cauchy problem with initial datum in the sense of distribution, which contains the dual space of a Gelfand-Shilov class. We also prove that this solution belongs to the Gelfand-Shilov space for any positive time.

Key words: Boltzmann equation, spectral decomposition, Gelfand-Shilov class, distribution initial datum

CLC Number:

35Q20

Haoguang LI. CAUCHY PROBLEM FOR LINEARIZED NON-CUTOFF BOLTZMANN EQUATION WITH DISTRIBUTION INITIAL DATUM[J].Acta mathematica scientia,Series B, 2015, 35(2): 459-476.

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References

[1] 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

[2] Cannone M, Karch G. Infinite energy solutions to the homogeneous Boltzmann equation. Communications on Pure and Applied Mathematics, 2010, 63(6): 747-778

[3] Cercignani C. The Boltzmann Equation and its Applications, Applied Mathematical Sciences, Vol 67. New York: Springer-Verlag, 1988

[4] Desvillettes L. About the regularizing properties of the non-cut-off Kac equation. Communications in Mathematical Physics, 1995, 168(2): 417-440

[5] Desvillettes L, Furioli G, Terraneo E. Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules. Transactions of the American Mathematical Society, 2009, 361(4): 1731-1747

[6] Desvillettes L, Wennberg B. Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm Partial Differential Equations, 2004, 29(1/2): 133-155

[7] Dolera E. On the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules. Boll UMI, 2010, 46: 67-105

[8] Erdélyi A, Magnus W, Oberhettinger F, et al. Higher Trascendental Functions. New York: McGrawHill, 1953

[9] Glangetas L, Najeme M. Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation. Kinet Relat Models, 2013, 6(2): 407-427

[10] Gramchev T, Pilipovi? S, Rodino L. Classes of degenerate elliptic operators in Gelfand-Shilov spaces. New Developments in Pseudo-Differential Operators. Birkhäuser Basel, 2009: 15-31

[11] Lekrine N, Xu C J. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation. Kinet Relat Models, 2009, 2(4): 647-444

[12] Lerner N, Morimoto Y, Pravda-Starov K, et al. Spectral and phase space analysis of the linearized non-cutoff Kac collision operator. Journal de Mathématiques Pures et Appliquées, 2013, 100(6): 832-867

[13] Lerner N, Morimoto Y, Pravda-Starov K, et al. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinet Relat Models, 2013, 6(3): 625-648

[14] Lerner N, Morimoto Y, Pravda-Starov K, et al. Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff. Journal of Differential Equations, 2014, 256(2): 797-831

[15] Lu Xuguang, Mouhot C. On measure solutions of the Boltzmann ]equation, part I: moment production and stability estimates. Journal of Differential Equations, 2012, 252(4): 3305-3363

[16] Morimoto Y, Yang T. Smoothing effect of the homogeneous Boltzmann equation with measure initial datum. Annales de l'Institut Henri Poincare Non Linear Analysis. Elsevier Masson, 2014. http://dx.doi.org/10.1016/j.anibpc.2013.12.004

[17] Morimoto Y, Ukai S. Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff. Journal of Pseudo-Differential Operators and Applications, 2010, 1(1): 139-159

[18] Morimoto Y, Ukai S, Xu C J, et al. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete and Continuous Dynamical Systems-Series A, 2009, 24(1): 187-212

[19] Reed, M C, Barry S. Methods of Modern Mathematical Physics: Functional Analysis. 1. Vol 1. Access Online via Elsevier, 1980

[20] Sansone G. Orthogonal Functions. Pure and Applied Mathematics. Vol. IX. New York: Interscience Publishers, 1959; Reprinted by Dover Publications, 1991

[21] Taylor M E. Partial differential equations, II: Qualitative studies of linear equations, 2nd Ed. Applied Mathematical Sciences 116. New York: Springer, 2011

[22] Ukai S. Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan Journal of Applied Mathematics, 1984, 1(1): 141-156

[23] Chang W, Uhlenbeck G E. On the propagation of sound in monoatomic gases. Ann Arbor, Michigan: University of Michigan, 1952

[24] Zhang T F, Yin Z. Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff. Journal of Differential Equations, 2012, 253(4): 1172-1190