THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL

doi:10.1007/s10473-024-0205-0

Abstract

Abstract: In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary $L^2$ weighted estimates.

Key words: Boltzmann equation, Gevrey regularity, non-cutoff, hard potential

CLC Number:

35B65

Lvqiao LIU, Juan ZENG. THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL[J].Acta mathematica scientia,Series B, 2024, 44(2): 455-473.

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References

[1] Alexandre R, Desvillettes L, Villani C, Wennberg B. Entropy dissipation and long-range interactions. Arch Ration Mech Anal, 2000, 152: 327-355
[2] Alexandre R, Herau F, Li W X. Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff. J Math Pures Appl, 2019, 126(9): 1-71
[3] Alexandre R, Morimoto Y, Ukai S, et al. Regularizing effect and local existence for the non-cutoff Boltzmann equation. Arch Ration Mech Anal, 2010, 198: 39-123
[4] Alexandre R, Morimoto Y, Ukai S, et al. The Boltzmann equation without angular cutoff in the whole space: qualitative properties of solutions. Arch Ration Mech Anal, 2011, 202: 599-661
[5] Alexandre R, Morimoto Y, Ukai S, et al. The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J Funct Anal, 2012, 262: 915-1010
[6] Alexandre R, Morimoto Y, Ukai S, et al. Local existence with mild regularity for the Boltzmann equation. Kinet Relat Models, 2013, 6: 1011-1041
[7] Alexandre R, Safadi M. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations I: Non-cutoff case and Maxwellian molecules. Math Models Methods Appl Sci, 2005, 15: 907-920
[8] Alexandre R, Safadi M. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II: Non cutoff case and non Maxwellian molecules. Discrete Contin Dyn Syst, 2009, 24: 1-11
[9] Barbaroux J M, Hundertmark D, Ried T, Vugalter S. Gevrey smoothing for weak solutions of the fully nonlinear homogeneous boltzmann and kac equations without cutoff for maxwellian molecules. Arch Ration Mech Anal, 2017, 225: 601-661
[10] Cao H, Li W X, Xu C J. Analytic smoothing effect of the spatially inhomogeneous Landauequations for hard potentials. J Math Pures Appl, 2023, 176: 138-182
[11] Chen H, Hu X, Li W X, Zhan J. The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off. Sci China Math, 2022, 65: 443-470
[12] Chen H, Li W X, Xu C J. Analytic smoothness effect of solutions for spatially homogeneous Landau equation. J Differential Equations, 2010, 248: 77-94
[13] Chen Y, He L. Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case. Arch Ration Mech Anal, 2011, 201: 501-548
[14] Desvillettes L. About the regularizing properties of the non-cut-off Kac equation. Comm Math Phys, 1995, 168: 417-440
[15] Desvillettes L. Regularization properties of the $2$-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules. Transport Theory Statist Phys, 1997, 26: 341-357
[16] Desvillettes L, Wennberg B. Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm Partial Differential Equations, 2004, 29: 133-155
[17] Duan W X, Li W X, Liu L. Gevrey regularity of mild solutions to the non-cutoff boltzmann equation. Advances in Mathematics, 2022, 395: 1-75
[18] Duan R, Liu S, Sakamoto S, Strain R M. Global mild solutions of the Landau and non-cutoff Boltzmann equations. Comm Pure Appl Math, 2021, 74: 932-1020
[19] Gressman P T, Strain R M. Global classical solutions of the Boltzmann equation without angular cut-off. J Amer Math Soc, 2011, 24: 771-847
[20] Gressman P T, Strain R M. Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production. Adv Math, 2011, 227: 2349-2384
[21] Hormander L.The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators.Berlin: Springer-Verlag, 1985
[22] Huo Z, Morimoto Y, Ukai S, Yang T. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinet Relat Models, 2008, 1: 453-489
[23] Lerner N. The Wick calculus of pseudo-differential operators and some of its applications. Cubo Mat Educ, 2003, 5: 213-236
[24] Lerner N.Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators. Basel: Birkhauser, 2010
[25] Lerner N, Morimoto Y, Pravda-Starov P, 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
[26] Li H G, Xu C J. The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect. J Differential Equations, 2017, 263: 5120-5150
[27] Lin S Y. Gevrey regularity for the noncutoff nonlinear homogeneous Boltzmann equation with strong singularity. Abstr Appl Anal, 2014, 2014: Art 584169
[28] Lions P L. Regularite et compacite pour des noyaux de collision de Boltzmann sans troncature angulaire. C R Acad Sci Paris Ser I Math, 1998, 326: 37-41
[29] 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
[30] Morimoto Y, Ukai S, Xu C J, Yang T. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete Contin Dyn Syst, 2009, 24: 187-212
[31] Morimoto Y, Xu C J. Ultra-analytic effect of Cauchy problem for a class of kinetic equations. J Differential Equations, 2009, 247: 596-617
[32] Morimoto Y, Xu C J. Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules.Kinet Relat Models, 2020, 13: 951-978
[33] Morimoto Y, Yang T. Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum. Ann Inst H Poincare Anal Non Lineaire, 2015, 32: 429-442
[34] Mouhot C. Explicit coercivity estimates for the Boltzmann and Landau operators. Comm Partial Differential Equations, 2006, 31: 1321-1348
[35] Mouhot C, Strain R. Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J Math Pures Appl, 2007, 87: 515-535
[36] Shubin M A.Pseudodifferential Operators and Spectral Theory. Berlin: Springer-Verlag, 1987
[37] Ukai S. Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan J Appl Math, 1984, 1: 141-156
[38] Villani C. Regularity estimates via the entropy dissipation for the spatially homogeneous boltzmann equation without cut-off. Rev Mat Iberoam, 1999, 15: 335-352