Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum

Morimoto, Yoshinori; Yang, Tong

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 429-442.

Nous justifions l'effet régularisant pour les solutions à valeurs mesures de l'équation de Boltzmann spatialement homogène dans le cas des molécules maxwelliennes. Il s'agit de la première preuve rigoureuse de l'effet régularisant pour toutes données initiales à valeurs mesures sauf la masse de Dirac seule, ce qui donne la description optimale de la regularité des solutions en temps positif à causée par la singularité dans le noyau de collision. Le principal ingrédient nouveau dans la preuve est l'introduction d'unc inégalité de coercivité dégénérée par rapport au temps en utilisant l'analyse microlocale.

We justify the smoothing effect for measure valued solutions to the space homogeneous Boltzmann equation of Maxwellian type cross sections. This is the first rigorous proof of the smoothing effect for any measure valued initial data except the single Dirac mass, which gives the optimal description on the regularity of solutions for positive time, caused by the singularity in the cross section. The main new ingredient in the proof is the introduction of a time degenerate coercivity estimate by using the microlocal analysis.

• We discuss the Boltzmann equation with the angular singularity in the cross section. • The spatially homogeneous Cauchy problem is considered in the Maxwellian case. • The smoothing effect occurs for measures initial data except for a single Dirac mass. • The coercivity of the collision operator is derived from the angular singularity. • The time degenerate coercivity is available for the initial datum, two Dirac masses.

Classification : 35Q20, 76P05, 35H20, 82B40, 82C40
Mots clés : Boltzmann equation, Smoothing effect, Measure initial datum, Coercivity estimate

[1] R. Alexandre, L. Desvillettes, C. Villani, B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000), 327 -355 | Zbl

[2] R. Alexandre, M. Elsafadi, Littlewood Paley decomposition and regularity issues in Boltzmann homogeneous equations. I. Non cutoff and Maxwell cases, Math. Models Methods Appl. Sci. 15 (2005), 907 -920

[3] R. Alexandre, M. Elsafadi, Littlewood–Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules, Discrete Contin. Dyn. Syst. 24 (2009), 1 -11 | Zbl

[4] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math. 52 (2012), 433 -463 | Zbl

[5] L. Arkeryd, On the Boltzmann equation, Arch. Ration. Mech. Anal. 34 (1972), 1 -34

[6] M. Cannone, G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation, Commun. Pure Appl. Math. 63 (2010), 747 -778 | Zbl

[7] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1933), 91 -146 | JFM

[8] Y. Chen, L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case, Arch. Ration. Mech. Anal. 201 (2011), 501 -548 | Zbl

[9] L. Desvillettes, B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Commun. Partial Differ. Equ. 29 (2004), 133 -155 | Zbl

[10] Z.H. Huo, Y. Morimoto, S. Ukai, T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without Angular cutoff, Kinet. Relat. Models 1 (2008), 453 -489 | Zbl

[11] N. Jacob, Pseudo-Differential Operators and Markov Processes. Vol 1: Fourier Analysis and Semigroups, Imperial College Press, London (2001) | Zbl

[12] X. Lu, C. Mouhot, On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates, J. Differ. Equ. 252 (2012), 3305 -3363 | Zbl

[13] X. Lu, B. Wennberg, Solutions with increasing energy for the spatially homogeneous Boltzmann equation, Nonlinear Anal., Real World Appl. 3 (2002), 243 -258 | Zbl

[14] Y. Morimoto, A remark on Cannone–Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules, Kinet. Relat. Models 5 (2012), 551 -561 | Zbl

[15] Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst., Ser. A 24 (2009), 187 -212 | Zbl

[16] Y. Morimoto, S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseudo-Differ. Oper. Appl. 1 (2010), 139 -159 | Zbl

[17] A. Pulvirenti, G. Toscani, The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation, Ann. Mat. Pura Appl. 171 (1996), 181 -204 | Zbl

[18] H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrscheinlichkeitstheor. Verw. Geb. 46 (1978), 67 -105 | Zbl

[19] G. Toscani, C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas, J. Stat. Phys. 94 (1999), 619 -637 | Zbl

[20] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal. 143 (1998), 273 -307 | Zbl

[21] C. Villani, A review of mathematical topics in collisional kinetic theory, S. Friedlander, D. Serre (ed.), Handbook of Fluid Mathematical Fluid Dynamics, Elsevier Science (2002) | Zbl

[22] C. Villani, August 2008, Kyoto, private communication.