Sharp bounds for Boltzmann and Landau collision operators

He, Ling-Bing

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1253-1341.

L'objectif de ce travail est de fournir une méthode robuste pour obtenir des estimations précises pour les opérateurs de Boltzmann et de Landau dans des espaces de Sobolev à poids et des espaces anisotropes. Les résultats et leur démonstration font ressortir les innovations suivantes :

• Toutes les estimations précises concernent les opérateurs originaux de Boltzmann et de Landau. Le mot `précis' se réfère au fait que les estimations sont cohérentes avec le comportement des opérateurs linéarisés correspondants. Ceci est utile pour étudier le caractère bien posé des équations originales.
• En accord avec la formule de Bobylev, on introduit deux types de décomposition dyadique, dans l'espace des phases et dans celui des fréquences, afin d'utiliser au maximum les annulations. Cela nous permet de voir clairement quelle partie de l'opérateur se comporte comme un opérateur de type Laplacien, et quelle partie est dominée par la structure anisotrope.
• En se basant sur la structure géométrique des collisions élastiques, on fait une décomposition géométrique pour capturer la structure anisotrope de l'opérateur de collision. Plus précisément, on explicite le fait que l'opérateur de Laplace-Beltrami apparaît bien dans l'opérateur de collision. Cela nous permet d'obtenir des estimations précises dans des espaces anisotropes et de finaliser les estimations sur la dissipation d'entropie.
• Les structures mentionnées ci-dessus sont si robustes qu'on peut les retrouver dans la limite des collisions rasantes. On obtient ainsi des estimations précises pour le noyau de collision de Landau en passant à la limite. On remarque que la présente analyse éclaire le passage à la limite de l'équation de Boltzmann vers celle de Landau.

The aim of the work is to provide a stable method to get sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. The results and proofs have the following main features and innovations:

• All the sharp bounds are given for the original Boltzmann and Landau operators. The sharpness means the lower and upper bounds for the operators are consistent with the behavior of the linearized operators. Moreover, we make clear the difference between the bounds for the original operators and those for the linearized ones. It will be useful for the well-posedness of the original equations.
• According to the Bobylev's formula, we introduce two types of dyadic decompositions performed in both phase and frequency spaces to make full use of the interaction and the cancelation. It allows us to see clearly which part of the operator behaves like a Laplace type operator and which part is dominated by the anisotropic structure. It is the key point to get the sharp bounds in weighted Sobolev spaces and in anisotropic spaces.
• Based on the geometric structure of the elastic collision, we make a geometric decomposition to capture the anisotropic structure of the collision operator. More precisely, we make it explicit that the fractional Laplace-Beltrami operator really exists in the structure of the collision operator. It enables us to derive the sharp bounds in anisotropic spaces and then complete the entropy dissipation estimates.
• The structures mentioned above are so stable that we can apply them to the rescaled Boltzmann collision operator in the process of the grazing collisions limit. Then we get the sharp bounds for the Landau collision operator by passing to the limit. We remark that our analysis used here will shed light on the investigation of the asymptotics from Boltzmann equation to Landau equation.

Classification : 35Q20, 35A23, 35Q62.
Keywords: Boltzmann and Landau equations, anisotropic structure, grazing collisions limit.
Mot clés : Équations de Boltzmann et de Landau, structure anisotrope, limite des collisions rasantes.

Alexandre, R.; Desvillettes, L.; Villani, C.; Wennberg, B. Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., Volume 152 (2000), pp. 327-355 (ISSN: 0003-9527) | DOI | MR | Zbl

Alexandre, R.; Herau, F.; Li, W.-X. Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff (preprint arXiv:1212.4632 ) | MR

Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C.-J.; Yang, T. Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., Volume 198 (2010), pp. 39-123 (ISSN: 0003-9527) | DOI | MR | Zbl

Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C.-J.; Yang, T. The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential, Anal. Appl. (Singap.), Volume 9 (2011), pp. 113-134 (ISSN: 0219-5305) | DOI | MR | Zbl

Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C.-J.; Yang, T. The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., Volume 262 (2012), pp. 915-1010 (ISSN: 0022-1236) | DOI | MR | Zbl

Alexandre, R.; Villani, C. On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., Volume 55 (2002), pp. 30-70 (ISSN: 0010-3640) | DOI | MR | Zbl

Chen, Y.; He, L. Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case, Arch. Ration. Mech. Anal., Volume 201 (2011), pp. 501-548 (ISSN: 0003-9527) | DOI | MR | Zbl

Gressman, P. T.; Strain, R. M. Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., Volume 24 (2011), pp. 771-847 (ISSN: 0894-0347) | DOI | MR | Zbl

Gressman, P. T.; Strain, R. M. Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., Volume 227 (2011), pp. 2349-2384 (ISSN: 0001-8708) | DOI | MR | Zbl

He, L. Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction, Comm. Math. Phys., Volume 312 (2012), pp. 447-476 (ISSN: 0010-3616) | DOI | MR | Zbl

He, L. Asymptotic analysis of the spatially homogeneous Boltzmann equation: grazing collisions limit, J. Stat. Phys., Volume 155 (2014), pp. 151-210 (ISSN: 0022-4715) | DOI | MR | Zbl

He, L.; Jiang, J.-C. Well-posedness of the inhomogeneous Boltzmann equation with Hard potentials and Maxwellian molecules (in preparation)

Huo, Z.; Morimoto, Y.; Ukai, S.; Yang, T. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, Volume 1 (2008), pp. 453-489 (ISSN: 1937-5093) | DOI | MR | Zbl

Jiang, J. C.; Liu, T. P. Boltzmann collision operator for the infinite range potential (personal communication)

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, Volume 6 (2013), pp. 625-648 (ISSN: 1937-5093) | DOI | MR | Zbl

Lunardi, A., Appunti. Scuola Normale Superiore di Pisa, Edizioni della Normale, Pisa, 2009 | MR | Zbl

Morimoto, Y.; Sakamoto, S. Global solutions in the critical Besov space for the non-cutoff Boltzmann equation, J. Differential Equations, Volume 261 (2016), pp. 4073-4134 (ISSN: 0022-0396) | DOI | MR | Zbl

Villani, C. Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., Volume 234 (2003), pp. 455-490 (ISSN: 0010-3616) | DOI | MR | Zbl

Villani, C. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., Volume 143 (1998), pp. 273-307 (ISSN: 0003-9527) | DOI | MR | Zbl

Villani, C. On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., Volume 8 (1998), pp. 957-983 (ISSN: 0218-2025) | DOI | MR | Zbl

Wang Chang, C. S.; Uhlenbeck, G. E., Univ. of Michigan Press, 1952 (reprinted in 1970 in Studies in Statistical Mechanics. Vol. V)