CIRM - Videos & books Library - A De Giorgi argument for $L^{\infty}$ solution to the Boltzmann equation without angular cutoff

Virtualconference

Auteurs : Yang, Tong (Auteur de la conférence)
CIRM (Editeur )

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Résumé : In this talk, after reviewing the work on global well-posedness of the Boltzmann equation without angular cutoff with algebraic decay tails, we will present a recent work on the global weighted $L^{\infty}$-solutions to the Boltzmann equation without angular cutoff in the regime close to equilibrium. A De Giorgi type argument, well developed for diffusion equations, is crafted in this kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable $L^{p}$ estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Then we extend local solutions to global by using the spectral gap of the linearized Boltzmann operator with the convergence to the equilibrium state obtained as a byproduct. This result fill in the gap of well-posedness theory for the Boltzmann equation without angular cutoff in the $L^{\infty}$ framework. The talk is based on the joint works with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.

Mots-Clés : Boltzmann equation; De Giorgi argument; non-angular cutoff

Codes MSC :
35Q35 - PDEs in connection with fluid mechanics
47H20 - Semigroups of nonlinear operators and nonlinear evolution equations, See also {58D07}
76P05 - Rarefied gas flows, Boltzmann equation, See also {82B40, 82C40, 82D05}

Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Tong_YANG.pdf

Informations sur la Vidéo

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2021-03-23_Yang.mp4

Informations sur la Rencontre

Nom de la Rencontre : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et Analyse
Organisateurs de la Rencontre : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine
Dates : 22/03/2021 - 26/03/2021
Année de la rencontre : 2021
URL de la Rencontre : https://www.chairejeanmorlet.com/2355.html

Données de citation

DOI : 10.24350/CIRM.V.19735803
Citer cette vidéo: Yang, Tong (2021). A De Giorgi argument for $L^{\infty}$ solution to the Boltzmann equation without angular cutoff. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19735803
URI : http://dx.doi.org/10.24350/CIRM.V.19735803

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Bibliographie

ALONSO, R., MORIMOTO, Y., SUN, W., et al. De Giorgi argument for weighted $ L^ 2\cap L^\infty $ solutions to the non-cutoff Boltzmann equation. arXiv preprint arXiv:2010.10065, 2020. - https://arxiv.org/abs/2010.10065

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