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A De Giorgi argument for $L^{\infty}$ solution to the Boltzmann equation without angular cutoff
Virtualconference
Authors : Yang, Tong (Author of the conference)
CIRM (Publisher )
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}
Additional resources :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Tong_YANG.pdf
CIRM (Publisher )
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Abstract :
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.
Keywords : Boltzmann equation; De Giorgi argument; non-angular cutoff
MSC Codes : Keywords : Boltzmann equation; De Giorgi argument; non-angular cutoff
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}
Additional resources :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Tong_YANG.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 09/04/2021 Conference Date : 23/03/2021 Subseries : Research talks arXiv category : Analysis of PDEs Mathematical Area(s) : PDE ; Mathematical Physics Format : MP4 (.mp4) - HD Video Time : 00:38:38 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-03-23_Yang.mp4 
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Information on the Event
Event Title : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et AnalyseEvent Organizers : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine Dates : 22/03/2021 - 26/03/2021 Event Year : 2021 Event URL : https://www.chairejeanmorlet.com/2355.html
Citation Data
DOI : 10.24350/CIRM.V.19735803Cite this video as: 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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Bibliography
- 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
