Strong Harnack inequality for the Boltzmann equation
Amélie Loher1
1 DPMMS, University of Cambridge, Wilberforce road, Cambridge CB3 0WA, UK
Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 1, 15 p.

We review local regularity properties of the non-cutoff Boltzmann equation for moderately soft potentials. We explain how to view the Boltzmann equation as a non-local hypoelliptic equation. We show that despite its non-locality, we can derive a Strong Harnack inequality. To this end, we establish a linear First De Giorgi Lemma, which relates the local supremum to the local L 2 norm without non-local tail terms.

Publié le :
DOI : 10.5802/slsedp.164
Amélie Loher 1

1 DPMMS, University of Cambridge, Wilberforce road, Cambridge CB3 0WA, UK 
@article{SLSEDP_2023-2024____A1_0,
     author = {Am\'elie Loher},
     title = {Strong {Harnack} inequality for the {Boltzmann} equation},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:1},
     pages = {1--15},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2023-2024},
     doi = {10.5802/slsedp.164},
     zbl = {1543.35123},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.164/}
}
TY  - JOUR
AU  - Amélie Loher
TI  - Strong Harnack inequality for the Boltzmann equation
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:1
PY  - 2023-2024
SP  - 1
EP  - 15
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.164/
DO  - 10.5802/slsedp.164
LA  - en
ID  - SLSEDP_2023-2024____A1_0
ER  - 
%0 Journal Article
%A Amélie Loher
%T Strong Harnack inequality for the Boltzmann equation
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:1
%D 2023-2024
%P 1-15
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.164/
%R 10.5802/slsedp.164
%G en
%F SLSEDP_2023-2024____A1_0
Amélie Loher. Strong Harnack inequality for the Boltzmann equation. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 1, 15 p. doi : 10.5802/slsedp.164. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.164/

[1] Donald Gary Aronson. Bounds for the fundamental solution of a parabolic equation. Bull. . Amer. Math. Soc., 73:890–896, 1967. | DOI | MR | Zbl

[2] Richard F. Bass and David A. Levin. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc., 354(7):2933–2953, 2002. | DOI | MR | Zbl

[3] Eugene Barry Fabes and Daniel W. Stroock. A new proof of Moser’s parabolic Harnack inequality using old ideas of Nash. Arch. Rat. Mech. Anal., 96:327–338, 1986. | DOI | MR | Zbl

[4] Ennio De Giorgi. Sulla differenziabilità e l’analiticità delle estremaili degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Math. Nat., 3:25–43, 1957. | MR | Zbl

[5] François Golse, Cyril Imbert, Clément Mouhot, and Alexis F. Vasseur. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(1):253–295, 2019. | DOI | MR | Zbl

[6] Cyril Imbert, Clément Mouhot, and Luis Silvestre. Gaussian Lower Bounds for the Boltzmann Equation without Cutoff. SIAM J. Math. Anal., 52(3):2930–2944, 2020. | DOI | MR | Zbl

[7] Cyril Imbert and Luis Silvestre. The weak Harnack inequality for the Boltzmann equation without cut-off. J. Eur. Math. Soc., 22(2):507–592, 2020. | DOI | MR | Zbl

[8] Cyril Imbert and Luis Silvestre. Global regularity estimates for the Boltzmann equation without cut-off. J. Amer. Math. Soc., 2021. | DOI | MR | Zbl

[9] Cyril Imbert and Luis Silvestre. The Schauder estimate for kinetic integral equations. Anal. PDE, 14(1):171–204, 2021. | DOI | MR | Zbl

[10] Moritz Kassmann and Marvin Weidner. The parabolic Harnack inequality for nonlocal equations, 2023. | arXiv

[11] Amélie Loher. Quantitative De Giorgi methods in kinetic theory for non-local operators, 2022. | arXiv

[12] Amélie Loher. Local behaviour of non-local equations in kinetic theory, 2023. To appear.

[13] Amélie Loher. Quantitative Schauder estimates for hypoelliptic equations, 2023. | arXiv

[14] John Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math., 80:931–954, 1958. | DOI | MR | Zbl

[15] Luis Silvestre. A new regularization mechanism for the Boltzmann equation without cut-off. Comm. Math. Phys., 348(1):69–100, 2016. | DOI | MR | Zbl

[16] Cédric Villani. Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Mat. Iberoamericana, 15, 1999. | DOI | MR | Zbl

[17] Takashi Kumagai Zhen-Qing Chen. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl., 108(1):27–62, 2003. | DOI | MR | Zbl

Cité par Sources :