Entropy dissipation estimates for the Boltzmann equation without cut-off

Jamil Chaker; Luis Silvestre

doi:10.3934/krm.2023006

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2023, Volume 16, Issue 5: 748-763. Doi: 10.3934/krm.2023006

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Entropy dissipation estimates for the Boltzmann equation without cut-off

Jamil Chaker^1, , and
Luis Silvestre^2,

1.
Universität Bielefeld, Germany
2.
University of Chicago, USA

^*Corresponding author: Jamil Chaker
^*Corresponding author: Jamil Chaker

Received: August 2022

Revised: December 2022

Early access: February 2023

Published: October 2023

Jamil Chaker gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (SFB 1283/2 2021 - 317210226). Luis Silvestre is supported by NSF grants 2054888 and 1764285

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Abstract

We prove the the entropy production of the Boltzmann equation, in the non cutoff regime, is bounded from below by a weighted $ L^p $ norm of the solution. The estimate holds for a wide range of potentials including soft potentials as well as very soft potentials. We discuss applications of this estimate for weak solutions of the Boltzmann equation. In particular, we obtain that weak solutions must be belong to the space $ L^1([0,T],L^p_q( \mathbb R^d)) $ for some precise exponents $ p $ and $ q $.

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Mathematics Subject Classification: Primary: 76P05, 45P05.

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[1]	R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.
[2]	R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 55 (2002), 30-70. doi: 10.1002/cpa.10012.
[3]	R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 21 (2004), 61-95. doi: 10.1016/j.anihpc.2002.12.001.
[4]	J. Chaker and L. Silvestre, Coercivity estimates for integro-differential operators, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 106, 20 pp. doi: 10.1007/s00526-020-01764-y.
[5]	L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923.
[6]	L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation, Riv. Mat. Univ. Parma, 7 (2003), 1-99.
[7]	L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., 269 (2015), 1359-1403. doi: 10.1016/j.jfa.2015.05.009.
[8]	L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Inventiones Mathematicae, 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.
[9]	R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423.
[10]	F. Golse, M. P. Gualdani, C. Imbert and A. Vasseur, Partial regularity in time for the space homogeneous Landau equation with Coulomb potential, arXiv preprint, arXiv: 1906.02841, 2019.
[11]	T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Statist. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543.
[12]	P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Natl. Acad. Sci. USA, 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107.
[13]	P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.
[14]	P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384. doi: 10.1016/j.aim.2011.05.005.
[15]	C. Imbert and L. Silvestre, Regularity for the Boltzmann equation conditional to macroscopic bounds, EMS Surv. Math. Sci., 7 (2020), 117-172. doi: 10.4171/emss/37.
[16]	C. Imbert and L. E. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, J. Amer. Math. Soc., 35 (2022), 625-703. doi: 10.1090/jams/986.
[17]	C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348. doi: 10.1080/03605300600635004.
[18]	L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100. doi: 10.1007/s00220-016-2757-x.
[19]	C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.
[20]	C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off, Rev. Mat. Iberoamericana, 15 (1999), 335-352. doi: 10.4171/RMI/259.