Global regularity estimates for the Boltzmann equation without cut-off
HTML articles powered by AMS MathViewer
- by Cyril Imbert and Luis Enrique Silvestre;
- J. Amer. Math. Soc. 35 (2022), 625-703
- DOI: https://doi.org/10.1090/jams/986
- Published electronically: September 10, 2021
- HTML | PDF | Request permission
Abstract:
We derive $C^\infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to pointwise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution.References
- R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000), no. 4, 327–355. MR 1765272, DOI 10.1007/s002050000083
- Radjesvarane Alexandre and Mouhamad El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci. 15 (2005), no. 6, 907–920. MR 2149928, DOI 10.1142/S0218202505000613
- Radjesvarane Alexandre and Mouhamad Elsafadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules, Discrete Contin. Dyn. Syst. 24 (2009), no. 1, 1–11. MR 2476677, DOI 10.3934/dcds.2009.24.1
- Radjesvarane Alexandre, Yoshinore Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Regularity of solutions for the Boltzmann equation without angular cutoff, C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 747–752 (English, with English and French summaries). MR 2543976, DOI 10.1016/j.crma.2009.04.025
- Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 39–123. MR 2679369, DOI 10.1007/s00205-010-0290-1
- R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math. 55 (2002), no. 1, 30–70. MR 1857879, DOI 10.1002/cpa.10012
- Scott Armstrong and Jean-Christophe Mourrat, Variational methods for the kinetic Fokker-Planck equation, Preprint arXiv:1902.04037, 2019.
- Nathalie Ayi, Maxime Herda, Hélène Hivert, and Isabelle Tristani, A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium, C. R. Math. Acad. Sci. Paris 358 (2020), no. 3, 333–340 (English, with English and French summaries). MR 4125777, DOI 10.5802/crmath.46
- Claude Bardos, François Golse, and David Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys. 63 (1991), no. 1-2, 323–344. MR 1115587, DOI 10.1007/BF01026608
- Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597–638. MR 2494809, DOI 10.1002/cpa.20274
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- Stephen Cameron, Luis Silvestre, and Stanley Snelson, Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 3, 625–642 (English, with English and French summaries). MR 3778645, DOI 10.1016/j.anihpc.2017.07.001
- Villani Cédric, Théorème vivant, 2012.
- Jamil Chaker and Luis Silvestre, Coercivity estimates for integro-differential operators, Calc. Var. Partial Differential Equations 59 (2020), no. 4, Paper No. 106, 20. MR 4111815, DOI 10.1007/s00526-020-01764-y
- Yemin Chen and Lingbing He, Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case, Arch. Ration. Mech. Anal. 201 (2011), no. 2, 501–548. MR 2820356, DOI 10.1007/s00205-010-0393-8
- Yemin Chen and Lingbing He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 343–377. MR 2885564, DOI 10.1007/s00205-011-0482-3
- Zhen-Qing Chen and Xicheng Zhang, $L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators, J. Math. Pures Appl. (9) 116 (2018), 52–87 (English, with English and French summaries). MR 3826548, DOI 10.1016/j.matpur.2017.10.003
- L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2005), no. 2, 245–316. MR 2116276, DOI 10.1007/s00222-004-0389-9
- Laurent Desvillettes and Bernt Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations 29 (2004), no. 1-2, 133–155. MR 2038147, DOI 10.1081/PDE-120028847
- Bartłomiej Dyda and Moritz Kassmann, Comparability and regularity estimates for symmetric nonlocal dirichlet forms, arXiv preprint arXiv:1109.6812, 2011.
- Bartłomiej Dyda and Moritz Kassmann, Regularity estimates for elliptic nonlocal operators, Anal. PDE 13 (2020), no. 2, 317–370. MR 4078229, DOI 10.2140/apde.2020.13.317
- 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 (2019), no. 1, 253–295. MR 3923847
- Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc. 24 (2011), no. 3, 771–847. MR 2784329, DOI 10.1090/S0894-0347-2011-00697-8
- Philip T. Gressman and Robert M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math. 227 (2011), no. 6, 2349–2384. MR 2807092, DOI 10.1016/j.aim.2011.05.005
- Zimo Hao, Mingyan Wu, and Xicheng Zhang, Schauder estimates for nonlocal kinetic equations and applications, J. Math. Pures Appl. (9) 140 (2020), 139–184 (English, with English and French summaries). MR 4124429, DOI 10.1016/j.matpur.2020.06.003
- Christopher Henderson and Stanley Snelson, $C^\infty$ smoothing for weak solutions of the inhomogeneous Landau equation, Arch. Ration. Mech. Anal. 236 (2020), no. 1, 113–143. MR 4072211, DOI 10.1007/s00205-019-01465-7
- Christopher Henderson, Stanley Snelson, and Andrei Tarfulea, Local existence, lower mass bounds, and a new continuation criterion for the Landau equation, J. Differential Equations 266 (2019), no. 2-3, 1536–1577. MR 3906224, DOI 10.1016/j.jde.2018.08.005
- Christopher Henderson, Stanley Snelson, and Andrei Tarfulea, Local well-posedness of the Boltzmann equation with polynomially decaying initial data, Kinet. Relat. Models 13 (2020), no. 4, 837–867. MR 4112183, DOI 10.3934/krm.2020029
- Christopher Henderson, Stanley Snelson, and Andrei Tarfulea, Self-generating lower bounds and continuation for the Boltzmann equation, Calc. Var. Partial Differential Equations 59 (2020), no. 6, Paper No. 191, 13. MR 4163318, DOI 10.1007/s00526-020-01856-9
- Frédéric Hérau, Daniela Tonon, and Isabelle Tristani, Short time diffusion properties of inhomogeneous kinetic equations with fractional collision kernel, Preprint, arXiv:1709.09943, 2017.
- Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, and Tong Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models 1 (2008), no. 3, 453–489. MR 2425608, DOI 10.3934/krm.2008.1.453
- Cyril Imbert, Clément Mouhot, and Luis Silvestre, Decay estimates for large velocities in the Boltzmann equation without cutoff, J. Éc. polytech. Math. 7 (2020), 143–184 (English, with English and French summaries). MR 4033752, DOI 10.5802/jep.113
- Cyril Imbert, Clément Mouhot, and Luis Silvestre, Gaussian lower bounds for the Boltzmann equation without cutoff, SIAM J. Math. Anal. 52 (2020), no. 3, 2930–2944. MR 4112729, DOI 10.1137/19M1252375
- Cyril Imbert and Luis Silvestre, The Schauder estimate for kinetic integral equations, Anal. PDE 14 (2021), no. 1, 171–204. MR 4229202, DOI 10.2140/apde.2021.14.171
- Cyril Imbert and Luis Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 2, 507–592. MR 4049224, DOI 10.4171/jems/928
- Moritz Kassmann and Russell W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 1, 183–212. MR 3289601
- Frank Merle, Pierre Raphael, Igor Rodnianski, and Jeremie Szeftel, On smooth self similar solutions to the compressible Euler equations, arXiv:1912.10998, 2019.
- Frank Merle, Pierre Raphael, Igor Rodnianski, and Jeremie Szeftel, On the implosion of a three dimensional compressible fluid, arXiv:1912.11009, 2019.
- Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst. 24 (2009), no. 1, 187–212. MR 2476686, DOI 10.3934/dcds.2009.24.187
- Yoshinori Morimoto and Tong Yang, Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials, Anal. Appl. (Singap.) 13 (2015), no. 6, 663–683. MR 3376931, DOI 10.1142/S0219530514500079
- Lukas Niebel and Rico Zacher, Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation, arXiv, Preprint arXiv:2006.11531, 2020.
- Polidoro Sergio, Recent results on Kolmogorov equations and applications, Workshop on Second Order Subelliptic Equations and Applications, vol. 3, Graficom Edizioni, 2004, pp. 129–143.
- Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR 815196, DOI 10.1007/BF01210741
- Luis Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys. 348 (2016), no. 1, 69–100. MR 3551261, DOI 10.1007/s00220-016-2757-x
- Stanley Snelson, Gaussian bounds for the inhomogeneous Landau equation with hard potentials, SIAM J. Math. Anal. 52 (2020), no. 2, 2081–2097. MR 4091877, DOI 10.1137/19M1244275
- Logan F. Stokols, Hölder continuity for a family of nonlocal hypoelliptic kinetic equations, SIAM J. Math. Anal. 51 (2019), no. 6, 4815–4847. MR 4039522, DOI 10.1137/18M1234953
- Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. MR 1942465, DOI 10.1016/S1874-5792(02)80004-0
Bibliographic Information
- Cyril Imbert
- Affiliation: CNRS & DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France; and 45 rue d’Ulm, 75005 Paris, France
- MR Author ID: 639611
- Email: Cyril.Imbert@ens.psl.eu
- Luis Enrique Silvestre
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 757280
- Email: luis@math.uchicago.edu
- Received by editor(s): October 2, 2019
- Received by editor(s) in revised form: January 28, 2021, and February 12, 2021
- Published electronically: September 10, 2021
- Additional Notes: The second author was supported by NSF grant DMS-1764285
- © Copyright 2021 American Mathematical Society
- Journal: J. Amer. Math. Soc. 35 (2022), 625-703
- MSC (2020): Primary 35R09, 35Q20
- DOI: https://doi.org/10.1090/jams/986
- MathSciNet review: 4433077