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August  2021, 14(4): 705-724. doi: 10.3934/krm.2021020

## Lower bound for the Boltzmann equation whose regularity grows tempered with time

 1 Department of Mathematical Sciences, Tsinghua University, China 2 School of Mathematical Sciences, Center for Statistical Science, Peking University, China

* Corresponding author: Jie Ji

Received  January 2021 Revised  May 2021 Published  August 2021 Early access  June 2021

Fund Project: This work is supported by NSFC under Grant NO.11771236

As a first step towards the general global-in-time stability for the Boltzmann equation with soft potentials, in the present work, we prove the quantitative lower bounds for the equation under the following two assumptions, which stem from the available energy estimates, i.e. (ⅰ). the hydrodynamic quantities (local mass, local energy, and local entropy density) are bounded (from below or from above) uniformly in time, (ⅱ). the Sobolev regularity for the solution grows tempered with time.

Citation: Ling-Bing He, Jie Ji, Ling-Xuan Shao. Lower bound for the Boltzmann equation whose regularity grows tempered with time. Kinetic and Related Models, 2021, 14 (4) : 705-724. doi: 10.3934/krm.2021020
##### References:
 [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] M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.  doi: 10.1007/s00205-015-0874-x. [3] M. Briant, Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, 8 (2015), 281-308.  doi: 10.3934/krm.2015.8.281. [4] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Arch. Math., 60 (1933), 91-146.  doi: 10.1007/BF02398270. [5] T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler., 2. Uppsala: Almqvist & Wiksell, 1957. [6] E. A. Carlen and M. C. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Statist. Phys., 74 (1994), 743-782.  doi: 10.1007/BF02188578. [7] E. A. Carlen, M. C. Carvalho and X. Lu, On strong convergence to equilibrium for the Boltzmann equation with soft potentials, J. Stat. Phys, 135 (2009), 681–736. doi: 10.1007/s10955-009-9741-1. [8] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9. [9] L.-B. He and J. Ji, Uniqueness and global dynamics of spatially homogeneous non cutoff Boltzmann equation with moderate soft potentials, preprint. [10] L.-B. He, J.-C. Jiang and Y.-L. Zhou, On the cutoff approximation for the Boltzmann equation with long-range interaction, J. Stat. Phys., 181 (2020), 1817-1905.  doi: 10.1007/s10955-020-02646-5. [11] C. Henderson, S. Snelson and A. Tarfulea, Self-generating lower bounds and continuation for the Boltzmann equation, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 191, 13 pp. doi: 10.1007/s00526-020-01856-9. [12] C. Imbert, C. Mouhot and L. Silvestre, Gaussian lower bounds for the Boltzmann equation without cutoff, SIAM J. Math. Anal., 52 (2020), 2930-2944.  doi: 10.1137/19M1252375. [13] C. Mouhot, Quantitative lower bound for the full Boltzmann equation, part Ⅰ: Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917.  doi: 10.1081/PDE-200059299. [14] A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys., 183 (1997), 145-160.  doi: 10.1007/BF02509799. [15] G. Toscani and C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds, J. Statist. Phys., 98 (2000), 1279-1309.  doi: 10.1023/A:1018623930325. [16] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.

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##### References:
 [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] M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.  doi: 10.1007/s00205-015-0874-x. [3] M. Briant, Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, 8 (2015), 281-308.  doi: 10.3934/krm.2015.8.281. [4] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Arch. Math., 60 (1933), 91-146.  doi: 10.1007/BF02398270. [5] T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler., 2. Uppsala: Almqvist & Wiksell, 1957. [6] E. A. Carlen and M. C. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Statist. Phys., 74 (1994), 743-782.  doi: 10.1007/BF02188578. [7] E. A. Carlen, M. C. Carvalho and X. Lu, On strong convergence to equilibrium for the Boltzmann equation with soft potentials, J. Stat. Phys, 135 (2009), 681–736. doi: 10.1007/s10955-009-9741-1. [8] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9. [9] L.-B. He and J. Ji, Uniqueness and global dynamics of spatially homogeneous non cutoff Boltzmann equation with moderate soft potentials, preprint. [10] L.-B. He, J.-C. Jiang and Y.-L. Zhou, On the cutoff approximation for the Boltzmann equation with long-range interaction, J. Stat. Phys., 181 (2020), 1817-1905.  doi: 10.1007/s10955-020-02646-5. [11] C. Henderson, S. Snelson and A. Tarfulea, Self-generating lower bounds and continuation for the Boltzmann equation, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 191, 13 pp. doi: 10.1007/s00526-020-01856-9. [12] C. Imbert, C. Mouhot and L. Silvestre, Gaussian lower bounds for the Boltzmann equation without cutoff, SIAM J. Math. Anal., 52 (2020), 2930-2944.  doi: 10.1137/19M1252375. [13] C. Mouhot, Quantitative lower bound for the full Boltzmann equation, part Ⅰ: Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917.  doi: 10.1081/PDE-200059299. [14] A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys., 183 (1997), 145-160.  doi: 10.1007/BF02509799. [15] G. Toscani and C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds, J. Statist. Phys., 98 (2000), 1279-1309.  doi: 10.1023/A:1018623930325. [16] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.
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