October  2019, 12(5): 1163-1183. doi: 10.3934/krm.2019044

Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations

1. 

Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente 225, Rio de Janeiro, CEP 22451-900, Brazil

2. 

Université Clermont Auvergne, LMBP, UMR 6620 - CNRS, Campus des Cézeaux, 3, place Vasarely, TSA 60026, CS 60026, F-63178 Aubière Cedex, France

3. 

Università degli Studi di Torino & Collegio Carlo Alberto, Department ESOMAS, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy

* Corresponding author

Received  February 2019 Revised  April 2019 Published  July 2019

Fund Project: B. L gratefully acknowledges the financial support from the Italian Ministry of Education, University and Research (MIUR), "Dipartimenti di Eccellenza" grant 2018-2022.

In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in [24,26,15,23].

Citation: Ricardo J. Alonso, Véronique Bagland, Bertrand Lods. Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations. Kinetic & Related Models, 2019, 12 (5) : 1163-1183. doi: 10.3934/krm.2019044
References:
[1]

R. Alonso, V. Bagland and B. Lods, Convergence to self-similarity for ballistic annihilation dynamics, preprint, https://arXiv.org/abs/1804.06192, 2018. Google Scholar

[2]

R. AlonsoJ. A. CañizoI. M. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, 38 (2013), 155-169.  doi: 10.1080/03605302.2012.715707.  Google Scholar

[3]

R. AlonsoE. Carneiro and I. M. Gamba, Convolution inequalities for the Boltzmann collision operator, Comm. Math. Phys., 298 (2010), 293-322.  doi: 10.1007/s00220-010-1065-0.  Google Scholar

[4]

R. Alonso and I. M. Gamba, Gain of integrability for the Boltzmann collisional operator, Kinet. Relat. Models, 4 (2011), 41-51.  doi: 10.3934/krm.2011.4.41.  Google Scholar

[5]

R. Alonso, I. M. Gamba and M. Tasković, Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation, preprint, https://arXiv.org/abs/1711.06596v1, 2017. Google Scholar

[6]

R. Alonso and B. Lods, Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal., 42 (2010), 2499-2538.  doi: 10.1137/100793979.  Google Scholar

[7]

F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel, Revista Mat. Iberoam., 14 (1998), 47-61.  doi: 10.4171/RMI/233.  Google Scholar

[8]

E. A. Carlen and M. C. Carvalho, Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation, J. Stat. Phys., 67 (1992), 575-608.  doi: 10.1007/BF01049721.  Google Scholar

[9]

E. A. Carlen and M. C. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Stat. Phys., 74 (1994), 743-782.  doi: 10.1007/BF02188578.  Google Scholar

[10]

E. A. CarlenE. Gabetta and G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys., 199 (1999), 521-546.  doi: 10.1007/s002200050511.  Google Scholar

[11]

K. Carrapatoso, On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials, J. Math. Pures Appl., 104 (2015), 276-310.  doi: 10.1016/j.matpur.2015.02.008.  Google Scholar

[12]

J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations, Math. Methods Appl. Sci., 21 (1998), 1269-1286.  doi: 10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.3.CO;2-F.  Google Scholar

[13]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032.  Google Scholar

[14]

L. Desvillettes,, Entropy dissipation estimates for the Landau equation: General cross sections, in From Particle Systems to Partial Differential Equations III (eds. P. Gonçalves P., A. Soares), Springer Proceedings in Mathematics and Statistics, Springer, 162 (2016), 121–143. doi: 10.1007/978-3-319-32144-8_6.  Google Scholar

[15]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.  Google Scholar

[16]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part Ⅱ: H theorem and applications, Comm. Partial Differential Equations, 25 (2000), 261-298.  doi: 10.1080/03605300008821513.  Google Scholar

[17]

R. A. Fisher,, Theory of statistical estimation, Proc. Cambridge Philos. Soc., 22 (1925) 700–725. Google Scholar

[18]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, AMS, 2001. doi: 10.1090/gsm/014.  Google Scholar

[19]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rational Mech. Anal., 173 (2004), 169-212.  doi: 10.1007/s00205-004-0316-7.  Google Scholar

[20]

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.  Google Scholar

[21]

G. Toscani, New a priori estimates for the spatially homogeneous Boltzmann equation, Cont. Mech. Thermodyn., 4 (1992), 81-93.  doi: 10.1007/BF01125691.  Google Scholar

[22]

G. Toscani, Strong convergence in Lp for a spatially homogeneous Maxwell gas with cut-off, Transp. Theory Stat. Phys., 24 (1995), 319-328.  doi: 10.1080/00411459508205132.  Google Scholar

[23]

G. Toscani and C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a prior bounds, J. Statist. Phys., 98 (2000), 1279-1309.  doi: 10.1023/A:1018623930325.  Google Scholar

[24]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983.  doi: 10.1142/S0218202598000433.  Google Scholar

[25]

C. Villani, Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules, Math. Models Methods Appl. Sci., 10 (2000), 153-161.  doi: 10.1142/S0218202500000100.  Google Scholar

[26]

C. Villani, Fisher information estimates for Boltzmann's collision operator, J. Math. Pures Appl., 77 (1998), 821-837.  doi: 10.1016/S0021-7824(98)80010-X.  Google Scholar

[27]

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.  Google Scholar

[28]

B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066.  doi: 10.1007/BF02183613.  Google Scholar

show all references

References:
[1]

R. Alonso, V. Bagland and B. Lods, Convergence to self-similarity for ballistic annihilation dynamics, preprint, https://arXiv.org/abs/1804.06192, 2018. Google Scholar

[2]

R. AlonsoJ. A. CañizoI. M. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, 38 (2013), 155-169.  doi: 10.1080/03605302.2012.715707.  Google Scholar

[3]

R. AlonsoE. Carneiro and I. M. Gamba, Convolution inequalities for the Boltzmann collision operator, Comm. Math. Phys., 298 (2010), 293-322.  doi: 10.1007/s00220-010-1065-0.  Google Scholar

[4]

R. Alonso and I. M. Gamba, Gain of integrability for the Boltzmann collisional operator, Kinet. Relat. Models, 4 (2011), 41-51.  doi: 10.3934/krm.2011.4.41.  Google Scholar

[5]

R. Alonso, I. M. Gamba and M. Tasković, Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation, preprint, https://arXiv.org/abs/1711.06596v1, 2017. Google Scholar

[6]

R. Alonso and B. Lods, Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal., 42 (2010), 2499-2538.  doi: 10.1137/100793979.  Google Scholar

[7]

F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel, Revista Mat. Iberoam., 14 (1998), 47-61.  doi: 10.4171/RMI/233.  Google Scholar

[8]

E. A. Carlen and M. C. Carvalho, Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation, J. Stat. Phys., 67 (1992), 575-608.  doi: 10.1007/BF01049721.  Google Scholar

[9]

E. A. Carlen and M. C. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Stat. Phys., 74 (1994), 743-782.  doi: 10.1007/BF02188578.  Google Scholar

[10]

E. A. CarlenE. Gabetta and G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys., 199 (1999), 521-546.  doi: 10.1007/s002200050511.  Google Scholar

[11]

K. Carrapatoso, On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials, J. Math. Pures Appl., 104 (2015), 276-310.  doi: 10.1016/j.matpur.2015.02.008.  Google Scholar

[12]

J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations, Math. Methods Appl. Sci., 21 (1998), 1269-1286.  doi: 10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.3.CO;2-F.  Google Scholar

[13]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032.  Google Scholar

[14]

L. Desvillettes,, Entropy dissipation estimates for the Landau equation: General cross sections, in From Particle Systems to Partial Differential Equations III (eds. P. Gonçalves P., A. Soares), Springer Proceedings in Mathematics and Statistics, Springer, 162 (2016), 121–143. doi: 10.1007/978-3-319-32144-8_6.  Google Scholar

[15]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.  Google Scholar

[16]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part Ⅱ: H theorem and applications, Comm. Partial Differential Equations, 25 (2000), 261-298.  doi: 10.1080/03605300008821513.  Google Scholar

[17]

R. A. Fisher,, Theory of statistical estimation, Proc. Cambridge Philos. Soc., 22 (1925) 700–725. Google Scholar

[18]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, AMS, 2001. doi: 10.1090/gsm/014.  Google Scholar

[19]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rational Mech. Anal., 173 (2004), 169-212.  doi: 10.1007/s00205-004-0316-7.  Google Scholar

[20]

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.  Google Scholar

[21]

G. Toscani, New a priori estimates for the spatially homogeneous Boltzmann equation, Cont. Mech. Thermodyn., 4 (1992), 81-93.  doi: 10.1007/BF01125691.  Google Scholar

[22]

G. Toscani, Strong convergence in Lp for a spatially homogeneous Maxwell gas with cut-off, Transp. Theory Stat. Phys., 24 (1995), 319-328.  doi: 10.1080/00411459508205132.  Google Scholar

[23]

G. Toscani and C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a prior bounds, J. Statist. Phys., 98 (2000), 1279-1309.  doi: 10.1023/A:1018623930325.  Google Scholar

[24]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983.  doi: 10.1142/S0218202598000433.  Google Scholar

[25]

C. Villani, Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules, Math. Models Methods Appl. Sci., 10 (2000), 153-161.  doi: 10.1142/S0218202500000100.  Google Scholar

[26]

C. Villani, Fisher information estimates for Boltzmann's collision operator, J. Math. Pures Appl., 77 (1998), 821-837.  doi: 10.1016/S0021-7824(98)80010-X.  Google Scholar

[27]

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.  Google Scholar

[28]

B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066.  doi: 10.1007/BF02183613.  Google Scholar

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