-
Previous Article
Diffusion limit for a kinetic equation with a thermostatted interface
- KRM Home
- This Issue
-
Next Article
On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks
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 |
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 [
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. |
[2] |
R. Alonso, J. A. Cañizo, I. 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. |
[3] |
R. Alonso, E. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
E. A. Carlen, E. 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. |
[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. |
[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. |
[13] |
J. A. Carrillo, A. Jüngel, P. A. Markowich, G. 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. |
[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. |
[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. |
[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. |
[17] |
R. A. Fisher,, Theory of statistical estimation, Proc. Cambridge Philos. Soc., 22 (1925) 700–725. |
[18] |
E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, AMS, 2001.
doi: 10.1090/gsm/014. |
[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. |
[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. |
[21] |
G. Toscani,
New a priori estimates for the spatially homogeneous Boltzmann equation, Cont. Mech. Thermodyn., 4 (1992), 81-93.
doi: 10.1007/BF01125691. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
B. Wennberg,
Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066.
doi: 10.1007/BF02183613. |
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. |
[2] |
R. Alonso, J. A. Cañizo, I. 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. |
[3] |
R. Alonso, E. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
E. A. Carlen, E. 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. |
[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. |
[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. |
[13] |
J. A. Carrillo, A. Jüngel, P. A. Markowich, G. 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. |
[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. |
[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. |
[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. |
[17] |
R. A. Fisher,, Theory of statistical estimation, Proc. Cambridge Philos. Soc., 22 (1925) 700–725. |
[18] |
E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, AMS, 2001.
doi: 10.1090/gsm/014. |
[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. |
[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. |
[21] |
G. Toscani,
New a priori estimates for the spatially homogeneous Boltzmann equation, Cont. Mech. Thermodyn., 4 (1992), 81-93.
doi: 10.1007/BF01125691. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
B. Wennberg,
Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066.
doi: 10.1007/BF02183613. |
[1] |
Milana Pavić-Čolić, Maja Tasković. Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules. Kinetic and Related Models, 2018, 11 (3) : 597-613. doi: 10.3934/krm.2018025 |
[2] |
Immanuel Ben Porat. Local conditional regularity for the Landau equation with Coulomb potential. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022010 |
[3] |
Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic and Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1 |
[4] |
Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic and Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036 |
[5] |
Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204 |
[6] |
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 1011-1041. doi: 10.3934/krm.2013.6.1011 |
[7] |
Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13 |
[8] |
Hua Chen, Wei-Xi Li, Chao-Jiang Xu. Propagation of Gevrey regularity for solutions of Landau equations. Kinetic and Related Models, 2008, 1 (3) : 355-368. doi: 10.3934/krm.2008.1.355 |
[9] |
Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299 |
[10] |
Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic and Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453 |
[11] |
Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187 |
[12] |
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 |
[13] |
D. Blömker, S. Maier-Paape, G. Schneider. The stochastic Landau equation as an amplitude equation. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 527-541. doi: 10.3934/dcdsb.2001.1.527 |
[14] |
Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic and Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333 |
[15] |
Nicolas Rougerie. On two properties of the Fisher information. Kinetic and Related Models, 2021, 14 (1) : 77-88. doi: 10.3934/krm.2020049 |
[16] |
Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic and Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 |
[17] |
Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455 |
[18] |
Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145 |
[19] |
Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic and Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 |
[20] |
Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure and Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]