About the Landau-Fermi-Dirac equation with moderately soft potentials. (English) Zbl 1489.35278
Summary: We present some essential properties of solutions to the homogeneous Landau-Fermi-Dirac equation for moderately soft potentials. Uniform in time estimates for statistical moments, \(L^p\)-norm generation and Sobolev regularity are shown using a combination of techniques that include recent developments concerning level set analysis in the spirit of E. De Giorgi [Mem. Accad. Sci. Torino, P. I., III. Ser. 3, 25–43 (1957; Zbl 0084.31901)] and refined entropy-entropy dissipation functional inequalities for the Landau collision operator which are extended to the case in question here. As a consequence of the analysis, we prove algebraic relaxation of non degenerate distributions towards the Fermi-Dirac statistics under a weak non saturation condition for the initial datum. All quantitative estimates are uniform with respect to the quantum parameter. They therefore also hold for the classical limit, that is, the Landau equation.
MSC:
| 35Q82 | PDEs in connection with statistical mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 81V74 | Fermionic systems in quantum theory |
Citations:
Zbl 0084.31901References:
| [1] | Chapman, S.; Cowling, TG, The mathematical theory of non-uniform gases (1970), Cambridge: Cambridge University Press, Cambridge · Zbl 0063.00782 |
| [2] | Alonso, R.; Bagland, V.; Lods, B., Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials, J. Differ. Equ., 270, 596-663 (2021) · Zbl 1451.35153 · doi:10.1016/j.jde.2020.08.010 |
| [3] | Bobylev, AV; Pulvirenti, M.; Saffirio, C., From particle systems to the Landau equation: a consistency result, Comm. Math. Phys., 319, 683-702 (2013) · Zbl 1273.35281 · doi:10.1007/s00220-012-1633-6 |
| [4] | Alexandre, R.; Villani, C., On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 61-95 (2004) · Zbl 1044.83007 · doi:10.1016/j.anihpc.2002.12.001 |
| [5] | Desvillettes, L., On asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theor. Stat. Phys., 21, 259-276 (1992) · Zbl 0769.76059 · doi:10.1080/00411459208203923 |
| [6] | Villani, C., A review of mathematical topics in collisional kinetic theory, handbook of mathematical fluid dynamics, 71-305 (2002), Amsterdam: North-Holland, Amsterdam · Zbl 1170.82369 |
| [7] | Desvillettes, L.; Villani, C., On the spatially homogeneous Landau equation for hard potentials. Part I : existence, uniqueness and smoothness, Commun. Partial Differ. Equ., 25, 179-259 (2000) · Zbl 0946.35109 · doi:10.1080/03605300008821512 |
| [8] | Wu, K-C, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, J. Funct. Anal., 266, 3134-3155 (2014) · Zbl 1296.35112 · doi:10.1016/j.jfa.2013.11.005 |
| [9] | Desvillettes, L., Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., 269, 1359-1403 (2015) · Zbl 1325.35223 · doi:10.1016/j.jfa.2015.05.009 |
| [10] | Golding, W., Gualdani, M.P., Zamponi, N.: Existence of smooth solutions to the Landau-Fermi-Dirac equation with Coulomb potential, preprint, arxiv: 2107.10463, 2021 |
| [11] | Lu, X., On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles, J. Stat. Phys., 105, 353-388 (2001) · Zbl 1156.82380 · doi:10.1023/A:1012282516668 |
| [12] | Bagland, V., Well-posedness for the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, Proc. R. Soc. Edinb. Sect. A, 66, 415-447 (2004) · Zbl 1060.35115 · doi:10.1017/S0308210500003280 |
| [13] | Gualdani, M-P; Guillen, N., On \(A_p\) weights and the Landau equation, Calc. Var. Partial Differ. Equ., 58, 55 (2019) · Zbl 1404.35078 |
| [14] | Beckner, W., Pitt’s inequality with sharp convolution estimates, Proc. Am. Math. Soc., 136, 1871-1885 (2008) · Zbl 1221.42016 · doi:10.1090/S0002-9939-07-09216-7 |
| [15] | Alonso, R.; Bagland, V.; Desvillettes, L.; Lods, B., About the use of Entropy dissipation for the Landau-Fermi-Dirac equation, J. Stat. Phys., 183, 1-27 (2021) · Zbl 1466.76054 · doi:10.1007/s10955-021-02751-z |
| [16] | De Giorgi, E., Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3, 25-43 (1957) · Zbl 0084.31901 |
| [17] | Caffarelli, L.; Chan, CH; Vasseur, AF, Regularity theory for parabolic nonlinear integral operators, J. Am. Math. Soc., 24, 849-869 (2011) · Zbl 1223.35098 · doi:10.1090/S0894-0347-2011-00698-X |
| [18] | Vasseur, A.F.: The De Giorgi method for elliptic and parabolic equations and some applications. In: Lectures on the analysis of nonlinear partial differential equations, Part 4, 195-222, Morningside Lect. Math., 4, Int. Press, Somerville, MA, 2016. · Zbl 1353.35096 |
| [19] | Golse, F.; Imbert, C.; Mouhot, C.; Vasseur, AF, 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, 253-295 (2019) · Zbl 1431.35016 |
| [20] | Alonso, R., Emergence of exponentially weighted \(L^p\)-norms and Sobolev regularity for the Boltzmann equation, Commun. Partial Differ. Equ., 44, 416-446 (2019) · Zbl 1507.35054 |
| [21] | Alonso, R., Morimoto, Y., Sun, W., Yang, T.: De Giorgi argument for weighted \(L^2\cap L^{\infty }\) solutions to the non-cutoff Boltzmann equation, arXiv:2010.10065 |
| [22] | Carrapatoso, K.; Desvillettes, L.; He, L., Estimates for the large time behavior of the Landau equation in the Coulomb case, Arch. Ration. Mech. Anal., 224, 381-420 (2017) · Zbl 1390.35360 · doi:10.1007/s00205-017-1078-3 |
| [23] | Lu, X.; Wennberg, B., On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles, Arch. Ration. Mech. Anal., 168, 1-34 (2003) · Zbl 1044.76058 · doi:10.1007/s00205-003-0247-8 |
| [24] | Alexandre, R.; Liao, J.; Lin, C-J, Some a priori estimates for the homogeneous Landau equation with soft potentials, Kinet. Relat. Models, 8, 617-650 (2015) · Zbl 1320.35104 · doi:10.3934/krm.2015.8.617 |
| [25] | Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C-J; Yang, T., Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52, 433-463 (2012) · Zbl 1247.35085 · doi:10.1215/21562261-1625154 |
| [26] | Desvillettes, L.: Entropy dissipation estimates for the Landau equation: general cross sections. In: From particle systems to partial differential equations. III, pp. 121-143. Springer Proc. Math. Stat., 162, Springer, 2016. · Zbl 1351.35206 |
| [27] | Alonso, R.; Bagland, V.; Lods, B., Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations, Kinet. Relat. Models, 12, 1163-1183 (2019) · Zbl 1420.35170 · doi:10.3934/krm.2019044 |
| [28] | Desvillettes, L.; Mouhot, C., Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials, Asymptot. Anal., 54, 235-245 (2007) · Zbl 1141.35337 |
| [29] | Cañizo, J.; Einav, A.; Lods, B., On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials, J. Math. Anal. Appl., 462, 801-839 (2018) · Zbl 1394.35041 · doi:10.1016/j.jmaa.2017.12.052 |
| [30] | Ladyzenskaja, OA; Solonnikov, VV; Ural’ceva, NN, Linear and quasilinear equations of parabolic type (1968), Providence: American Mathematical Society, Providence · Zbl 0174.15403 · doi:10.1090/mmono/023 |
| [31] | Simon, J., Compact sets in the space \(L^p(0, T;B)\), Ann. Mat. Pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 |
| [32] | Temam, R., Navier-stokes equations. Theory and numerical analysis (1977), Amsterdam: North Holland, Amsterdam · Zbl 0383.35057 |
| [33] | Cameron, S.; Silvestre, L.; Snelson, S., Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35, 625-642 (2018) · Zbl 1407.35036 · doi:10.1016/j.anihpc.2017.07.001 |
| [34] | Carrapatoso, K., On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials, J. Math. Pures Appl., 104, 276-310 (2015) · Zbl 1320.47055 · doi:10.1016/j.matpur.2015.02.008 |
| [35] | Desvillettes, L.; Villani, C., On the spatially homogeneous Landau equation for hard potentials. Part II : H theorem and applications, Commun. Partial Differ. Equ., 25, 261-298 (2000) · Zbl 0951.35130 · doi:10.1080/03605300008821513 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
