# American Institute of Mathematical Sciences

December  2015, 8(4): 617-650. doi: 10.3934/krm.2015.8.617

## Some a priori estimates for the homogeneous Landau equation with soft potentials

 1 Arts et Metiers Paris Tech, Paris 75013, France 2 School of Science, East China University of Science and Technology, Shanghai, 200237 3 Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China

Received  May 2014 Revised  June 2015 Published  July 2015

This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in $L^2$ space hold true for moderately soft potential cases $\gamma \in[-2, 0)$ without any smallness assumption on the initial data. For very soft potential cases $\gamma \in[-3, -2)$, which cover in particular the Coulomb case $\gamma=-3$, we get local in time estimates of weak solutions in $L^{2}$.
In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $\gamma \in[-3, -2)$, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality.
Citation: Radjesvarane Alexandre, Jie Liao, Chunjin Lin. Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinetic and Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617
##### References:
 [1] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61-95. doi: 10.1016/S0294-1449(03)00030-1. [2] L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403. [3] A. A. Arsen'ev and O. E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478. doi: 10.1070/SM1991v069n02ABEH001244. [4] A. A. Arsen'ev and N. V. Peskov, On the existence of a generalized solution of Landau's equation, Z. Vycisl. Mat. i Mat. Fiz., 17 (1977), 1063-1068. doi: 10.1016/0041-5553(77)90125-2. [5] W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606. doi: 10.1515/FORUM.2008.030. [6] W. Beckner, Pitt's inequality with sharp convolution estimates, Proc. Amer. Math. Soc., 136 (2008), 1871-1885. doi: 10.1090/S0002-9939-07-09216-7. [7] W. Beckner, Pitt's inequality and the fractional Laplacian: sharp error estimates, Forum Math., 24 (2012), 177-209. doi: 10.1515/form.2011.056. [8] K. Carrapatoso, On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials, Journal de Mathématiques Pures et Appliquées, 104 (2015), 276-310. doi: 10.1016/j.matpur.2015.02.008. [9] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press., London, 1970. [10] L. Desvillettes, On asymptotic of the Boltzmann equation when the collisions become grazing, Transp. theory and stat. phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923. [11] L. Desvillettes, Plasma kinetic models: The Fokker-Planck-Landau equation, Chapter 6 of Modeling and Computational Methods for Kinetic Equations, pp. 171-193, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Boston, MA, 2004. [12] L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, preprint, arXiv:1408.6025. doi: 10.1016/j.jfa.2015.05.009. [13] L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part I: Existence, uniqueness and smoothness, Communications in Partial Differential Equations, 25 (2000), 179-259. doi: 10.1080/03605300008821512. [14] L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part II: H-theorem and applications, Communications in Partial Differential Equations, 25 (2000), 261-298. doi: 10.1080/03605300008821513. [15] N. Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Commun. Math. Phys., 299 (2010), 765-782. doi: 10.1007/s00220-010-1113-9. [16] N. Fournier and H. Guerin, Well-posedness of the spatially homogeneous Landau equation for soft potentials, Journal of Functional Analysis, 256 (2009), 2542-2560. doi: 10.1016/j.jfa.2008.11.008. [17] T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543. [18] H. Guerin, Solving Landau equation for some soft potentials through a probabilistic approach, Ann. Appl. Probab., 13 (2003), 515-539. doi: 10.1214/aoap/1050689592. [19] E. M. Lifschitz and L. P. Pitaevskii, Physical Kinetics, Perg. Press., Oxford, 1981. [20] P. L. Lions, On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond., A, 346 (1994), 191-204. doi: 10.1098/rsta.1994.0018. [21] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, New-York, 2nd ed., 2011. doi: 10.1007/978-1-4419-7049-7. [22] G. Toscani, Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles, Communications in Partial Differential Equations, 37 (2012), 77-87. doi: 10.1080/03605302.2011.592236. [23] C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Differential Equations, 1 (1996), 793-816. [24] C. Villani, Contribution à L'étude Mathématique Des Équations de Boltzmann et de Landau en Théorie Cinétique des Gaz et Des Plasmas, PhD Thesis, Universite Paris 9-Dauphine, 1998. [25] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rat. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106. [26] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Meth. Mod. Appl. Sci., 8 (1998), 957-983. doi: 10.1142/S0218202598000433. [27] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0. [28] K. C. Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, Journal of Functional Analysis, 266 (2014), 3134-3155. doi: 10.1016/j.jfa.2013.11.005. [29] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag Vol 120, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

##### References:
 [1] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61-95. doi: 10.1016/S0294-1449(03)00030-1. [2] L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403. [3] A. A. Arsen'ev and O. E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478. doi: 10.1070/SM1991v069n02ABEH001244. [4] A. A. Arsen'ev and N. V. Peskov, On the existence of a generalized solution of Landau's equation, Z. Vycisl. Mat. i Mat. Fiz., 17 (1977), 1063-1068. doi: 10.1016/0041-5553(77)90125-2. [5] W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606. doi: 10.1515/FORUM.2008.030. [6] W. Beckner, Pitt's inequality with sharp convolution estimates, Proc. Amer. Math. Soc., 136 (2008), 1871-1885. doi: 10.1090/S0002-9939-07-09216-7. [7] W. Beckner, Pitt's inequality and the fractional Laplacian: sharp error estimates, Forum Math., 24 (2012), 177-209. doi: 10.1515/form.2011.056. [8] K. Carrapatoso, On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials, Journal de Mathématiques Pures et Appliquées, 104 (2015), 276-310. doi: 10.1016/j.matpur.2015.02.008. [9] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press., London, 1970. [10] L. Desvillettes, On asymptotic of the Boltzmann equation when the collisions become grazing, Transp. theory and stat. phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923. [11] L. Desvillettes, Plasma kinetic models: The Fokker-Planck-Landau equation, Chapter 6 of Modeling and Computational Methods for Kinetic Equations, pp. 171-193, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Boston, MA, 2004. [12] L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, preprint, arXiv:1408.6025. doi: 10.1016/j.jfa.2015.05.009. [13] L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part I: Existence, uniqueness and smoothness, Communications in Partial Differential Equations, 25 (2000), 179-259. doi: 10.1080/03605300008821512. [14] L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part II: H-theorem and applications, Communications in Partial Differential Equations, 25 (2000), 261-298. doi: 10.1080/03605300008821513. [15] N. Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Commun. Math. Phys., 299 (2010), 765-782. doi: 10.1007/s00220-010-1113-9. [16] N. Fournier and H. Guerin, Well-posedness of the spatially homogeneous Landau equation for soft potentials, Journal of Functional Analysis, 256 (2009), 2542-2560. doi: 10.1016/j.jfa.2008.11.008. [17] T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543. [18] H. Guerin, Solving Landau equation for some soft potentials through a probabilistic approach, Ann. Appl. Probab., 13 (2003), 515-539. doi: 10.1214/aoap/1050689592. [19] E. M. Lifschitz and L. P. Pitaevskii, Physical Kinetics, Perg. Press., Oxford, 1981. [20] P. L. Lions, On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond., A, 346 (1994), 191-204. doi: 10.1098/rsta.1994.0018. [21] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, New-York, 2nd ed., 2011. doi: 10.1007/978-1-4419-7049-7. [22] G. Toscani, Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles, Communications in Partial Differential Equations, 37 (2012), 77-87. doi: 10.1080/03605302.2011.592236. [23] C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Differential Equations, 1 (1996), 793-816. [24] C. Villani, Contribution à L'étude Mathématique Des Équations de Boltzmann et de Landau en Théorie Cinétique des Gaz et Des Plasmas, PhD Thesis, Universite Paris 9-Dauphine, 1998. [25] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rat. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106. [26] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Meth. Mod. Appl. Sci., 8 (1998), 957-983. doi: 10.1142/S0218202598000433. [27] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0. [28] K. C. Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, Journal of Functional Analysis, 266 (2014), 3134-3155. doi: 10.1016/j.jfa.2013.11.005. [29] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag Vol 120, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
 [1] Ricardo J. Alonso, Véronique Bagland, Bertrand Lods. Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations. Kinetic and Related Models, 2019, 12 (5) : 1163-1183. doi: 10.3934/krm.2019044 [2] Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic and Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405 [3] Fei Meng, Fang Liu. On the inelastic Boltzmann equation for soft potentials with diffusion. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5197-5217. doi: 10.3934/cpaa.2020233 [4] Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483 [5] Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1857-1873. doi: 10.3934/dcdsb.2017110 [6] Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control and Related Fields, 2020, 10 (1) : 113-140. doi: 10.3934/mcrf.2019032 [7] Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174 [8] 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 [9] Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure and Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031 [10] Nicolas Rougerie. On two properties of the Fisher information. Kinetic and Related Models, 2021, 14 (1) : 77-88. doi: 10.3934/krm.2020049 [11] Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199 [12] Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895 [13] Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731 [14] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [15] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [16] Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic and Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015 [17] Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691 [18] Lijuan Wang, Weike Wang. Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2835-2854. doi: 10.3934/cpaa.2019127 [19] Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic and Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919 [20] N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647

2021 Impact Factor: 1.398