# 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 & Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617
##### References:
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Beckner, Pitt's inequality with sharp convolution estimates,, Proc. Amer. Math. Soc., 136 (2008), 1871. doi: 10.1090/S0002-9939-07-09216-7. Google Scholar [7] W. Beckner, Pitt's inequality and the fractional Laplacian: sharp error estimates,, Forum Math., 24 (2012), 177. doi: 10.1515/form.2011.056. Google Scholar [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. doi: 10.1016/j.matpur.2015.02.008. Google Scholar [9] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases,, Cambridge Univ. Press., (1970). Google Scholar [10] L. Desvillettes, On asymptotic of the Boltzmann equation when the collisions become grazing,, Transp. theory and stat. phys., 21 (1992), 259. doi: 10.1080/00411459208203923. Google Scholar [11] L. Desvillettes, Plasma kinetic models: The Fokker-Planck-Landau equation,, Chapter 6 of Modeling and Computational Methods for Kinetic Equations, (2004), 171. Google Scholar [12] L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications,, preprint, (). doi: 10.1016/j.jfa.2015.05.009. Google Scholar [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. doi: 10.1080/03605300008821512. Google Scholar [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. doi: 10.1080/03605300008821513. Google Scholar [15] N. Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential,, Commun. Math. Phys., 299 (2010), 765. doi: 10.1007/s00220-010-1113-9. Google Scholar [16] N. Fournier and H. Guerin, Well-posedness of the spatially homogeneous Landau equation for soft potentials,, Journal of Functional Analysis, 256 (2009), 2542. doi: 10.1016/j.jfa.2008.11.008. Google Scholar [17] T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions,, J. Stat. Phys., 89 (1997), 751. doi: 10.1007/BF02765543. Google Scholar [18] H. Guerin, Solving Landau equation for some soft potentials through a probabilistic approach,, Ann. Appl. Probab., 13 (2003), 515. doi: 10.1214/aoap/1050689592. Google Scholar [19] E. M. Lifschitz and L. P. Pitaevskii, Physical Kinetics,, Perg. Press., (1981). Google Scholar [20] P. L. Lions, On Boltzmann and Landau equations,, Phil. Trans. R. Soc. Lond., 346 (1994), 191. doi: 10.1098/rsta.1994.0018. Google Scholar [21] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations,, Springer, (2011). doi: 10.1007/978-1-4419-7049-7. Google Scholar [22] G. Toscani, Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles,, Communications in Partial Differential Equations, 37 (2012), 77. doi: 10.1080/03605302.2011.592236. Google Scholar [23] C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence,, Adv. Differential Equations, 1 (1996), 793. Google Scholar [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, (1998). Google Scholar [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. doi: 10.1007/s002050050106. Google Scholar [26] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules,, Math. Meth. Mod. Appl. Sci., 8 (1998), 957. doi: 10.1142/S0218202598000433. Google Scholar [27] C. Villani, A review of mathematical topics in collisional kinetic theory,, Handbook of mathematical fluid dynamics, I (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar [28] K. C. Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials,, Journal of Functional Analysis, 266 (2014), 3134. doi: 10.1016/j.jfa.2013.11.005. Google Scholar [29] W. P. Ziemer, Weakly Differentiable Functions,, Springer-Verlag Vol 120, 120 (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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##### References:
 [1] R. Alexandre and C. Villani, On the Landau approximation in plasma physics,, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61. doi: 10.1016/S0294-1449(03)00030-1. Google Scholar [2] L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,, Arch. Rational Mech. Anal., 77 (1981), 11. doi: 10.1007/BF00280403. Google Scholar [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. doi: 10.1070/SM1991v069n02ABEH001244. Google Scholar [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. doi: 10.1016/0041-5553(77)90125-2. Google Scholar [5] W. Beckner, Weighted inequalities and Stein-Weiss potentials,, Forum Math., 20 (2008), 587. doi: 10.1515/FORUM.2008.030. Google Scholar [6] W. Beckner, Pitt's inequality with sharp convolution estimates,, Proc. Amer. Math. Soc., 136 (2008), 1871. doi: 10.1090/S0002-9939-07-09216-7. Google Scholar [7] W. Beckner, Pitt's inequality and the fractional Laplacian: sharp error estimates,, Forum Math., 24 (2012), 177. doi: 10.1515/form.2011.056. Google Scholar [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. doi: 10.1016/j.matpur.2015.02.008. Google Scholar [9] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases,, Cambridge Univ. Press., (1970). Google Scholar [10] L. Desvillettes, On asymptotic of the Boltzmann equation when the collisions become grazing,, Transp. theory and stat. phys., 21 (1992), 259. doi: 10.1080/00411459208203923. Google Scholar [11] L. Desvillettes, Plasma kinetic models: The Fokker-Planck-Landau equation,, Chapter 6 of Modeling and Computational Methods for Kinetic Equations, (2004), 171. Google Scholar [12] L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications,, preprint, (). doi: 10.1016/j.jfa.2015.05.009. Google Scholar [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. doi: 10.1080/03605300008821512. Google Scholar [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. doi: 10.1080/03605300008821513. Google Scholar [15] N. Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential,, Commun. Math. Phys., 299 (2010), 765. doi: 10.1007/s00220-010-1113-9. Google Scholar [16] N. Fournier and H. Guerin, Well-posedness of the spatially homogeneous Landau equation for soft potentials,, Journal of Functional Analysis, 256 (2009), 2542. doi: 10.1016/j.jfa.2008.11.008. Google Scholar [17] T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions,, J. Stat. Phys., 89 (1997), 751. doi: 10.1007/BF02765543. Google Scholar [18] H. Guerin, Solving Landau equation for some soft potentials through a probabilistic approach,, Ann. Appl. Probab., 13 (2003), 515. doi: 10.1214/aoap/1050689592. Google Scholar [19] E. M. Lifschitz and L. P. Pitaevskii, Physical Kinetics,, Perg. Press., (1981). Google Scholar [20] P. L. Lions, On Boltzmann and Landau equations,, Phil. Trans. R. Soc. Lond., 346 (1994), 191. doi: 10.1098/rsta.1994.0018. Google Scholar [21] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations,, Springer, (2011). doi: 10.1007/978-1-4419-7049-7. Google Scholar [22] G. Toscani, Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles,, Communications in Partial Differential Equations, 37 (2012), 77. doi: 10.1080/03605302.2011.592236. Google Scholar [23] C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence,, Adv. Differential Equations, 1 (1996), 793. Google Scholar [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, (1998). Google Scholar [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. doi: 10.1007/s002050050106. Google Scholar [26] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules,, Math. Meth. Mod. Appl. Sci., 8 (1998), 957. doi: 10.1142/S0218202598000433. Google Scholar [27] C. Villani, A review of mathematical topics in collisional kinetic theory,, Handbook of mathematical fluid dynamics, I (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar [28] K. C. Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials,, Journal of Functional Analysis, 266 (2014), 3134. doi: 10.1016/j.jfa.2013.11.005. Google Scholar [29] W. P. Ziemer, Weakly Differentiable Functions,, Springer-Verlag Vol 120, 120 (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar
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