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

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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

Radjesvarane Alexandre ^1, , Jie Liao ^2, and Chunjin Lin ^3,

1.	Arts et Metiers 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

Abstract
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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.

Keywords: entropy dissipation, weighted Fisher information, Landau equation with soft potentials, weak solutions, moment estimates..

Mathematics Subject Classification: Primary: 35B45, 35D30; Secondary: 82C4.

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.

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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), 61-95. 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.

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