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 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

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:
[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

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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

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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

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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

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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

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.  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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