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December  2010, 3(4): 645-667. doi: 10.3934/krm.2010.3.645

Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials

Yemin Chen 1,

1. 

Department of Mathematics, Beijing Institute of Technology, Beijing 100081

Received  April 2010 Revised  September 2010 Published  October 2010

In this paper, we consider the regularity of solutions to the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. In particular, we get the analytic smoothing effects for solutions obtained by Bagland if we assume all the moments for the initial datum are finite.
Keywords: spatially homogeneous Landau-Fermi-Dirac equation, Sobolev embedding theorem., Analytic regularity, Gagliardo-Nirenberg's inequality.
Mathematics Subject Classification: Primary: 35B65, 35Q82; Secondary: 82C4.
Citation: 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
References:
[1]

A. A. Arsen'ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478.

[2]

V. Bagland, Well-posedness for the spatially homogeneous Landan-fermi-Dirac equation for hard potentials, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 415-447.

[3]

V. Bagland and M. Lemou, Equilibrium states for the Landau-Fermi-Dirac equation, Banach Center Publ., 66 (2004), 29-37.

[4]

F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel, Rev. Mat. Iberoamericana, 14 (1998), 47-61.

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambridge University Press, New York, 1970.

[6]

H. Chen, W. Li and C. Xu, Gevrey regularity for solutions of the spatially homogeneous Landau equation, Acta Math. Scientia Ser. B, 29 (2009), 673-686.

[7]

H. Chen, W. Li and C. Xu, Analytic smoothness effect for solutions for spatially homogeneous Landau equation, J. Differ. Equ., 248 (2010), 77-94.

[8]

Y. Chen, Smoothness of classical solutions to the Vlasov-Poisson-Landau system, Kinet. Relat. Models, 1 (2008), 369-386.

[9]

Y. Chen, Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians, Discrete Contin. Dyn. Syst., 20 (2008), 889-910.

[10]

Y. Chen, Smoothing effects for weak solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, to appear in Acta Appl. Math.

[11]

Y. Chen, L. Desvillettes and L. He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal., 193 (2009), 21-55.

[12]

L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation, Riv. Mat. Univ. Parma, 7 (2003), 1-99 (special issue).

[13]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials, Part 1: Existence, uniqueness and smoothness, Comm. P. D. E., 25 (2000), 179-259.

[14]

M. El Safadi, Smoothness of weak solutions of the spatially homogeneous Landau equation, Anal. Appl. (Singap.), 5 (2007), 29-49.

[15]

E. M. Lifshitz and L. P. Pitaevskiĭ, "Course of Theoretical Physics ["Landau-Lifshitz"], Vol. 10," (translated from the Russian by J. B. Sykes and R. N. Franklin), Pergamon Press, Oxford, 1981.

[16]

P. L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461.

[17]

X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation, J. Math. Anal. Appl., 228 (1998), 409-435.

[18]

Y. Morimoto and C. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differ. Equ., 247 (2009), 596-617.

[19]

C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics, Vol. 1," North-Holland, (2002), 71-305.

[20]

B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. P. D. E., 19 (1994), 2057-2074.

show all references

References:
[1]

A. A. Arsen'ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478.

[2]

V. Bagland, Well-posedness for the spatially homogeneous Landan-fermi-Dirac equation for hard potentials, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 415-447.

[3]

V. Bagland and M. Lemou, Equilibrium states for the Landau-Fermi-Dirac equation, Banach Center Publ., 66 (2004), 29-37.

[4]

F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel, Rev. Mat. Iberoamericana, 14 (1998), 47-61.

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambridge University Press, New York, 1970.

[6]

H. Chen, W. Li and C. Xu, Gevrey regularity for solutions of the spatially homogeneous Landau equation, Acta Math. Scientia Ser. B, 29 (2009), 673-686.

[7]

H. Chen, W. Li and C. Xu, Analytic smoothness effect for solutions for spatially homogeneous Landau equation, J. Differ. Equ., 248 (2010), 77-94.

[8]

Y. Chen, Smoothness of classical solutions to the Vlasov-Poisson-Landau system, Kinet. Relat. Models, 1 (2008), 369-386.

[9]

Y. Chen, Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians, Discrete Contin. Dyn. Syst., 20 (2008), 889-910.

[10]

Y. Chen, Smoothing effects for weak solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, to appear in Acta Appl. Math.

[11]

Y. Chen, L. Desvillettes and L. He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal., 193 (2009), 21-55.

[12]

L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation, Riv. Mat. Univ. Parma, 7 (2003), 1-99 (special issue).

[13]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials, Part 1: Existence, uniqueness and smoothness, Comm. P. D. E., 25 (2000), 179-259.

[14]

M. El Safadi, Smoothness of weak solutions of the spatially homogeneous Landau equation, Anal. Appl. (Singap.), 5 (2007), 29-49.

[15]

E. M. Lifshitz and L. P. Pitaevskiĭ, "Course of Theoretical Physics ["Landau-Lifshitz"], Vol. 10," (translated from the Russian by J. B. Sykes and R. N. Franklin), Pergamon Press, Oxford, 1981.

[16]

P. L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461.

[17]

X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation, J. Math. Anal. Appl., 228 (1998), 409-435.

[18]

Y. Morimoto and C. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differ. Equ., 247 (2009), 596-617.

[19]

C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics, Vol. 1," North-Holland, (2002), 71-305.

[20]

B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. P. D. E., 19 (1994), 2057-2074.

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