American Institute of Mathematical Sciences

Advanced
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.

[1]

Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation. Kinetic and Related Models, 2013, 6 (4) : 715-727. doi: 10.3934/krm.2013.6.715
[2]

Maria J. Esteban. Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2101-2114. doi: 10.3934/cpaa.2022051
[3]

Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic and Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1
[4]

Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299
[5]

Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic and Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453
[6]

Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187
[7]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[8]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[9]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413
[10]

Yoshinori Morimoto, Chao-Jiang Xu. Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules. Kinetic and Related Models, 2020, 13 (5) : 951-978. doi: 10.3934/krm.2020033
[11]

Léo Glangetas, Mohamed Najeme. Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 407-427. doi: 10.3934/krm.2013.6.407
[12]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227
[13]

Immanuel Ben Porat. Local conditional regularity for the Landau equation with Coulomb potential. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022010
[14]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[15]

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
[16]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[17]

Marcel Braukhoff. Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation. Kinetic and Related Models, 2020, 13 (1) : 187-210. doi: 10.3934/krm.2020007
[18]

Joseph J Kohn. Nirenberg's contributions to complex analysis. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 537-545. doi: 10.3934/dcds.2011.30.537
[19]

Alexei Shadrin. The Landau--Kolmogorov inequality revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1183-1210. doi: 10.3934/dcds.2014.34.1183
[20]

Zheng-Chao Han, YanYan Li. On the local solvability of the Nirenberg problem on $\mathbb S^2$. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 607-615. doi: 10.3934/dcds.2010.28.607

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy