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 & 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.   Google Scholar

[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.   Google Scholar

[3]

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

[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.   Google Scholar

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", Cambridge University Press, (1970).   Google Scholar

[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.   Google Scholar

[7]

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

[8]

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

[9]

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

[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., ().   Google Scholar

[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.   Google Scholar

[12]

L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation,, Riv. Mat. Univ. Parma, 7 (2003), 1.   Google Scholar

[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.   Google Scholar

[14]

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

[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), (1981).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71.   Google Scholar

[20]

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

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.   Google Scholar

[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.   Google Scholar

[3]

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

[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.   Google Scholar

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", Cambridge University Press, (1970).   Google Scholar

[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.   Google Scholar

[7]

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

[8]

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

[9]

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

[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., ().   Google Scholar

[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.   Google Scholar

[12]

L. Desvillettes, About the use of the Fourier transform for the Boltzmann equation,, Riv. Mat. Univ. Parma, 7 (2003), 1.   Google Scholar

[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.   Google Scholar

[14]

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

[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), (1981).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71.   Google Scholar

[20]

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

[1]

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

Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic & Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1
[3]

Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic & Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299
[4]

Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic & Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453
[5]

Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187
[6]

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

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413
[9]

Léo Glangetas, Mohamed Najeme. Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 407-427. doi: 10.3934/krm.2013.6.407
[10]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[11]

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
[12]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[13]

Marcel Braukhoff. Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation. Kinetic & Related Models, 2020, 13 (1) : 187-210. doi: 10.3934/krm.2020007
[14]

Joseph J Kohn. Nirenberg's contributions to complex analysis. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 537-545. doi: 10.3934/dcds.2011.30.537
[15]

Zheng-Chao Han, YanYan Li. On the local solvability of the Nirenberg problem on $\mathbb S^2$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 607-615. doi: 10.3934/dcds.2010.28.607
[16]

Alexei Shadrin. The Landau--Kolmogorov inequality revisited. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1183-1210. doi: 10.3934/dcds.2014.34.1183
[17]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1
[18]

Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165
[19]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[20]

Francesco Solombrino. Quasistatic evolution for plasticity with softening: The spatially homogeneous case. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1189-1217. doi: 10.3934/dcds.2010.27.1189

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences