On \(A_p\) weights and the Landau equation

  • Maria GualdaniEmail author
  • Nestor Guillen


In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of very soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for very soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrödinger operator whose coefficients are controlled by \(A_p\)-weights.

Mathematics Subject Classification

35B65 35K57 35B44 35K61 35Q20 35P15 



MPG is supported by NSF DMS-1412748 and DMS-1514761. NG is partially supported by NSF-DMS 1700307. NG would like to thank the Fields Institute for Research in Mathematical Sciences, where part of the work in this manuscript was carried out in the Fall of 2014. MPG would like to thank NCTS Mathematics Division Taipei for their kind hospitality. The authors thank Luis Silvestre for many fruitful communications, as well as the anonymous referee for many useful remarks that helped improve this paper.


  1. 1.
    Alexandre, R., Liao, J., Lin, C.: Some a priori estimates for the homogeneous Landau equation with soft potentials. KRM 8(4), 617–650 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Uncertainty principle and kinetic equations. J. Funct. Anal. 225(8), 2013–2066 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baras, Pierre, Goldstein, Jerome A.: The heat equation with a singular potential. Trans. Am. Math. Soc. 284(1), 121–139 (1984)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cameron, S., Silvestre, L., Snelson, S.: Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials. Ann. Inst. H. Poincare Anal. Non Lineaire (3), 625642 (2018)zbMATHGoogle Scholar
  5. 5.
    Carrapatoso, K., Desvillettes, L., He, L.: Estimates for the large time behavior of the Landau equation in the Coulomb case. Arch. Ration. Mech. Anal. (2), 381420 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Carrapatoso, K., Tristani, I., Wu, K.-C.: Cauchy problem and exponential stability for the inhomogeneous Landau equation. Arch. Ration. Mech. Anal. 223(2), 1035–1037 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chanillo, S., Wheeden, R.: L-p estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators. Commun. Part. Differ. Equ. 10(9), 1077–1116 (1985)CrossRefGoogle Scholar
  8. 8.
    Chanillo, S., Wheeden, R.: Weighted Poincare and Sobolev inequalities and estimates for weighted Peano maximal functions. Am. J. Math. 107(5), 1191–1226 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, Y., Desvillettes, L., He, L.: Smoothing effects for classical solutions of the full Landau equation. Arch. Ration. Mech. Anal. 193(1), 21–55 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Desvillettes, L.: Entropy dissipation estimates for the Landau equation in the Coulomb case and applications. J. Funct. Anal. 269(5), 1359–1403 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Commun. Part. Differ. Equ. 25(1–2), 179–259 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. II. \(H\)-theorem and applications. Commun. Part. Differ. Equ. 25(1–2), 261–298 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. PDE (7), 77–116 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fefferman, C.: The uncertainty principle. Am. Math. Soc. 9(2), 129–206 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fournier, N., Guérin, H.: Well-posedness of the spatially homogeneous Landau equation for soft potentials. J. Funct. Anal. 256(8), 2542–2560 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fournier, Nicolas: Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential. Commun. Math. Phys. 299(3), 765–782 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    García-Cuerva, José, Rubio De Francia, J.L.: Weighted Norm Inequalities and Related Topics. Elsevier, Amsterdam (1985)zbMATHGoogle Scholar
  18. 18.
    Giga, Y., Kohn, R.V.: Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38(3), 297–319 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Golse, F., Imbert, C., Mouhot, C., Vasseur, A.: Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation. Annali della Scuola Normale Superiore di Pisa (2017) (accepted)Google Scholar
  20. 20.
    Gressman, P., Krieger, J., Strain, R.: A non-local inequality and global existence. Adv. Math. 230(2), 642–648 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gualdani, M., Guillen, N.: Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential. Anal. PDE 9(8), 1772–1809 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gutierrez, C., Wheeden, R.: Harnack’s inequality for degenerate parabolic equations. Commun. PDE 4,5(16), 745–770 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Henderson, C., Snelson, S.: \(\text{C}^{\wedge }\backslash \)infty smoothing for weak solutions of the inhomogeneous landau equation. arXiv preprint arXiv:1707.05710 (2017)
  25. 25.
    Henderson, C., Snelson, S., Tarfulea, A.: Local existence, lower mass bounds, and smoothing for the landau equation. arXiv preprint arXiv:1712.07111 (2017)
  26. 26.
    Imbert, C., Silvestre, L.: Weak Harnack inequality for the Boltzmann equation without cut-off. J. Eur. Math. Society (2018) (accepted)Google Scholar
  27. 27.
    Kim, J., Guo, Y., Hwang, H.J.: A \({L}^2\) to \({L}^\infty \) approach for the Landau equation. arXiv preprint arXiv:1610.05346 (2016)
  28. 28.
    Krieger, J., Strain, R.: Global solutions to a non-local diffusion equation with quadratic non-linearity. Commun. Part. Differ. Equ. 37(4), 647–689 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pascucci, Andrea, Polidoro, Sergio: The moser’s iterative method for a class of ultraparabolic equations. Commun. Contemp. Math. 6(03), 395–417 (2004)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sawyer, E., Wheeden, R.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Silvestre, L.: Upper bounds for parabolic equations and the Landau equation. J. Differ. Equ. 262(3), 3034–3055 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Silvestre, L.: A new regularization mechanism for the Boltzmann equation without cut-off. Commun. Math. Phys. 348(1), 69–100 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Turesson, B.O.: Nonlinear potential theory and weighted Sobolev spaces. Lecture Notes in Mathematics, vol. 1736. Springer, Berlin (2000)CrossRefGoogle Scholar
  34. 34.
    Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273–307 (1998)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wu, K.-C.: Global in time estimates for the spatially homogeneous Landau equation with soft potentials. J. Funct. Anal. 266(5), 3134–3155 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsGeorge Washington UniversityWashingtonUSA
  2. 2.Royal Institute of Technology (KTH)StockholmSweden
  3. 3.Department of MathematicsUniversity of MassachusettsAmherstUSA

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