A Review for an Isotropic Landau Model

  • Maria Gualdani
  • Nicola Zamponi
Part of the Springer INdAM Series book series (SINDAMS, volume 28)


We consider the equation
$$\displaystyle u_t = \mathrm{div}\,(a[u]\nabla u - u\nabla a[u]),\qquad -\Delta a = u. $$
This model has attracted some attention in the recent years and several results are available in the literature. We review recent results on existence and smoothness of solutions and explain the open problems.


Landau equation Coulomb potential Isotropic model Even solutions Weighted Poincaré and Sobolev inequalities Regularity estimates 



MPG is supported by NSF DMS-1514761. MPG would like to thank NCTS Mathematics Division Taipei for their kind hospitality. NZ acknowledges support from the Austrian Science Fund (FWF), grants P22108, P24304, W1245.


  1. 1.
    Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. Henri Poincaré, C Anal. Non Linéaire 21(1), 61–95 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cameron, S., Silvestre, L., Snelson, S.: Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials. Ann. Inst. H. Poincaré C Anal. Non Linéaire 35(3), 625–642 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang, S.-Y.A., Wilson, J.M., Wolff, T.H.: Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60(2), 217–246 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chanillo, S., Wheeden, R.: Lp-estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators. Commun. Partial Differ. Equ. 10(9), 1077–1116 (1985)CrossRefGoogle Scholar
  5. 5.
    Chanillo, S., Wheeden, R.: Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Am. J. Math. 107(5), 1191–1226 (1985)CrossRefGoogle Scholar
  6. 6.
    Desvillettes, L.: Entropy dissipation estimates for the Landau equation in the Coulomb case and applications. J. Funct. Anal. 269, 1359–1403 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fefferman, C.: The uncertainty principle. Bull. Am. Math. Soc. 9, 129–206 (1983)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fefferman, C., Phong, D.H.: On positivity of pseudo-differential operators. Proc. Natl. Acad. Sci. U. S. A. 75(10), 4673–4674 (1978)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    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, To appear)Google Scholar
  11. 11.
    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
  12. 12.
    Gualdani, M., Guillen, N.: On A p weights and the homogeneous Landau equation. Calc. Var. Partial Differ. Equ. (2018, to appear)Google Scholar
  13. 13.
    Gualdani, M., Zamponi, N.: Global existence of weak even solutions for an isotropic Landau equation with Coulomb potential. SIAM J. Math. Anal. 50(4), 3676–3714 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gressman, P., Krieger, J., Strain, R.: A non-local inequality and global existence. Adv. Math. 230(2), 642–648 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Handerson, C., Snelson, S.: C -smoothing for weak solutions of the inhomogeneous Landau equation (Preprint)Google Scholar
  16. 16.
    Krieger, J., Strain, R.: Global solutions to a non-local diffusion equation with quadratic nonlinearity. Commun. Partial Differ. Equ. 37(4), 647–689 (2012)CrossRefGoogle Scholar
  17. 17.
    Sawyer, E., Wheeden, R.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Silvestre, L.: Upper bounds for parabolic equations and the Landau equation. J. Differ. Equ. 262(3), 3034–3055 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, vol. 1, pp. 71–74. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsGeorge Washington UniversityWashingtonUSA
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria

Personalised recommendations