Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 8 (2016), 1773-1810.

Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential

Maria Gualdani and Nestor Guillen

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/apde.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow-up in the L norm at some finite time T occurs only if a certain quotient involving f and its Newtonian potential concentrates near zero, which implies blow-up in more standard norms, such as the L32 norm. This quotient is shown to be always less than a universal constant, suggesting that the problem of regularity for the Landau equation is in some sense critical.

The bounds are obtained using the comparison principle both for the Landau equation and for the associated mass function. In particular, the method provides long-time existence results for a modified version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain.

Article information

Source
Anal. PDE, Volume 9, Number 8 (2016), 1773-1810.

Dates
Received: 4 May 2015
Revised: 15 June 2016
Accepted: 28 August 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843374

Digital Object Identifier
doi:10.2140/apde.2016.9.1772

Mathematical Reviews number (MathSciNet)
MR3599518

Zentralblatt MATH identifier
1378.35325

Subjects
Primary: 35B65: Smoothness and regularity of solutions 35K57: Reaction-diffusion equations 35B44: Blow-up 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 35Q20: Boltzmann equations

Keywords
Landau equation Coulomb potential homogeneous solutions upper bounds barriers regularity

Citation

Gualdani, Maria; Guillen, Nestor. Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential. Anal. PDE 9 (2016), no. 8, 1773--1810. doi:10.2140/apde.2016.9.1772. https://projecteuclid.org/euclid.apde/1510843374


Export citation

References

  • R. Alexandre, J. Liao, and C. Lin, “Some a priori estimates for the homogeneous Landau equation with soft potentials”, Kinet. Relat. Models 8:4 (2015), 617–650.
  • A. A. Arsen'ev and N. V. Peskov, “The existence of a generalized solution of Landau's equation”, Ž. Vyčisl. Mat. i Mat. Fiz. 17:4 (1977), 1063–1068, 1096. In Russian; translated in U.S.S.R. Computational Math. and Math. Phys. 17:4 (1977), 241–246.
  • L. Desvillettes and C. Villani, “On the spatially homogeneous Landau equation for hard potentials, I: Existence, uniqueness and smoothness”, Comm. Partial Differential Equations 25:1-2 (2000), 179–259.
  • L. Desvillettes and C. Villani, “On the spatially homogeneous Landau equation for hard potentials, II: $H$-theorem and applications”, Comm. Partial Differential Equations 25:1-2 (2000), 261–298.
  • N. Fournier, “Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential”, Comm. Math. Phys. 299:3 (2010), 765–782.
  • N. Fournier and H. Guérin, “Well-posedness of the spatially homogeneous Landau equation for soft potentials”, J. Funct. Anal. 256:8 (2009), 2542–2560.
  • Y. Giga and R. V. Kohn, “Asymptotically self-similar blow-up of semilinear heat equations”, Comm. Pure Appl. Math. 38:3 (1985), 297–319.
  • P. T. Gressman, J. Krieger, and R. M. Strain, “A non-local inequality and global existence”, Adv. Math. 230:2 (2012), 642–648.
  • M. Gualdani and N. Guillen, “On Ap weights and regularization effects for homogeneous Landau equations”, in preparation.
  • Y. Guo, “The Landau equation in a periodic box”, Comm. Math. Phys. 231:3 (2002), 391–434.
  • J. Krieger and R. M. Strain, “Global solutions to a non-local diffusion equation with quadratic non-linearity”, Comm. Partial Differential Equations 37:4 (2012), 647–689.
  • O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, RI, 1968.
  • E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001.
  • G. M. Lieberman, Second order parabolic differential equations, World Sci. Pub., River Edge, NJ, 1996.
  • R. W. Schwab and L. Silvestre, “Regularity for parabolic integro-differential equations with very irregular kernels”, Anal. PDE 9:3 (2016), 727–772.
  • L. Silvestre, “A new regularization mechanism for the Boltzmann equation without cut-off”, Comm. Math. Phys. 348:1 (2016), 69–100.
  • C. Villani, “On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations”, Arch. Rational Mech. Anal. 143:3 (1998), 273–307.
  • C. Villani, “A review of mathematical topics in collisional kinetic theory”, pp. 71–305 in Handbook of mathematical fluid dynamics, vol. I, edited by S. Friedlander and D. Serre, North-Holland, Amsterdam, 2002.
  • K.-C. Wu, “Global in time estimates for the spatially homogeneous Landau equation with soft potentials”, J. Funct. Anal. 266:5 (2014), 3134–3155.