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2016 Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential
Maria Gualdani, Nestor Guillen
Anal. PDE 9(8): 1773-1810 (2016). DOI: 10.2140/apde.2016.9.1772


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.


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Maria Gualdani. Nestor Guillen. "Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential." Anal. PDE 9 (8) 1773 - 1810, 2016.


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

zbMATH: 1378.35325
MathSciNet: MR3599518
Digital Object Identifier: 10.2140/apde.2016.9.1772

Primary: 35B44 , 35B65 , 35K57 , 35K61 , 35Q20

Keywords: barriers , Coulomb potential , homogeneous solutions , Landau equation , regularity , Upper bounds

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.9 • No. 8 • 2016
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