Vol. 9, No. 8, 2016

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

Maria Pia Gualdani and Nestor Guillen

Vol. 9 (2016), No. 8, 1773–1810

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.

Landau equation, Coulomb potential, homogeneous solutions, upper bounds, barriers, regularity
Mathematical Subject Classification 2010
Primary: 35B65, 35K57, 35B44, 35K61, 35Q20
Received: 4 May 2015
Revised: 15 June 2016
Accepted: 28 August 2016
Published: 11 December 2016
Maria Pia Gualdani
Department of Mathematics
George Washington University
2115 G Street NW
Monroe Hall 272
Washington, DC 20052
United States
Nestor Guillen
Department of Mathematics and Statistics
University of Massachussetts Amherst
Amherst, MA 01003-9305
United States