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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
L 3 ∕ 2
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.
Keywords
Landau equation, Coulomb potential, homogeneous solutions,
upper bounds, barriers, regularity
Mathematical Subject Classification 2010
Primary: 35B65, 35K57, 35B44, 35K61, 35Q20
Milestones
Received: 4 May 2015
Revised: 15 June 2016
Accepted: 28 August 2016
Published: 11 December 2016