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 norm at some finite time occurs only if a certain quotient involving and its Newtonian potential concentrates near zero, which implies blow-up in more standard norms, such as the 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.
"Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential." Anal. PDE 9 (8) 1773 - 1810, 2016. https://doi.org/10.2140/apde.2016.9.1772