Abstract

This talk is motivated by the long-time open question of global well-posedness for the Landau equation with Coulomb potential. We present recent results on upper bounds for radially symmetric and monotone solutions. The estimates say that blow up in the 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 norm. 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 isotropic version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain. This is a joint work with Nestor Guillen.