Global in time estimates for the spatially homogeneous Landau equation with soft potentials

This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials γ∈[−2,0). For the first result, we obtain the estimate of weak solutions in LtαLv3−ε for α=2(3−ε)3(2−ε) and 0<ε<1, which is an improvement over estimates by Fournier and Guerin [10]. For the second result, we have the estimate of weak solutions in Lt∞Lvp, p>1, which extends part of results by Fournier and Guerin [10] and Alexandre, Liao and Lin [1]. As an application, we deduce some global well-posedness results for γ∈[−2,0). Our estimates include the case γ=−2, which is the key point in this paper.