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.