Global in Time Estimates for the Spatially Homogeneous Landau Equation with Soft Potentials

Abstract

This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials $\ga\in[-2,0)$. For the first result, we obtain the estimate of weak solutions in $L^{\alpha}_{t}L_{v}^{3-\eps}$ for $\alpha=\frac{2(3-\eps)}{3(2-\eps)}$ and $0<\eps<1$, which is an improvement over estimates by Fournier-Guerin [N. Fournier; H. Guerin, Well-posedness of the spatially homogeneous Landau equation for soft potentials. J. Funct. Anal. 25(2009), no. 8, 2542--2560]. Foe the second result, we have the estimate of weak solutions in $L_{t}^{\infty}L^{p}_{v}$, $p>1$, which extends part of results by Fournier-Guerin and Alexandre-Liao-Lin [R. Alexandre, J. Liao, and C. Lin, Some a priori estimates for the homogeneous Landau equation with soft potentials, arXiv:1302.1814]. As an application, we deduce some global well-posedness results for $\ga\in [-2,0)$. Our estimates include the critical case $\ga=-2$, which is the key point in this paper.