## Global in Time Estimates for the Spatially Homogeneous Landau Equation for Soft Potentials    [PDF]

Kung-Chien Wu
This paper deals with some global in time estimates of the spatially homogeneous Landau equation for soft potentials. Assume that $\ga\in[-2,0)$, 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]. For $\ga=-2$, we can apply the entropy production estimate to get the estimate of weak solutions in $L_{t}^{\infty}L^{2}_{v}$, which extends the result by 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].
View original: http://arxiv.org/abs/1306.1220