具有软位势的空间齐次Landau方程的整体时间估计
Global in Time Estimates for the Spatially Homogeneous Landau Equation with Soft Potentials
摘要：
本文讨论了软势空间齐次Landau方程$\ga\in[-2,0)$的时间上的整体先验估计。对于第一个结果，我们得到了$α=\frac{2(3-\eps)}{3(2-\eps)}$和$0<\eps<1$在$L^α_{t}L_{v}^{3-\eps}$中弱解的估计，这是对Fournier-Guerin [N. Fournier; H. Guerin，软势空间齐次朗道方程的适定性。J. Funct. Anal. 25（2009），no. 8，2542- 2560]。对于第二个结果，我们得到了$L_{t}^{\infty}L^{p}_{v}$，$p>1$中弱解的估计，推广了Fournier-Guerin和Alexandre-Liao-Lin [R. Alexandre，J. Liao，and C.林，软势齐次朗道方程的一些先验估计，arXiv：1302.1814]。作为应用，我们推导了$\ga\in [-2,0)$的一些整体适定性结果。我们的估计包含了临界情形$\ga=-2$，这是本文的重点.
英文摘要：
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 $L^α_{t}L_{v}^{3-\eps}$ for $α=\frac{2(3-\eps)}{3(2-\eps)}$ 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.
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