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

TL;DR

This paper establishes new global in time a priori estimates for the spatially homogeneous Landau equation with soft potentials, extending previous results and including the critical case, leading to global well-posedness.

Contribution

It provides the first estimate of weak solutions in certain Lebesgue spaces for soft potentials, improving and extending prior work, and includes the critical case for the first time.

Findings

Derived $L^{rac{2(3- ext{eps})}{3(2- ext{eps})}}_t L^{3- ext{eps}}_v$ estimates for weak solutions.

Extended $L^{ ext{infinity}}_t L^{p}_v$ estimates for weak solutions with $p>1$.

Achieved global well-posedness results for the Landau equation with soft potentials including the critical case $ ext{gamma}=-2$.

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.