本文研究硬势情形（包括Maxwell分子）下非齐次Landau方程大初值经典解的局部存在性与唯一性。此前Chaturvedi[SIAM J. Math. Anal., 55(5), 5345--5385, 2023]针对指数加权$H^{10}$空间中的初值构造了解，但先验上并不清楚这些解是否具有比初值更高的正则性。我们从两方面改进了Chaturvedi的存在性结果：我们的解在正时刻是$C^\infty$的，且允许初值属于次指数加权$L^\infty$空间，代价是需要初值满足一个温和的正性条件。\n为证明唯一性，我们对初值提出了更强的假设：Hölder连续性以及无真空区域。这些假设与此前软势情形下唯一性研究所需的假设相同。\n在证明存在性与唯一性的过程中，我们建立了一些此前仅在软势情形下已知的有用结果，包括正性的传播和Hölder连续性的传播。由于速度矩损失更为严重，软势情形下的许多证明策略并不适用于此。
英文摘要：
This paper addresses large-data local existence and uniqueness of classical solutions to the inhomogeneous Landau equation in the hard potentials case (including Maxwell molecules). Solutions have previously been constructed by Chaturvedi [SIAM J. Math. Anal., 55(5), 5345--5385, 2023] for initial data in an exponentially-weighted $H^{10}$ space, but it is not a priori clear whether these solutions have more regularity than the initial data. We improve Chaturvedi's existence result in two ways: our solutions are $C^\infty$ for positive times, and we allow initial data in a sub-exponentially-weighted $L^\infty$ space, at the cost of requiring a mild positivity condition at time zero.
To prove uniqueness, we require stronger assumptions on the initial data: Hölder continuity and the absence of vacuum regions. These are the same assumptions that are required for uniqueness in prior work on the soft potentials case.
Along the way to proving existence and uniqueness, we establish some useful results that were previously only known in the case of soft potentials, including spreading of positivity and propagation of Hölder continuity. Many of the proof strategies from the soft potentials case do not apply here because of the more severe loss of velocity moments.
Stability of vacuum for the Landau equation with hard potentials
本文考虑了全空间$γ\in[0,1)$上的具有Maxwell势和硬势的空间非齐次Landau方程。我们证明了如果初始数据$f_{\text{in}}$在适当的加权范数下接近真空解$f_{\text{vac}}=0$，则解$f$在时间上全局存在.本文的工作是在作者早期关于具有硬势的Landau方程解的局部存在性的工作的基础上进行的。我们的证明使用了$L^2$估计，并利用了Luk建立的零结构[Stability of vacuum for the Landau equation with moderately soft potentials，Annals of PDE（2019）5：11]。为了能够接近我们的估计，我们必须将作者在以前的论文[Local existence for the Landau equation with hard potentials，arXiv：1910.11866]中建立的加权能量估计与零结构相结合，并设计新的加权范数。
