SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2081-2097, January 2020.
We consider weak solutions of the spatially inhomogeneous Landau equation with hard potentials ($\gamma \in (0,1]$), under the assumption that mass, energy, and entropy densities are under control. In this regime, with arbitrary initial data, we show that solutions satisfy pointwise Gaussian upper and lower bounds in the velocity variable. This is different from the behavior in the soft potentials case ($\gamma <0$), where Gaussian estimates are known not to hold without corresponding assumptions on the initial data. Our upper bounds imply that weak solutions are $C^\infty$ in all three variables, and that continuation of solutions is governed only by the mass, energy, and entropy.

中文翻译：

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具有硬势的非均匀Landau方程的高斯界

SIAM数学分析杂志，第52卷，第2期，第2081-2097页，2020年1月
。在假设的前提下，我们考虑具有硬势（$ \ gamma \ in（0,1] $）的空间不均匀Landau方程的弱解。质量，能量和熵密度都处于受控状态。在这种情况下，利用任意初始数据，我们证明解决方案满足了速度变量中点状高斯上下界的要求。这与软势情况下的行为（$ \ gamma <0 $），其中高斯估计值在没有初始数据的相应假设的情况下就不成立。我们的上限意味着弱解在所有三个变量中均为$ C ^ \ infty $，并且解的连续性仅受控制由质量，能量和熵决定。