For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in the whole space \({\mathbb {R}}^3_x\), we establish pointwise lower bounds that appear instantaneously even if the initial data contains vacuum regions. Our lower bounds depend only on the initial data and upper bounds for the mass and energy densities of the solution. As an application, we improve the weakest known continuation criterion for large-data solutions, by removing the assumptions of mass bounded below and entropy bounded above.

中文翻译：

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自生下界和玻尔兹曼方程的延续

对于摆在整个空间\（{{mathbb {R}} ^ 3_x \）中的空间不均匀，非截断的Boltzmann方程，即使初始数据包含真空区域，我们也建立了瞬时出现的点下界。我们的下限仅取决于初始数据，上限取决于解决方案的质量和能量密度。作为一种应用，我们通过消除质量限制在下面和熵限制在上面的假设，改进了大数据解决方案的最弱的已知延续准则。