As a first step towards the general global-in-time stability for the Boltzmann equation with soft potentials, in the present work, we prove the quantitative lower bounds for the equation under the following two assumptions, which stem from the available energy estimates, i.e. (ⅰ). the hydrodynamic quantities (local mass, local energy, and local entropy density) are bounded (from below or from above) uniformly in time, (ⅱ). the Sobolev regularity for the solution grows tempered with time.

中文翻译：

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玻尔兹曼方程的下界，其规律性随时间而变缓

作为具有软势的玻尔兹曼方程的一般全局时间稳定性的第一步，在目前的工作中，我们在以下两个假设下证明了方程的定量下界，这些假设源于可用的能量估计，即(ⅰ)。流体力学量（局部质量、局部能量和局部熵密度）在时间上（从下或从上）均匀地有界，(ⅱ)。解的 Sobolev 规律性随着时间的推移而增长。