We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed $ L^2 $ and $ L^\infty $ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.

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多项式衰减初始数据的玻尔兹曼方程的局部适定性

我们考虑具有速度变量的多项式衰减初始数据的空间不均匀非截止Boltzmann方程的柯西问题。我们通过在混合的$ L ^ 2 $和$ L ^ \ infty $空间中工作来建立任何初始数据在五阶Sobolev空间中的这种衰变的短时存在性，该空间可以补偿潜在的矩量生成并获得非常适合此空间的碰撞算子。我们的结果改善了在这种衰减状态下Boltzmann方程适用的参数范围，以及放宽了对初始规律性的限制。作为一种应用程序，我们可以将我们的存在结果与Imbert-Silvestre的最新条件正则性估计值结合起来（arXiv：1909.12729 [math.AP]），以得出只要质量可以继续求解的结论，能量和熵密度仍处于控制之中。该延续准则以前仅在多项式衰减初始数据的先前适定性结果的参数限制范围内可用。