In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

中文翻译：

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临界 Besov 空间中非齐次非截止 Kac 方程的柯西问题

在这项工作中，我们研究了空间非均匀非截止 Kac 方程的柯西问题。如果初始数据属于空间临界 Besov 空间，我们可以证明微扰框架下弱解的适定性。此外，还表明该解具有关于速度变量的 Gelfand-Shilov 正则化特性和关于位置变量的 Gevrey 正则化特性。与 [18] 中最近的结果相比，Gelfand-Shilov 正则性指数改进为最优。据我们所知，我们的工作是第一个对 Besov 空间中的动力学方程表现出平滑效应的工作。