论文标题
具有不规则初始数据的Boltzmann方程的经典解决方案
Classical solutions of the Boltzmann equation with irregular initial data
论文作者
论文摘要
本文考虑了空间不均匀的非切割玻尔兹曼方程。我们在速度变量中构建了一个大型DATA经典解决方案，具有有界的，可测量的初始数据，具有均匀的多项式衰变。我们的结果不需要对初始数据进行严格阳性的假设，除非在相空间中的某些小球中进行局部。当我们对初始数据的衰减和阳性假设放松时，我们还获得了弱解决方案的存在结果。由于我们的解决方案的规律性可能会退化为$ t \ rightarrow 0 $，因此独特性是一个艰巨的问题。我们在额外的假设中建立了弱量的唯一性，即初始数据没有真空区域，并且是Hölder的连续。作为我们短期存在定理的应用，我们证明了在有限的速度多项式速率下衰减的有限的，可测量的初始数据的全局存在。
This article considers the spatially inhomogeneous, non-cutoff Boltzmann equation. We construct a large-data classical solution given bounded, measurable initial data with uniform polynomial decay of mild order in the velocity variable. Our result requires no assumption of strict positivity for the initial data, except locally in some small ball in phase space. We also obtain existence results for weak solutions when our decay and positivity assumptions for the initial data are relaxed. Because the regularity of our solutions may degenerate as $t \rightarrow 0$, uniqueness is a challenging issue. We establish weak-strong uniqueness under the additional assumption that the initial data possesses no vacuum regions and is Hölder continuous. As an application of our short-time existence theorem, we prove global existence near equilibrium for bounded, measurable initial data that decays at a finite polynomial rate in velocity.
