发表机构
Yonsei University（延世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文针对平板中具有物理碰撞频率的BGK方程，在仅假设有限质量、动能和熵的初始数据下，通过补偿估计和熵重整化方法，首次建立了全局大数据存在性理论。
AI 中文摘要
我们建立了平板中具有由局部密度给出的物理碰撞频率的Bhatnagar-Gross-Krook方程的全局大数据存在性理论。我们考虑流入和Maxwell边界条件，并且仅假设初始数据的有限质量、动能和熵，以及相应的流入通量界。数据不需要是小量或逐点有界的，我们也不对密度或温度施加正下界，也不规定高于二阶的速度矩。据我们所知，这是在该假设下该初边值问题的第一个非微扰全局存在性结果。核心障碍是未重整化碰撞算子的能量紧致性和可积性的同时丧失。仅有限能量不能给出第二速度矩的一致可积性，而密度依赖的碰撞频率阻止了增益和损失项的分别局部$L^1$控制。我们通过补偿估计（对碰撞频率截断一致）、由数据确定的de la Vallée Poussin权重以及使用熵产生的两次连续重整化来克服这些困难。这产生了宏观能量、局部Maxwellian和非线性碰撞项的紧致性，从而得到全局重整化解。
英文摘要
We establish a global large-data existence theory for the Bhatnagar--Gross--Krook equation in a slab with the physical collision frequency given by the local density. We consider both inflow and Maxwell boundary conditions and assume only finite mass, kinetic energy, and entropy of the initial data, together with the corresponding inflow flux bounds. The data need not be small or pointwise bounded, and we impose neither positive lower bounds on the density or temperature nor prescribed velocity moments above order two. To our knowledge, this is the first nonperturbative global existence result for this initial and boundary value problem under these assumptions. The central obstacle is the simultaneous loss of energy compactness and integrability of the unrenormalized collision operator. Finite energy alone does not give uniform integrability of the second velocity moment, while the density dependent collision frequency prevents separate local $L^1$ control of the gain and loss terms. We overcome these difficulties through a compensated estimate uniform in the truncation of the collision frequency, a de la Vallée Poussin weight determined by the data, and two successive renormalizations using entropy production. This yields the compactness of the macroscopic energy, local Maxwellians, and nonlinear collision term, and hence global renormalized solutions.
