自由边界问题的弱形式及其在带选择的粒子系统流体动力学极限中的应用
A weak formulation of free boundary problems and its application to hydrodynamic limits of particle systems with selection
AI总结：
本文提出了一类抛物型自由边界问题的弱形式，并将其应用于带选择的粒子系统的流体动力学极限，证明了在经典自由边界不存在时该弱形式解的唯一性与收敛性。
摘要：
本文提出了一类抛物型自由边界问题（FBP）的弱形式，该形式不涉及自由边界的概念，但在存在经典解时退化为 FBP。它旨在用于带选择的粒子系统的流体动力学极限（HDL），适用于宏观模型不具备（或难以证明具备）经典意义下正则自由边界的情况。该形式涉及粒子的宏观密度和一个用于表示选择的测度。它由密度满足的由该测度驱动的二阶抛物型方程，以及密度-测度对满足的互补条件组成。该方法被应用于在 $\R$ 上的扩散注入-分支-选择粒子系统，其注入和移除率可任意变化，而对应的 FBP 通常无法已知是否经典可解。HDL 被刻画为该弱形式的唯一解。收敛性证明基于 PDE 唯一性，而后者依赖于障碍方法。
英文摘要：
A weak formulation for a class of parabolic free boundary problems (FBP) is proposed that does not involve the notion of a free boundary but reduces to a FBP when classical solutions exist. It is aimed at hydrodynamic limits (HDL) of particle systems with selection in circumstances where the macroscopic model does not possess (or is hard to prove to possess) a regular free boundary in the classical sense. The formulation involves the macroscopic density of particles and a measure that accounts for selection. It consists of a second order parabolic equation satisfied by the density and driven by the measure, coupled with a complementarity condition satisfied by the density-measure pair. The approach is applied to an injection-branching-selection particle system of diffusion on $\R$ under arbitrarily varying injection and removal rates, for which the corresponding FBP is not in general known to be classically solvable. The HDL is characterized as the unique solution to the weak formulation. The proof of convergence is based on PDE uniqueness, which in turn relies on the barrier method.
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