论文标题
自由边界问题的薄弱表述及其应用于粒子系统的流体动力限制
A weak formulation of free boundary problems and its application to hydrodynamic limits of particle systems with selection
论文作者
论文摘要
提出了一类抛物线自由边界问题（FBP）的弱公式，它不涉及自由边界的概念，而是在存在经典解决方案时会减少到FBP。它针对粒子系统的流体动力极限（HDL），在宏观模型不具备（或很难证明具有）经典意义上的常规自由边界的情况下。该配方涉及颗粒的宏观密度和一个占选择的度量。它由密度满足的二阶抛物线方程组成，并由度量驱动，并与密度测量对满足的互补条件相结合。该方法适用于在任意变化的注入和去除速率下，在$ \ r $上进行扩散的注入分支隔离粒子系统，对此，相应的FBP通常不可能在经典上解决。 HDL的特征是弱公式的独特解决方案。收敛的证明是基于PDE唯一性，这反过来依赖于屏障方法。
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.
