A weak formulation of free boundary problems and its application to hydrodynamic limits of particle systems with selection

Abstract

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.