Linear and Nonlinear Fractional PDEs from interacting particle systems

2024/07/02 by Cardoso, Pedro, Gonçalves, Patrícia
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2407.02246

Abstract

In these notes, we describe the strategy for the derivation of the hydrodynamic limit for a family of long range interacting particle systems of exclusion type with symmetric rates. For m ∈ ℕ:=\1, 2, …\ fixed, the hydrodynamic equation is ∂_t ρ(t,u)= [-(-Δ)^γ/2 ρ^m](t,u) . For m=1, this is the fractional equation, which is linear. On the other hand, for m ≥ 2, this is the fractional porous medium equation (which is nonlinear), obtained by choosing a rate which depends on the number of particles next to the initial and final position of a jump.

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