Turing instability in a model with two interacting Ising lines: hydrodynamic limit

This is the first of two articles on the study of a particle system model that exhibits a Turing instability type effect. The model is based on two discrete lines (or toruses) with Ising spins, that evolve according to a continuous time Markov process defined in terms of macroscopic Kac potentials and local interactions. For fixed time, we prove that the density fields weakly converge to the solution of a system of partial differential equations involving convolutions. The presence of local interactions results in the lack of propagation of chaos, reason why the hydrodynamic limit cannot be obtained from previous results.