The gradient flow of the double well potential and its appearance in interacting particle systems

Abstract

In this work we are interested in the existence of solutions to parabolic partial differential equations associated to gradient flows which involve the so-called double well potential, which is a nonconvex and nonconcave functional. Therefore the formal L²-gradient flow of the double well potential leads to a so-called forward-backward parabolic equation, which is not well-posed: it may fail to admit local in time classical solutions, at least for a large class of initial data.<br /> We discretize this forward-backward parabolic equation in space and prove convergence of the scheme for a suitable class of initial data. Moreover we identify the limit equation and characterize the long-time behavior of the limit solutions.<br /> Then we view such discrete-in-space schemes as systems of particles driven by the double-well potential and add a perturbation by independent Brownian motions to their dynamics. We describe the behaviour of a particle system with long-range interactions, in which the range of interactions is allowed to depend on the size of the system. We give conditions on the interaction strength under which the scaling limit of the particle system is a well-posed stochastic PDE and characterize the long-time behavior of this stochastic PDE.