Global existence and uniqueness of solutions to a nonlocal phase-field system

摘要

From a microscopic description of a two phase material with long range interaction (for example an Ising system with Kac potential), by taking the continuum limit one may arrive at the following equation for the evolution of the order parameter, PHI : (1.1) PHI _t chemical bounds J * PHI - PHI + f ( PHI ) + l theta _0, Where J is a nonnegative convolution kernel with unit integral arising from the pairwise interaction within the lattice, f is a bistable nonlinearity, theta _0 represents (constant) temperature, and l is a latent heat conefficient. Various parameters are set to be unity either by rescaling or for simplicity. For a derivation of (1.1) and its variants and for its analysis, we refer to (DOPT), (KS), (S), (BR), (BFGJ), and (BFRW). We note that (1.1) can be viewed as an isothermal gradient flow in L~2 of the energy functional (1.2) E_1 ( PHI ) chemical bounds integral from to x=R~d integral-(R~d) (1/4 integral-(R~d) J(x-y)( PHI (x) - PHI (y))~2!@dy - F ( PHI ) - l theta _0 PHI ) dx, where F'chemical bounds f. We take f to have stable zeros at PHI chemical bounds +- 1, these represents the two pure phases.