Multi–component Cahn–Hilliard Systems with Singular Potentials: Theoretical Results

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of N >= 2 components occupying a bounded region Omega subset of R-d, d <= 3. Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if d <= 2, we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d = 3, we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d = 3, this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.