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

Abstract

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of Nge 2 components occupying a bounded region Omega subset mathbb {R}^{d},dle 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 dle 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.