On the unique solvability of a nonlocal phase separation problem for multicomponent systems
- Griepentrog, Jens André
2010 Mathematics Subject Classification
- 35K45 45K05 47J35 35D10
- Nonlocal phase separation models; Cahn--Hilliard equation; Initial boundary value problems; Nonlinear evolution equations;Regularity theory
A nonlocal model of phase separation in multicomponent systems is presented. It is derived from conservation principles and minimization of free energy containing a nonlocal part due to particle interaction. In contrast to the classical Cahn--Hilliard theory with higher order terms this leads to an evolution system of second order parabolic equations for the particle densities, coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction potential differences. Applying fixed-point arguments and comparison principles we prove the existence of variational solutions in standard Hilbert spaces for evolution systems. Moreover, using some regularity theory for parabolic boundary value problems in Hoelder spaces we get the unique solvability of our problem. We conclude our considerations with the presentation of simulation results for a ternary system.
- Banach Center Publ., 66 (2004) pp. 153-164.