WIAS Preprint No. 1932, (2014)

On regularity, positivity and long-time behavior of solutions to an evolution system of nonlocally interacting particles



Authors

  • Griepentrog, Jens André

2010 Mathematics Subject Classification

  • 35K51 35R09 47J35 35B65 35B09 35B40

Keywords

  • Nonlocal Cahn-Hilliard equations, nonconvex functionals, Sobolev-Morrey spaces, regularity theory, Łojasiewicz-Simon gradient inequality, asymptotic behavior

Abstract

An analytical model for multicomponent systems of nonlocally interacting particles is presented. Its derivation is based on the principle of minimization of free energy under the constraint of conservation of particle number and justified by methods established in statistical mechanics. In contrast to the classical Cahn-Hilliard theory with higher order terms, the nonlocal theory leads to an evolution system of second order parabolic equations for the particle densities, weakly 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 suitable Hilbert spaces for evolution systems. Moreover, using maximal regularity for nonsmooth parabolic boundary value problems in Sobolev-Morrey spaces and comparison principles, we show uniqueness, global regularity and uniform positivity of solutions under minimal assumptions on the regularity of interaction. Applying a refined version of the Łojasiewicz-Simon gradient inequality, this paves the way to the convergence of solutions to equilibrium states. We conclude our considerations with the presentation of simulation results for a phase separation process in ternary systems.

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