On a monotone operator approach to Cahn-Hilliard equation with singular potentials

  • Helmut Abels (MPI MiS, Leipzig)
A3 01 (Sophus-Lie room)


We consider the instationary Cahn-Hilliard equation, which describes the phase separation of a two-component alloys. In contrast to most mathematical treatments, which deal with a smooth free energy, we consider a logarithmic or similar singular free energy. Such a free energy naturally occurs in physics and has the mathematical property that it ensures that the concentration difference stays in the physical reasonable regime. In contrast to known proofs for existence of unique weak solutions, which are based on a truncation of the singularity in the free energy, we present a direct approach via Lipschitz perturbations of monotone operators. Based on these results, we prove that a solution of the instationary system converges to a solution of the stationary as time goes to infinity. Finally, we discuss how this method can be used to solve Navier-Stokes-Cahn-Hilliard type equations arising in phase field models for two-phase flows of viscous, incompressible, immiscible fluids.