WIAS Preprint No. 1941, (2014)

A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility


  • Frigeri, Sergio Pietro
  • Grasselli, Maurizio
  • Rocca, Elisabetta
    ORCID: 0000-0002-9930-907X

2010 Mathematics Subject Classification

  • 35Q30 37L30 45K05 76D03 76T99


  • Navier-Stokes equations, nonlocal Cahn-Hilliard equations, degenerate mobility, incompressible, binary fluids, weak solutions, global attractors




We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.

Appeared in

  • Nonlinearity, 28 (2015) pp. 1257--1293.

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