Abstract
We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier–Stokes equations coupled with a convective nonlocal Cahn–Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak–strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential.
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Acknowledgments
The authors thank the reviewers for their remarks and suggestions. The first author was supported by FP7-IDEAS-ERC-StG Grant \( \sharp \)256872 (EntroPhase). The first and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Communicated by Paul Newton.
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Frigeri, S., Gal, C.G. & Grasselli, M. On Nonlocal Cahn–Hilliard–Navier–Stokes Systems in Two Dimensions. J Nonlinear Sci 26, 847–893 (2016). https://doi.org/10.1007/s00332-016-9292-y
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DOI: https://doi.org/10.1007/s00332-016-9292-y
Keywords
- Incompressible binary fluids
- Navier–Stokes equations
- Nonlocal Cahn–Hilliard equations
- Weak solutions
- Uniqueness
- Strong solutions
- Global attractors
- Exponential attractors