| Abstract: |
| In the talk we shall present the last results on some diffuse-interface models for flow and phase separation of binary
fluids which are based on the coupling of the Navier-Stokes equations with the nonlocal Cahn-Hilliard equation.
The nonlocal Cahn-Hilliard/Navier-Stokes system has been studied analytically in a series
of papers (cf. \cite{CFG,FG1,FG2,FGK,FGR,FGG,FRS}).
The attention will be focused on the more recent results concerning
the case where the two densities are different (weak solutions, cf.\cite{F}),
where the mobility is degenerate and the potential is singular (regularity and optimal control, cf.
\cite{FGGS} and \cite{FGS}), and where the
two-phase fluid is non-Newtonian (weak solutions and uniqueness, cf. \cite{FGP}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{11}
\bibitem{CFG} P. Colli, S. Frigeri and M. Grasselli, {\itshape Global existence of weak solutions to
a nonlocal Cahn-Hilliard-Navier-Stokes system}, J. Math. Anal. Appl. \textbf{386} (2012), 428-444.
\bibitem{F}
S. Frigeri, {\itshape Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities}, Math. Models Methods Appl. Sci. \textbf{26} (2016), 1955-1993.
\bibitem{FGG} S. Frigeri, C. Gal and M. Grasselli, {\itshape On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions}, J. Nonlinear Sci. \textbf{26} (2016), 847-893.
\bibitem{FGGS} S. Frigeri S, C.G. Gal, M. Grasselli and J. Sprekels, {\itshape Strong solutions to nonlocal 2D Cahn-Hilliard-Navier-Stokes systems with nonconstant viscosity}, WIAS Preprint 2309 (2016), 1-58 (submitted).
\bibitem{FG1} S. Frigeri and M. Grasselli, {\it Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system}, J. Dynam Differential Equations \textbf{24} (2012), 827-856.
\bibitem{FG2} S. Frigeri and M. Grasselli, {\it Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials}, Dyn. Partial Differ. Equ. \textbf{9} (2012), 273-304.
\bibitem{FGK} S. Frigeri, M. Grasselli and P. Krej\v{c}\`{i}, {\itshape Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems}, J. Differential Equations \textbf{255} (2013), 2587-2614.
\bibitem{FGP} S. Frigeri, M. Grasselli and D. Pra\v{z}\`{a}k, {\itshape Nonlocal Cahn-Hilliard-Navier-Stokes systems with shear dependent viscosity}, J. Math. Anal. Appl. \textbf{459} (2018), 753-777.
\bibitem{FGR} S. Frigeri, M. Grasselli and E. Rocca, {\itshape A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility}, Nonlinearity \textbf{28} (2015), 1257-1293.
\bibitem{FGS} S. Frigeri, M. Grasselli and J. Sprekels, {\itshape Optimal distributed control of two-dimensional nonlocal Cahn--Hilliard--Navier--Stokes systems with degenerate mobility and singular potential}, WIAS Preprint 2473 (2018), 1-32 (submitted).
\bibitem{FRS} S. Frigeri, E. Rocca and J. Sprekels, {\itshape Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes system in 2D}, SIAM J. Control Optim. \textbf{54} (2016), 221-250. |
|