Sergio Frigeri

Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System

The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.

On Nonlocal Cahn–Hilliard–Navier–Stokes Systems in Two Dimensions

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.

On a diffuse interface model of tumour growth

We consider a diffuse interface model of tumor growth proposed by A.~Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction

Optimal Distributed Control of a Nonlocal Cahn--Hilliard/Navier--Stokes System in Two Dimensions

\varphi

A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility

nonlinearly coupled with a reaction-diffusion equation for

Trajectory attractors for the Sun?Liu model for nematic liquid crystals in 3D

\psi

Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential

, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function

On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities

p(\varphi)

Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

multiplied by the differences of the chemical potentials for

Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials

\varphi