2016, Frigeri, Sergio, Gal, Ciprian G., Grasselli, Maurizio, Sprekels, Jürgen

We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.

2014, Frigeri, Sergio, Gal, Cipian G., Grasselli, Maurizio

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 strong-weak uniqueness in the case of viscosity depending on the order parameter, provided that the mobility is constant and the potential is regular. 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.

2010, Ehrt, Julia

This article investigates the properties of the global attractor of hyperbolic balance laws on the circle, given by : u_t+f(u)_x=g(u). The new tool of sub-attractors is introduced. They contain all solutions on the global attractor up to a given number of zeros. The article proves finite dimensionality of all sub-attractors, provides a full parameterization of all sub-attractors and derives a system of ODEs for the embedding parameters that describes the full PDE dynamics on the sub-attractor

2014, Frigeri, Sergio, Grasselli, Maurizio, Rocca, Elisabetta

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.

2014, Frigeri, Sergio, Grasselli, Maurizio, Rocca, Elisabetta

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 φ nonlinearly coupled with a reaction-diffusion equation for ψ which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation functionp(φ) multiplied by the differences of the chemical potentials for φ and ψ. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of φ+ψ. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

2006, Pata, Vittorino, Zelik, Sergey

In this note, we establish a general result on the existence of global attractors for semigroups S(t) of operators acting on a Banach space X, where the strong continuity S(t) [Elemenz von] C(X,X) is replaced by the much weaker requirement that S(t) be a closed map.