Antonio Segatti

ON THE HYPERBOLIC RELAXATION OF THE CAHN–HILLIARD EQUATION IN 3D: APPROXIMATION AND LONG TIME BEHAVIOUR

In this paper we consider the hyperbolic relaxation of the Cahn–Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain Ω of ℝ3. This equation is characterized by the presence of the additional inertial term ∊utt that accounts for the relaxation of the diffusion flux. For this equation we address the problem of the long time stability from the point of view of global attractors. The main difficulty in dealing with this system is the low regularity of its weak solutions, which prevents us from proving a uniqueness result and a proper energy identity for the solutions. We overcome this difficulty by using a density argument based on a Faedo–Galerkin approximation scheme and the recent J. M. Balls theory of generalized semiflows. Moreover, we address the problem of the approximation of the attractor of the continuous problem with the one of Faedo–Galerkin scheme. Finally, we show that the same type of results hold also for the damped semilinear wave equation when the nonlinearity ϕ is not Lipschitz continuous and has a super critical growth.

Equilibrium configurations of nematic liquid crystals on a torus.

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy and how new configurations emerge. In particular, our analysis predicts the existence of stable equilibria with complex windings.

Analysis of a variational model for nematic shells

We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the gradient flow of the energy; iii) in the case of a parametrized axisymmetric torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

Finite Dimensional Reduction and Convergence to Equilibrium for Incompressible Smectic-A Liquid Crystal Flows

We consider a hydrodynamic system that models smectic-A liquid crystal flow. The model consists of the Navier–Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in two dimensions, the problem possesses a global attractor

A gradient flow approach to the porous medium equation with fractional pressure

\mathcal{A}

On a singular heat equation with dynamic boundary conditions

in a certain phase space. Then we establish the existence of an exponential attractor

Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential

\mathcal{M}

Error Estimates for a Variable Time-Step Discretization of a Phase Transition Model with Hyperbolic Momentum

, which entails that the global attractor

Variational Analysis of Nematic Shells

\mathcal{A}

Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach

has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium by means of a suitable Łojasiewicz–Simon inequality. Corresponding results in three dimensions are also discussed.