Elisabetta Rocca

ANALYSIS OF A PHASE-FIELD MODEL FOR TWO-PHASE COMPRESSIBLE FLUIDS

A model describing the evolution of a binary mixture of compressible, viscous, and macroscopically immiscible fluids is investigated. The existence of global-in-time weak solutions for the resulting system coupling the compressible Navier–Stokes equations governing the motion of the mixture with the Allen–Cahn equation for the order parameter is proved without any restriction on the size of initial data.

A new approach to non-isothermal models for nematic liquid crystals

We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical thermodynamics. The resulting system of equations captures all essential features of physically relevant models; in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data.

On a non-isothermal model for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature , the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier–Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field d, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg–Landau type, with a suitable penalization term to relax the constraint |d| = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fouriers law, depending also on the director field d. The proposed model is shown to be compatible with first and second laws of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.

A degenerating PDE system for phase transitions and damage

In this paper, we analyze a PDE system arising in the modeling of phase transition and damage phenomena in thermoviscoelastic materials. The resulting evolution equations in the unknowns ϑ (absolute temperature), u (displacement), and χ (phase/damage parameter) are strongly nonlinearly coupled. Moreover, the momentum equation for u contains χ-dependent elliptic operators, which degenerate at the pure phases (corresponding to the values χ = 0 and χ = 1), making the whole system degenerate. That is why, we have to resort to a suitable weak solvability notion for the analysis of the problem: it consists of the weak formulations of the heat and momentum equation, and, for the phase/damage parameter χ, of a generalization of the principle of virtual powers, partially mutuated from the theory of rate-independent damage processes. To prove an existence result for this weak formulation, an approximating problem is introduced, where the elliptic degeneracy of the displacement equation is ruled out: in the framework of damage models, this corresponds to allowing for partial damage only. For such an approximate system, global-in-time existence and well-posedness results are established in various cases. Then, the passage to the limit to the degenerate system is performed via suitable variational techniques.

Optimal Distributed Control of a Nonlocal Convective Cahn--Hilliard Equation by the Velocity in Three Dimensions

In this paper we study a distributed optimal control problem for a nonlocal convective Cahn--Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are, however, quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this paper, we employ recently proved existence, uniqueness, and regularity results f...

``Entropic” Solutions to a Thermodynamically Consistent PDE System for Phase Transitions and Damage

In this paper we analyze a PDE system modeling (nonisothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in

On a diffuse interface model of tumour growth

L^1

WELL-POSEDNESS OF A PHASE TRANSITION MODEL WITH THE POSSIBILITY OF VOIDS

. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as “entropic,” where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics, as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time-discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its...

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

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

A nonlocal phase-field model with nonconstant specific heat †

\varphi