Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity

Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic lattice; in the balance equations of microforces and microenergy, the two unknowns are the order parameter and the chemical potential. A simpler version of the same system has recently been discussed in arXiv:1103.4585v1. In this paper, a fairly more general phase-field equation is coupled with a genuinely nonlinear diffusion equation. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of a constant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.Comment: Key words: phase-field model, nonlinear laws, existence of solutions, new uniqueness proo

The EU-Mercosur agreement: towards integrated climate policy? Egmont European Policy Brief No. 57 November 2019

The recently signed EU-Mercosur agreement has met with criticism from civil society, farmers and politicians around the EU. These criticisms have been amplified by recent forest fires in the Amazon. Although the Von der Leyen Commission’s strategic documents highlight the importance of mainstreaming climate change and environment throughout all policies, including trade, the EU-Mercosur agreement lacks enforceable measures to this end. In light of recent events, ratification of the EU-Mercosur agreement by all member states seems unlikely. However, the EU itself could also use this opportunity to send a clear message as to where its priorities lie by taking unified action to shift the terms of the trade agreement

Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems

An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all of these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved

Global existence for a singular phase field system related to a sliding mode control problem

In the present contribution we consider a singular phase field system located in a smooth and bounded three-dimensional domain. The entropy balance equation is perturbed by a logarithmic nonlinearity and by the presence of an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. The second equation of the system accounts for the phase dynamics, and it is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear; the main difficulties lie in the contemporary presence of two nonlinearities, one of which under time derivative, in the entropy balance equation. Consequently, we are able to prove only the existence of solutions. To this aim, we will introduce a backward finite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step.Comment: Key words: Phase field system; maximal monotone nonlinearities; nonlocal terms; initial and boundary value problem; existence of solution

Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary. A very general class of double-well like potentials is allowed. Moreover, some further regularity is obtained to guarantee the strong solution

Global existence for a phase separation system deduced from the entropy balance

This paper is concerned with a thermomechanical model describing phase separation phenomena in terms of the entropy balance and equilibrium equations for the microforces. The related system is highly nonlinear and admits singular potentials in the phase equation. Both the viscous and the non-viscous cases are considered in the Cahn--Hilliard relations characterizing the phase dynamics. The entropy balance is written in terms of the absolute temperature and of its logarithm, appearing under time derivative. The initial and boundary value problem is considered for the system of partial differential equations. The existence of a global solution is proved via some approximations involving Yosida regularizations and a suitable time discretization

The end of ‘business as usual’? COVID-19 and the European Green Deal. Egmont European Policy Brief No. 60 May 2020

The COVID-19 pandemic has had a clear and drastic effect on our daily lives and political priorities. But what implications does it have for the EU’s climate action and the Von der Leyen Commission’s flagship policy, the European Green Deal? The crisis may be a ‘make or break’ moment for the EU to act on climate change through its recovery plan

The EU’s Just Transition: three challenges and how to overcome them. Egmont European Policy Brief No. 59 March 2020

The EU’s ‘Just Transition Mechanism’ proposal has become highly contentious, bringing up issues of redistribution between countries. It faces three main challenges: overcoming a focus on national allocations; expanding the transition from energy to other sectors; and including the private sector and civil society in the transition. By effectively mainstreaming the idea of a just transition, the Commission can ensure that the current proposal not only becomes less sensitive, but also more effectively supports a fair shift to a zero-carbon society