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Modeling and analysis of a phase field system for damage and phase separation processes in solids
In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.
Infinite harmonic chain with heavy mass
Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to $1$, except for one heavy particle which has mass $M$. We investigate the macroscopic behaviour of such a system when $M$ is large, and time and space are scaled accordingly. As main result we derive a PDE for the light particles that is coupled with an ODE for the heavy particle.
Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations
A doubly nonlinear parabolic equation of the form [alpha](ut)-[delta]u+W'(u)=f, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function [alpha] and by the derivative W' of a smooth but possibly nonconvex potential W; f is a given known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for t>0. Such regularizing solutions contitute a semiflow S for which unqueness is satisfied for strictly positive times and we can study long time behaviour properties,. In particular, we can prove existence of both global and exponential attractors and investigate the structure of [omega]-limits of single trajectories.