Conference Speakers & Presentations
Professor Alain Miranville of the University of Poitiers (France) will deliver the series of ten main lectures. Professor Miranville is an international expert in Cahn-Hilliard type equations and applications and a world renowned expositor and lecturer. His lectures will take the participants on a journey through the subject beginning with the derivation of the equation and associated boundary conditions, through the mathematical analysis of the problems, numerical methods, and ending with cutting edge applications of Cahn-HIlliard type equations in fluid dynamics, image inpainting, and tumor growth.
The main lectures were supported by four additional speakers and panel discussions that heavily emphasize open research problems across multiple scientific fields.
Ciprian Gal (Florida International University)
“Doubly nonlocal Cahn-Hilliard equations”
Abstract: The Cahn-Hilliard equation was proposed in the late 1950’s and has become nowadays central in understanding phase transition phenomena in many complex materials. The equation aims to describe the process of phase separation, by which the two components of a binary material spontaneously separate and form domains that are pure in each material component. After we briefly revisit much of the history behind the classical form of the Cahn-Hilliard equation we move onto the modern approach which ultimately gives a generalized form of the Cahn-Hilliard equation that can be applied in more general situations (for instance, when the phase separation takes place in a heterogeneous environment). The latter equation reduces to the classical form under certain conditions or assumptions. Interesting mathematics is to be discovered in this new setting and surprisingly a better understanding of the classical form may be also accomplished within this setting.
Elisabetta Rocca (University of Pavia)
“Recent topics in the modeling and analysis of diffuse interface tumor growth”
Abstract: In this talk we would like to address some recent research issues on the modeling and analysis of diffuse interface tumor growth problems. In particular we will address some relevant questions regarding the well-posedness of two-phase and multi-phase variants models of tumor growth described by the evolution of a tumor phase (possibly vectorial) parameter, a nutrient proportion and, in some cases, the velocity field. The variables satisfy initial-boundary value problem for coupled Cahn-Hilliard/Reaction-Diffusion/Darcy type equations. We will moreover discuss about related optimal control problems and we will report about recent results on the long-time behavior of solutions.
Roger Temam (Indiana University)
“Uniqueness and Regularity of a Diffuse Interface Model for Binary Fluid Flows”
Abstract: The flow of two contiguous fluids gives rise to topological transitions at interfaces. A typical example is the break-up or coalescence of droplets. As opposed to sharp interface models, diffuse interface methods provide a different way to describe such deformations and transitions. The interface between the two fluids is herein assumed to be a narrow layer with finite thickness. Molecular force interactions between the fluids are described by the gradient theory. The energy of the non-uniform mixture is the sum of a free energy density and a gradient term. The latter one induces Korteweg reactive stresses, which become surface stress in the zero thickness limit. In this context, the evolution of two incompressible and viscous fluids is described by the Navier-Stokes-Cahn-Hilliard system, also called model H. In this talk I will present some recent results of uniqueness and regularity of weak and strong solutions. This is a joint work with A. Giorgini and A. Miranville.
(Unfortunately Temam was unable to attend the conference. A. Giorgini delivered the presentation in his absence.)
Steven Wise (University of Tennessee, Knoxville)
"Convergence Proof of a Generalized FAS Multigrid Solver for Nonlinear Problems"
Abstract: The Full Approximation Storage (FAS) scheme is a widely used multigrid method for nonlinear problems. In this talk, we shall provide a new framework to analyze generalized FAS algorithms for convex optimization problems, showing an improvement over the original method. We view FAS as an inexact version of nonlinear multigrid methods based on space decomposition and subspace correction. The local problem in each subspace can be simplified to be linear, and one gradient decent iteration is enough to ensure a linear convergence. We will discuss how to use this framework to solve the 4th-order stationary Can-Hilliard equation, with possibly singular potentials. This is joint work with Long Chen (University of California, Irvine) and Xiaozhe Hu (Tuffs University).