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Lectures 2010
Franco FLANDOLI Random perturbation of PDEs and fluid dynamic models Some linear and nonlinear Partial Differential Equations related to fluid dynamics lack uniqueness of solutions or present blow-up phenomena. The simplest and more explicit example is the linear transport equation, or the dual continuity equation. More difficult examples are Navier-Stokes, Euler equations and systems of conservation laws. Interesting phenomena can also be observed for simplified models, like certain infinite systems of nonlinear ordinary equations (dyadic and shell models of turbulence). The aim of these lectures is to investigate whether random perturbations restore uniqueness and prevent blow-up, as it happens for certain classes of ODEs. When the answer is positive, we would like to know what happens in the zero-noise limit. The best framework to investigate these problems is the case when the PDE has some flow of ODEs in the background. For linear transport equations and few examples of nonlinear dynamics it is possible to answer a number of questions. Certain multiplicative random perturbations of dyadic models can also be studied in great detail. For more difficult PDEs the state of the art is less complete, but there are intermediate results.
The number of the open questions in this field is larger than the known results,
so the hope of this series of lectures is to stimulate research and new questions.
Giambattista GIACOMIN Disorder and critical phenomena through basic probability models What is the effect of disorder on critical phenomena? In more probabilistic terms: what happens if we perturb a system exhibiting a phase transition by introducing a random environment? The physical community has approached this very wide issue aiming at general criteria telling whether or not the addition of disorder changes the critical properties of a model: some of the predictions are really striking and mathematically challenging. I will approach this domain of ideas by focusing on a specific class of models, the "pinning models". I will aim at giving full details in this restricted framework, as well as the gist, or at least a flavor, of the "general picture", that is mostly unexplored ground for mathematicians. Pinning models are Gibbs measures built on discrete renewal processes with Hamiltonian containing only one-body interactions (or external fields). In equivalent terms, these models can be built by "penalizing" or "rewarding" the visits of a Markov chain to a given state, possibly in a time dependent way, by drawing at random the sequence of penalizations/rewards. The phase transition one observes can be characterized in terms of the frequency of visits to this special state. Much attention has been devoted to pinning models (with and without disorder) in the applied sciences because they come up naturally in a variety of contexts and several non rigorous results can be found in the literature. A series of recent mathematical works has put on firm grounds essentially all the main predictions of the physical community and, in some cases, they have gone beyond, settling in particular some controversial issues. These works will be at the heart of the lectures whose preliminary plan is:
Takashi KUMAGAI Random walks on disordered media and their scaling limits The main theme of these lectures is to analyze heat conduction on disordered media such as fractals and percolation clusters using both probabilistic and analytic methods, and to study the scaling limits of Markov chains on the media. The problem of random walk on a percolation cluster `the ant in the labyrinth' has received much attention both in the physics and the mathematics literature. In 1986, H. Kesten showed an anomalous behavior of a random walk on a percolation cluster at critical probability for trees and for Z2. (To be precise, the critical percolation cluster is finite, so the random walk is considered on an incipient infinite cluster (IIC), namely a critical percolation cluster conditioned to be infinite.) Partly motivated by this work, analysis and diffusion processes on fractals have been developed since the late eighties. As a result, various new methods have been produced to estimate heat kernels on disordered media, and these turn out to be useful to establish quenched estimates on random media. Recently, it has been proved that random walks on IICs are sub-diffusive on Zd when d is high enough, on trees, and on the spread-out oriented percolation for d>6. Throughout the lectures, I will survey the above mentioned developments in a compact way. In the first part of the lectures, I will summarize some classical and non-classical estimates for heat kernels, and discuss stability of the estimates under perturbations of operators and spaces. Here Nash inequalities and equivalent inequalities will play a central role. In the latter part of the lectures, I will give various examples of disordered media and obtain heat kernel estimates for Markov chains on them. In some models, I will also discuss scaling limits of the Markov chains. Examples of disordered media include fractals, percolation clusters, random conductance models and random graphs.
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Dernière mise à jour le 06/10/2008 |
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