Weak and strong convergence methods for Partial Differential

 Formation de spécialité en mathématiques et informatique
Année universitaire 2015-2016
Weak and strong convergence methods for
Partial Differential Equations
Intervenant(e) : Christophe Prange Invitant(e) : ‐ Date, horaires et salle : le 01/02/2016 de 16h‐18h, salle 385, IMB le 03/02/2016 de 16h‐18h, salle 385, IMB le 08/02/2016 de 16h‐18h, salle 385, IMB le 10/02/2016 de 16h‐18h, salle 385, IMB le 15/02/2016 de 16h‐18h, salle 385, IMB le 17/02/2016 de 16h‐18h, salle 385, IMB Durée du cours : 12 heures Langue: Anglais Public : Etudiants en Master 2 ou en doctorat Matériel nécessaire : Aucun Compétences requises pour participer au cours : Niveau de M1 en analyse fonctionnelle Compétences acquises à la fin du cours : weak convergence, compensated compactness, div‐curl lemma, homogenization, H‐convergence, multiscale asymptotic expansions, correctors, error estimaters, boundary layers Résumé: This course is devoted to a fundamental problem in the theory of linear and nonlinear Partial Differential Equations (PDEs). How can we pass to the limit in a product of weakly convergent sequences? This problem arises naturally when proving existence of weak solutions to nonlinear PDEs, or when studying asymptotic problems in linear PDEs (homogenization theory of composite materials). Unlike strong convergence, weak convergence is in general incompatible with products. The limit of the product may be the product of the limits, but it also may not. What can then be said about the limit? Are there some new interesting effects arising from the interaction of two terms in the product? The goal of the course is to develop the following aspects: (i) div‐curl lemma, compensated compactness, defect measures to use the structure of the equation, (ii) construction of multiscale expansions, interior and boundary layer correctors to get refined information about the limit and precise error estimates, (iii) compactness methods to study regularity theory in homogenization. The course will be based mainly on examples coming from homogenization theory and fluid mechanics. Two possible references are: 1. Weak Convergence Methods for Nonlinear Partial Differential Equations, by Evans. 2. An Introduction to Homogenization, by Cioranescu and Donato. The course is open to any Master or Ph.D. student with a background in Analysis. A previous experience of PDEs is good but not necessary.