Mini-course: Controllability of Partial Differential Equations and Related Topics
Time:
Venue/Location: Vietnam Institute for Advanced Study in Mathematics, 157 Chua Lang street, Hanoi
Time: Morning 03 April 2023 to 05 April 2023
Venue: Vietnam Institute for Advanced Study in Mathematics, 157 Chua Lang street, Hanoi
Organizing committee
Vu Ngoc Phat, Vietnam Academy of Science and Technology
Le Van Hien, Hanoi National University of Education
Le Minh Ha, Vietnam Institute for Advanced Study in Mathematics
Lecturer
Prof. Franck Sueur
Institut de Mathématiques de Bordeaux UMR 5251
Université de Bordeaux
351, cours de la Libération-F 33 405 TALENCE
Abstract
This mini-course consists of three lectures. In the first lecture entitled Controllability of the incompressible Euler equations, we will review some results obtained by J.M. Coron and O. Glass on the controllability of the incompressible Euler equations. We will start with the case where the fluid occupies a rectangle to focus on the explanation of Coron's return method. Then we will see how to deal with more general 2D domains thanks to some tools from complex analysis. Finally, we will review some extensions by Glass to the 3D case, as well as some adaptations to close systems. In the second lecture, Controllability of the incompressible Navier-Stokes equations, we will tackle the controllability problem in the case of the incompressible Navier-Stokes equations. We will start with the case where the fluid occupies a 2D boundaryless compact manifold, for which Coron and Imanuvilov succeeded to adapt the return method. Then, we will examine the case where Navier slip-with-friction boundary conditions are prescribed, for which boundary layers appear. We will see some results for weak solutions by Coron, Marbach and Sueur, and for strong solutions by Liao, Sueur and Zhang. Finally, we will see the hardest case where no-slip boundary conditions are prescribed, which corresponds to a famous open problem by J.-L. Lions dating back to 1989. For this case, we will see some partial results obtained by Coron, Liao, Marbach, Sueur and Zhang. In the third lecture entitled Lagrangian controllability: the case of a patch of fluid and of an immersed rigid body, we will consider another type of controllability, namely the Lagrangian controllability, where one aims at driving, by a remote action, a patch of fluid particles from an initial place to a final one. We will start with reviewing some results by Glass, Horsin and Kavian on the case of the incompressible Euler equations, as well as some adaptations to the case where, rather than a patch of particles, one addresses the issue to remotely drive an immersed rigid body along a given path, reviewing some results by Glass, Kolumban and Sueur.
Time Table
9:30 to 11:00, Mon day, 03 April 2023
Lecture 1: Controllability of the incompressible Euler equations
Lecture Hall/Room: C.101
9:30 to 11:00, Tuesday, 04 April 2023
Lecture 2: Controllability of the incompressible Navier-Stokes equations
Lecture Hall/Room: C.101
9:30 to 11:00, Wednesday, 05 April 2023
Lecture 3: Lagrangian controllability: the case of a patch of fluid and of an immersed rigid body
Lecture Hall/Room: C.101
Registration link: https://docs.google.com/forms/d/e/1FAIpQLSe4CWlX-sxkeP3s6PtADi37cyUZoIQ7sFfeMakZijFRC37DFg/viewform
Languague: English
Registration: Online/Link
Link zoom: 98529394122
Password:2023
Contact: Le Van Hien (hienlv@hnue.edu.vn, +84912494391)