CIMPA Research School - Algeria

Enrique Zuazua
BCAM – Basque Center for Applied Mathematics
Alameda Mazarredo, 14
E-48009 Bilbao
Basque Country
Spain

Abstract: Flow control is an important topic for its many applications and its potential economic,social and environmental impact: water quality and supply, oild recovery, pollution, traffic,...
From a mathematical viewpoint this is a challenging topic in between the theory of Partial Differential Equations (PDE), Control Theory, Optimal Design and Numerical Analysis. It utilizes models from Fluid Mechanics such as Navier-Stokes and Euler equations, hyperbolic systems of conservations laws, whose understanding still nowadays constitutes an important challenge of the theory of PDE. Indeed, some of the main issues concerning existence, uniqueness and regularity of solutions for these systems of PDE are still open in this field.
The theories of Control and Optimal Design also face some added difficulties when addressing these issues because of the possible presence of singularities on solutions, which makes often classical approaches fail. These challenges are also reflected at the numerical and computational level, when developing algorithms able to mimic at the discrete level the picture predicted by Continuum Mechanics.
In this series of lectures we shall describe the state of the art in this field, and illustrate some possible directions of research and relevant applications mentioned above. We shall mainly focus on scalar hyperbolic conservation laws for which entropy solutions often present shock discontinuities.
The course is oriented to researchers with a basic background in the theory of Partial Differential Equations and their numerical analysis by means of finite differences and volumes.

Lecture notes: first part and second part.