"Shocks in fluids : formation, propagation and microscopic description" by Laure Saint-Raymond

The goal of this series of lectures is to show how to capture with mathematical models some physical phenomena exhibiting singularities : wave breaking, sound barrier... The equation we will study, the so-called Hopf equation, is not - strictly speaking - an equation coming from physics, but it reproduces in a simple way the formation and propagation of singularities, and can be therefore considered as a prototype. We will first describe its smooth solutions. To do that, we will introduce the method of characteristics, which reduces to solving ordinary differential equations. This explicit solution exhibits in general a singularity in finite time, time beyond which there is no more smooth solution. From the technical point of view, this session will aim at familiarizing with functions of several variables, and basics of differential calculus. The second part of the course will be devoted to the dynamics in the presence of singularities. A natural question is to know how to define a weak notion of solution for differential equations, or in other words how to give sense to the derivative of a function which is not even continuous. In order to answer this question, we will give some elements about the theory of distributions. We will then discuss the uniqueness of weak solutions and physical admissibility criteria. The corresponding discussion session should consist in handling some classical distributions : Heaviside function, Dirac mass, principal value,...