Partial differential equations

It's a theory motivated by the description of distributed phenomena. This means there is at least one (and often several) variable(s) of space and time. In contrast to the dynamics of the material point elaborated by Newton and Leibniz in the second half of the 17th century, this theory was (probably) born with Euler and d'Alembert some 70 years later, and Laplace another 40 years later.

These are called evolutionary equations when time is present, and stationary equations otherwise. As with differential equations, the unknowns (solutions to be found) are not just numerical values, but functions. Functions which themselves depend on functions: for example, for problems described in domains other than whole space, boundary conditions, themselves realized by functions defined on the edge, play an essential role.

We can more or less classify partial differential equations (PDEs) into elliptic, parabolic and hyperbolic categories, but this classification, which will not appear in our presentation, is only really rigorous for linear equations with constant coefficients. So we think it's best to bear in mind that there are a small number of model problems, and that equations that resemble them are given the same names. Finally, it's important to note that the richness of an equation corresponds to the variety of fields in which it can be applied.

In our view, it is therefore a special feature of the theory that a "small" number of equations are present.

The question is, is there a reason for this? It should be noted straight away that PDEs are in some way coupled to a phenomenology, either underlying (microscopic or other models) or asymptotic (compatibility with macroscopic models), which means that the real driving force behind their development is generally based on a number of principles of symmetries and conservations, which persist from one model to the next.

Naturally, explicit solutions were first sought (Poisson kernel, heat kernel, use of Laplace and Fourier transforms...). But we soon realized that, even more than for ordinary differential equations, the cases where solutions could be written explicitly were exceptional. Nevertheless, these examples remain instructive, despite two new tools that have introduced different points of view: on the one hand, the emergence of functional analysis, which provides information on the existence, uniqueness and stability of solutions without having to resort to their explicit calculation; and on the other, the appearance of computer calculations, which also make up for the absence of analytical information.

Although Euler foresaw the notion of a weak solution for fluid mechanics, it is with the Lebesgue integral and Schwartz...