Partial differential equations first appeared in the 18th century, when L. Eulerintroducedthe first models of fluid motion. The use of mathematical models based on partial differential equations (usually non-linear) has long been the prerogative of physicists, mechanics and mathematicians. In the 20th century, the field of mathematical modelling expanded considerably, enabling the study of a wide range of problems in physics and mechanics, of course, but also in economics, meteorology, biology, finance, the engineering sciences, geometry, probability theory, climatology...

This profusion of applications is matched by a profusion of mathematical tools, theories and methods. Progress has been and continues to be considerable, even if many problems remain poorly understood from a mathematical point of view. For example, the equations introduced by L. Eulerin 1755, for which he reports " a certain mathematical difficulty ", are still not fully understood from a mathematical point of view.

Thanks to the emergence and development of computers, mathematical models are increasingly used in applications, thanks to the possibilities offered by digital simulations, whose mathematical justification joins the analysis of equations.

The big questions that arise for each of these equations or classes of equations are always : existence, uniqueness, stability and numerical approximation of solutions.

The lecture deals with these questions by successively tackling different models or equations. The associated seminar, with presentations by mathematicians as well as engineers and scientists, explores this broad field from both a mathematical and an application point of view.