This session of the “ États de la Recherche SMF “ will focus on partial differential equations from kinetic theory and fluid mechanics. More specifically, particular attention will be paid to complex systems coupling kinetic and fluid equations, and to asymptotic limits aiming at rigorously recovering such PDEs from first principle physics.

Fluid-kinetic couplings were introduced some forty years ago to describe the evolution of a spray, that is to say a dispersed phase (a large number of particles) within a continuous one (a fluid). They are the subject of numerous research programs, both for the mathematical questions they raise and for the many concrete applications they may find. One important issue is the rigorous derivation of such systems, starting from a microscopic model (N particles in a fluid, in the limit N → +∞).

On the other hand, fluid equations can be derived from kinetic equations in some physically relevant regimes. This is an old question that dates back to the beginning of the 20th century, and which can be seen as an important part in the program of axiomatization of physics initiated by Hilbert in 1900. A great deal of work has led to very significant advances in the 1990s and 2000s, but important questions have been left open and this is still a very active field of research.