An introduction to mathematical analysis of incompressible fluid flow

Au sujet de ce cours

These lectures concern the Euler and Navier-Stokes equations for incompressible fluid flow. The focus is the mathematical analysis of these partial differential equations. For completeness’ sake we will begin with the derivation of the equations. We will then discuss local-in-time well-posedness of classical solutions, the problem of singularity formation and the Beale-Kato-Majda criterion and the issue of vortex stretching in three dimensions.

We then begin discussing weak solutions. We will explain the construction and proof of global-in-time existence of Leray-Hopf weak solutions of the 3D Navier-Stokes
equations and the weak-strong uniqueness theorem due to Prodi-Serrin, followed by Serrin’s uniqueness criterion. We briefly touch on weak solutions for 3D Euler and the method of convex integration for fluid dynamics.

Lastly we consider the special case of 2D flows, for which more analysis is possible. We will discuss results for weak solutions, beginning with the Yudovich existence and uniqueness theorem. We will then discuss rougher initial data, including vortex sheets. Pending time we will talk about recent results for weak solutions for 2D Euler.

Lieu : Amphi Darboux à l’Institut Henri Poincaré.

Dates et horaires :
– lundi 4 mai, 10h-12h et 16h15-17h45
– mardi 5 mai, 13h30-14h30
– lundi 11 mai, 10h-12h et 16h15-17h45
– mardi 12 mai, 13h30-14h30

This course will only be offered this year. It is also open to students in L3 or M2.