An introduction to mathematical analysis of incompressible fluid flow

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These lectures concern the Euler and Navier-Stokes equations, which are models for incompressible fluid flow. The focus is the mathematical analysis of these partial differential equations. We will discuss the main steps and ideas for local-in-time well-posedness of classical solutions, the problem of singularity formation and the Beale-Kato-Majda criterion and, finally, 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. Lastly we consider the special case of 2D flows and we will discuss results for weak solutions, in particular the Yudovich uniqueness theorem.

Cours prévu en avril 2026.