Non-smooth solutions in incompressible fluid dynamics

This work is devoted to the study of the main models which describe the motion of incompressible fluids, namely the Navier-Stokes, together with their hypodissipative version, and the Euler equations. We will mainly focus on the analysis of non-smooth weak solutions to those equations. Most of the results have been obtained by using the convex integration techniques introduced by Camillo De Lellis and László Székelyhidi in the context of the Euler equations, which recently led to the proof of the Onsager's conjecture on the anomalous dissipation of the kinetic energy. With various refinements of those iterative schemes we prove ill-posedness of Leray-Hopf weak solutions of the hypodissipative Navier-Stokes equations, sharpness of the kinetic energy regularity for Euler, typicality results in the sense of Baire's category for both Euler and Navier-Stokes, estimate on the dimension of the singular set in time of non-conservative Hölder weak solutions of the Euler equations. Moreover, building on different techniques, we also address some regularizing effects of those equations in various classes of weak solutions with some fractional differentiability in terms of Hölder, Sobolev and Besov regularity. The latter make use of new abstract interpolation results for multilinear operators which we developed for our specific context but which may also have independent interests.