Anomalous dissipation and other non-smooth phenomena in fluids

The thesis is dedicated to the study of two main partial differential equations (PDEs) in fluid dynamics: the Navier-Stokes equations, which describe the motion of incompressible fluids, and the transport equation with divergence-free velocity fields, which describes how a scalar quantity is transported within a fluid. Our main focus lies in the analysis of non-smooth weak solutions to these equations. In recent years, the majority of advancements within this framework have been obtained by using the convex integration techniques introduced by De Lellis and Székelyhidi for Euler equations, which recently led to the proof of the Onsager's conjecture. Part of the thesis aims to establish new results using refinements of the convex integration scheme for Navier--Stokes and transport equations, as well as for the finite state problem for a general linear differential operator. In the aforementioned context, it is challenging to determine whether the solutions of the Euler equations constructed with convex integration are physical solutions, i.e. vanishing viscosity solutions, and satisfy the Kolmogorov 0-th law of turbulence. Mathematically, this fundamental law in turbulence has led to the definition of anomalous dissipation. This extremely difficult problem finds its first approachable mathematical model in the transport equation, which is of independent interests. Recently, some successful results have been proven for vanishing viscosity solutions to the transport equation, based on different techniques rather than convex integration. We provide explicit constructions of divergence free velocity fields for which solutions to the transport diffusion equation exhibit anomalous dissipation in the so called full supercritical Obukhov-Corrsin regularity regime. For such velocity fields, we prove that vanishing viscosity can not be a selection criteria for the transport equation with divergence free Hölder regular velocity fields. We apply these ideas to the forced Navierâ Stokes equations to prove anomalous dissipation of solutions in the full Onsager supercritical regularity regime. Finally, we study the optimal time of dissipation of more regular Hamiltonian autonomous flows near non-degenerate elliptic points.