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Publication# Anomalous dissipation and other non-smooth phenomena in fluids

Abstract

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 SzÃ©kelyhidi 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 HÃ¶lder regular velocity fields. We apply these ideas to the forced NavierâStokes 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.

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Navier–Stokes equations

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids.

Fluid dynamics

In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Euler equations (fluid dynamics)

In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity. The Euler equations can be applied to incompressible or compressible flow. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field.

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