fluids Article On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations Luigi C. Berselli 1, * and Stefano Spirito 2   Citation: Berselli, L.C.; Spirito, S. On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations. Fluids 2021, 6, 42. https://doi.org/10.3390/fluids 6010042 Received: 15 December 2020 Accepted: 5 January 2021 Published: 13 January 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- ms in published maps and institutio- nal affiliations. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 1/c, I-56127 Pisa, Italy 2 DISIM—Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, I-67100 L’Aquila, Italy; stefano.spirito@univaq.it * Correspondence: luigi.carlo.berselli@unipi.it Abstract: We give a rather short and self-contained presentation of the global existence for Leray- Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions. Keywords: Navier-Stokes equations; Leray-Hopf weak solutions; existence 1. Introduction Let T > 0 be an arbitrary finite number representing the time, Ω ⊂ R 3 be a domain to be specified later, and ν > 0 be a positive number representing the kinematic viscosity. The incompressible Navier-Stokes equations model the dynamic of a viscous and incom- pressible fluid at constant temperature and with constant density. They are given by the following system of PDE’s posed in (0, T) × Ω:  ∂ t u +(u ·∇) u + ∇p − νΔu = f in (0, T) × Ω, div u = 0 in (0, T) × Ω. (1) The vector field u ∈ R 3 is the velocity, p ∈ R is the scalar pressure, and to avoid inessential complications, we set the external force f = 0 (but all results presented here can be easily extended to the case of a non vanishing external force, see Remark 4). The first equation is the conservation of linear momentum and the second equation, also called the incompressibility constraint, can be considered as the conservation of the mass, since the density is assumed to be constant. The system (1) has to be supplemented with initial and boundary conditions. Regarding the initial condition we impose that u| t=0 = u 0 , in Ω, with u 0 satisfying the compatibility condition div u 0 = 0 in Ω. For the boundary conditions we need to specify the assumptions on the domain. We consider three cases, Ω = R 3 , Ω = T 3 with T 3 being the three-dimensional flat torus, and Ω ⊂ R 3 being a bounded domain, whose boundary will be denoted by ∂Ω; we refer to Assumption 1 for the precise hypotheses on Ω. Fluids 2021, 6, 42. https://doi.org/10.3390/fluids6010042 https://www.mdpi.com/journal/fluids