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.
Highlights
Let T > 0 be an arbitrary finite number representing the time, Ω ⊂ R3 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 incompressible fluid at constant temperature and with constant density
Due to the limited regularity which can be generally inferred on weak solutions, the validity of any energy balance on the weak solutions to the 3D Navier-Stokes equations is obtained with a limiting process on the approximate solutions and not using the solution u itself as a test function as done to obtain (3), since this argument is only formal and not justified when dealing with genuine Leray-Hopf weak solutions
In this short note we provide a rather self-contained account on the global existence of weak solutions for the three-dimensional incompressible Navier-Stokes equations and some of the approximation methods used in the literature
Summary
Let T > 0 be an arbitrary finite number representing the time, Ω ⊂ R3 be a domain to be specified later, and ν > 0 be a positive number representing the kinematic viscosity. Due to the limited regularity which can be generally inferred on weak solutions, the validity of any energy balance on the weak solutions to the 3D Navier-Stokes equations is obtained with a limiting process on the approximate solutions and not using the solution u itself as a test function as done to obtain (3), since this argument is only formal and not justified when dealing with genuine Leray-Hopf weak solutions In this short note we provide a rather self-contained account on the global existence of weak solutions for the three-dimensional incompressible Navier-Stokes equations and some of the (several) approximation methods used in the literature.
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