scholarly journals Weak Solutions and Their Kinetic Energy Regarding Time-Periodic Navier–Stokes Equations in Three Dimensional Whole-Space

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1528
Author(s):  
Mads Kyed
Keyword(s):  
Kinetic Energy ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Periodic Forcing ◽  
Navier Stokes Equations ◽  
Whole Space ◽  
Time Periodic Solutions ◽  
Time Periodic

The existence of weak time-periodic solutions to Navier–Stokes equations in three dimensional whole-space with time-periodic forcing terms are established. The solutions are constructed in such a way that the structural properties of their kinetic energy are obtained. No restrictions on either the size or structure of the external force are required.

scholarly journals On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations

Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 42
Author(s):  
Luigi C. Berselli ◽  
Stefano Spirito
Keyword(s):  
Weak Solutions ◽  
Stokes Equations ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Approximation Schemes ◽  
Navier Stokes ◽  
Constant Density ◽  
Flat Torus ◽  
Navier Stokes Equations ◽  
Whole Space

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.

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