Suitable weak solutions to the 3D Navier-Stokes equations are constructed with the Voigt Approximation

TL;DR

This paper constructs suitable weak solutions to the 3D Navier-Stokes equations using the Voigt approximation, demonstrating convergence and energy inequality satisfaction under specific boundary conditions and parameter choices.

Contribution

It introduces a method to obtain suitable weak solutions via the Voigt approximation and Fourier-Galerkin methods, extending the understanding of solution construction for Navier-Stokes equations.

Findings

Weak solutions satisfy the local energy inequality.

Suitable weak solutions can be constructed through Fourier-Galerkin approximation.

Results hold under specific boundary conditions and parameter choices.

Abstract

In this paper we consider the Navier-Stokes equations supplemented with either the Dirichlet or vorticity-based Navier boundary conditions. We prove that weak solutions obtained as limits of solutions to the Navier-Stokes-Voigt model satisfy the local energy inequality. Moreover, in the periodic setting we prove that if the parameters are chosen in an appropriate way, then we can construct suitable weak solutions trough a Fourier-Galerkin finite-dimensional approximation in the space variables.