Suitable weak solutions of the Navier-Stokes equations constructed by a space-time numerical discretization

TL;DR

This paper demonstrates that weak solutions of the Navier-Stokes equations, derived from specific numerical space-time discretizations, satisfy the local energy inequality, confirming their suitability in the mathematical sense.

Contribution

The paper proves that numerical discretizations using finite elements and the theta-method produce solutions that are suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg.

Findings

Weak solutions satisfy the local energy inequality.

Discretization methods produce solutions consistent with theoretical criteria.

Results apply in the space-periodic setting.

Abstract

We prove that weak solutions obtained as limits of certain numerical space-time discretizations are suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, in the space-periodic setting, we consider a full discretization in which the theta-method is used to discretize the time variable, while in the space variables we consider appropriate families of finite elements. The main result is the validity of the so-called local energy inequality.