A New Proof of the Existence of Suitable Weak Solutions and Other Remarks for the Navier-Stokes Equations

Enrique Fernández-Cara,Irene Marín-Gayte

doi:10.4236/am.2018.94029

Abstract

We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.

Highlights

We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]
This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]
The main objective of this paper is to provide a new proof of the existence of suitable weak solutions to the Navier-Stokes equations

Summary

Introduction

The main objective of this paper is to provide a new proof of the existence of suitable weak solutions to the Navier-Stokes equations. We show that the semi-discrete and the completely discrete semi-implicit Euler schemes lead to families of approximate solutions that converge to a weak solution that is suitable in the sense of Caffarelli, Kohn and Nirenberg [2]. The key concept of suitable weak solution was introduced in [2]. Our results can be compared to other previous proofs of existence: the one in the Appendix in [2] (based on the construction of a family of time delayed linear approximations), the main result in Da Veiga [15] (relying on regularization with vanishing fourth-order terms), the main result in Guermond [3] (where Faedo-Galerkin techniques are employed) and, the results by Berselli and. We explain why suitable solutions are relevant in the context of the regularity problem. In Section 3, we recall the Euler approximation schemes and we establish the convergence to a suitable solution of the Navier-Stokes equations. Section 4 is devoted to some additional comments and open questions

The Main Properties of Suitable Solutions

On the Existence of Suitable Weak Solutions

The Convergence of the Fully Discretized Problems

The Same Results Hold for the Boussinesq System

Possible Extensions to Other Systems

Extensions to Other Approximation Schemes for the Navier-Stokes Equations