A new proof of the existence of suitable weak solutions and other remarks for the Navier-Stokes equations

Abstract

We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations upplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg. Our results are similar to the main result in Guermond, J.-L. (2007) Faedo-Galerkin Weak Solutions of the Navier-Stokes Equations with Dirichlet Boundary Conditions Are Suitable. Journal de Mathématiques Pures et Appliquées, 88, 87-106. We also present some additional remarks and open questions on suitable solutions.