Weak Solutions of Navier–stokes Equations Constructed by Artificial Compressibility Method Are Suitable

Abstract

In this paper we prove that weak solution constructed by artificial compressibility method are suitable in the sense of Scheffer, [17], [18]. Using Hilbertian setting and Fourier transform with respect to the time we obtain nontrivial estimates of the pressure and the time derivate which allow us to pass into the limit.

On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method

“…In addition, the case where the term −ε ∆p ε in equations (1.5) is replaced by ε ∂ t p ε has been considered by different authors, see [32,15]. Moreover, the convergence when ε → 0 to a suitable weak solution in the whole space was considered in [16] for the scheme with the time derivative of the pressure.…”

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

“…seem to produce, in the limit, solutions satisfying the local energy inequality. We started a systematics study of this question and, even if technicalities could be rather different, we obtained several positive answers, see [8,9,12]. The technical problems related with discrete (numerical, finite dimensional) approximations are of a different nature.…”

On the convergence of a fully discrete scheme of LES type to physically relevant solutions of the incompressible Navier–Stokes

“…The first existence result of suitable weak solutions is due to Caffarelli-Kohn-Nirenberg [8]. Then, the convergence to suitable weak solutions has been proved for different methods, see [1,2,6,12], but the approximation methods are of all of "infinite dimensional type", that is obtained by approximating the NSE (1.1) by another system of partial differential equations, and few results are available when the approximation methods are finite dimensional as in numerical methods. In [13,14] Guermond proved the convergence to a suitable weak solution for numerical solutions obtained by using some finite element Galerkin methods (only with respect to the space variables), while some conditional results on Fourier based Galerkin methods on the torus are proved in [7].…”