Faedo–Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable

Abstract

Faedo–Galerkin weak solutions of the three-dimensional Navier–Stokes equations supplemented with Dirichlet boundary conditions in bounded domains are suitable in the sense of Scheffer [V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Comm. Math. Phys. 55 (2) (1977) 97–112] provided they are constructed using finite-dimensional approximation spaces having a discrete commutator property and satisfying a proper inf-sup condition. Finite element and wavelet spaces appear to be acceptable for this purpose. This result extends that of [J.-L. Guermond, Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable, J. Math. Pures Appl. (9) 85 (3) (2006) 451–464] where periodic boundary conditions were assumed.