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

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.

RésuméOn s'intéresse aux solutions faibles des équations de Navier–Stokes avec conditions aux limites de Dirichlet homogènes en dimension trois qui sont construites comme limites d'approximation de Faedo–Galerkin. Ces solutions sont admissibles (suitable) au sens de Scheffer [V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Comm. Math. Phys. 55 (2) (1977) 97–112] si les espaces d'approximation jouissent d'une propriété de commutateur discret et satisfont une certaine condition de compatibilité. Les éléments finis et les ondelettes satisfont ces hypothèses. Ce résultat étend celui de [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] qui a été démontré pour des conditions aux limites périodiques.