W^{2,p} -estimates of the Stokes system with traction boundary conditions

Paul Deuring (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

The article deals with W^{2,p}-estimates of solutions to the 3D stationary Stokes system under traction boundary conditions. Existence, uniqueness and W^{2,p}-estimates up to the boundary are established for solutions in interior and exterior domains. The proofs are based on the method of integral equations. With this well known approach, solutions to the boundary value problem are constructed by solving certain integral equations on the boundary of the domain under consideration. This access simultaneously yields an integral representation of the solution of the boundary value problem. The difficulty then is to derive W^{2,p}-estimates of the integrals appearing in this representation. Ultimately such estimates are reduced to the W 2,p-theory of the Stokes system in bounded domains, under Dirichlet boundary conditions.

Mots clés: Stokes system; traction boundary conditions; Neumann boundary conditons; L^p-regularity; method of integral equations

Domaines: Mathématiques [math]

Paul Deuring. W^{2,p} -estimates of the Stokes system with traction boundary conditions. 2023. ⟨hal-04189890v2⟩