INCOMPRESSIBLE FLUIDS IN THIN DOMAINS WITH NAVIER FRICTION BOUNDARY CONDITIONS (I) By

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ωε, whose thickness is of order O(ε) as ε → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ωε, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ε3/4) as ε → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H(Ωε), respectively, L(Ωε), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ε, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.