Incompressible fluids in thin domains with navier friction boundary conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier-Stokes equations in a three-dimensional domain Ω_ε, 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 Ω_ε, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ε^3/4) as ε → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H ¹(Ω_ε), respectively, L ²(Ω_ε), 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.