Fully nonhomogeneous problem of two‐dimensional second grade fluids

Abstract

This article studies the solutions of a two‐dimensional grade‐two fluid model with a fully nonhomogeneous boundary condition for velocity u. Compared to problems with a homogeneous or tangential boundary condition, studied by many authors, we must add a boundary condition, otherwise the problem is no longer well posed. We propose two conditions on z = curl (u − αΔu), which differ according to the regularity of z, on the portion of ∂Ω where αu . n < 0. Following the approach of V. Girault and L.R. Scott in the tangential boundary case, we split the problem into a system with a generalized Stokes problem and a transport problem. However, compared to the study of these authors, we are now led to solve transport problems with boundary conditions. In two previous articles, we studied these transport problems. The results obtained in these articles allow us, by a fixed‐point argument, to establish existence of the solutions for the fully nonhomogeneous grade‐two problem. Uniqueness requires the boundary condition with z in H1.