Problem of Second Grade Fluids in Convex Polyhedrons

Abstract

This paper studies the solutions of a three-dimensional grade-two fluid model with a tangential boundary condition in a polyhedron. We begin to split the problem into a system with a generalized Stokes problem and a transport equation, as Girault and Scott have done in the two-dimensional case. But, compared to the two-dimensional problem, we have an additional term that is difficult to bound which requires regularity of the solutions and we have to prove that the solutions of the transport equation, which is no longer scalar, are divergence-free. In order to deal with these specific difficulties of the three dimensions, we establish a new system which implies the previous one without being equivalent. A substantial part of the article is devoted to proving that any solution in $L^2(\Omega)^3$ of the transport equation is divergence-free if the right-hand side is divergence-free and the velocity small enough in $W^{1,\infty}(\Omega)^3$. Existence is proven in a convex polyhedron with adequate restrictions on the size of the data and parameters. Uniqueness requires, in addition, inner angles smaller than $\frac{3\pi}{4}$.