Existence, Uniqueness and Regularity Issues for a Viscoelastic Fluid System

ABSTRACT This paper deals with a coupled system of nonlinear partial differential equations describing unsteady 3D flows of a viscoelastic fluid (with the constitutive law of differential type) through a porous medium. Using a modified Faedo‐Galerkin approximation procedure with special basis eigenfunctions and the Aubin compactness theorem, we prove a theorem about the existence of a global‐in‐time weak solution in a bounded Lipschitz domain under natural assumptions on model data. The proof schema of this theorem can be used for semi‐analytical and numerical solving of viscoelastic flow problems in tubes and channels filled with a porous medium. Moreover, in our paper, a new Serrin‐type criterion for regularity and uniqueness of a weak solution is established in terms of the velocity field and the elastic part of the extra stress tensor. Our results hold for both the no‐slip and perfect‐slip boundary conditions.