On the Dynamic Shear Flow Problem for Viscoelastic Liquids

Abstract

nitial-boundary value problems for a third order nonlinear integro-differential equation describing dynamic simple shear flow for viscoelastic liquids are studied on bounded one-dimensional spatial domains. Local and global existence results for arbitrary forces and initial data are given under suitable assumptions on the constitutive relations. Conditions on the forces and on the constitutive equations are formulated that imply that solutions of the equations tend to a rest state, and the convergence rates are estimated in terms of the force decay and of dissipation rates that can be derived from the constitutive equations.