  Viscoelastic rate-type fluids are popular models of choice in many
applications involving flows of fluid-like materials with complex
micro-structure. A well-developed mathematical theory for the most of these
classical fluid models is however missing. The main purpose of this study is to
provide a complete proof of long-time and large-data existence of weak
solutions to unsteady internal three-dimensional flows of Giesekus fluids
subject to a no-slip boundary condition. As a new auxiliary tool, we provide
the identification of certain biting limits in the parabolic setting, presented
here within the framework of evolutionary Stokes problems. We also generalize
the long-time and large-data existence result to higher dimensions, to
viscoelastic models with multiple relaxation mechanisms and to viscoelastic
models with different type of dissipation.
