HOMOGENIZATION OF A COUPLED STOKES VISCOELASTIC SYSTEM IN A PERIODIC COMPOSITE WITH MEMORY EFFECTS

Abstract

This paper studies the homogenization of a coupled Stokes–viscoelastic system posed in a three-dimensional periodically perforated domain. At the microscopic level, an incompressible Stokes fluid occupies a periodic network of channels, while a viscoelastic solid with an internal damping term fills the complement. The two phases interact through kinematic and dynamic transmission conditions on the interface. By combining the method of multiple scales with a Laplace-transform approach in time, a homogenized integro–differential model with memory is derived and rigorously justified by the energy method. The effective macroscopic equation features a tensor-valued convolution kernel that encodes both the spatial microstructure and the viscoelastic time dependence of the solid phase. A first-order corrector is constructed, and strong convergence in  is obtained. In addition, novel structural properties of the homogenized memory kernel are proved: symmetry and positive definiteness, a quantitative exponential decay rate, and optimal time-regularity in terms of the data. These results yield uniqueness, stability, and continuous dependence on the microscopic geometry for the macroscopic model, and thus provide a robust mathematical framework for the effective description of such composite media.