Existence of periodic weak solutions to the three-dimensional nonlinear viscoelastic system

In this paper, we discuss the time-periodic solutions to a general three-dimensional nonlinear viscoelastic system with Dirichlet boundary condition. Employing the singularity of the integral kernel, we obtain the energy estimates in Sobolev spaces with fractional index and then show the existence of time-periodic solutions to the problem for viscoelastic solid and liquid models, respectively.