Well posedness of a linearized fractional derivative fluid model

The one-dimensional fractional derivative Maxwell model (e.g. Palade, et al., Rheol. Acta 35 (1996) 265), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade, et al., Internat. J. Engrg. Sci. 37 (1999) 315, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade, J. Math. Phys. 49 (2008) 043101, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results.