Solvability of integro-differential problem for a viscoelastic fluid

The presentation is devoted to investigating the weak solubility of the alpha-model for a fractional viscoelastic Voigt medium. The model involves the Voigt rheological relation with a left Riemann–Liouville fractional derivative, which accounts for the fluid’s memory. The memory is considered along the trajectories of fluid particles determined by the velocity field. Since the velocity field is not smooth enough to uniquely determine the trajectories for every initial value, we introduce weak solutions of this problem using regular Lagrangian flows. Based on the approach to the study of hydrodynamics problems, developed in the Voronezh mathematical school (the so-called topological approximation approach) we prove the existence of weak solutions of the alpha-model and establish the convergence of solutions of the alpha-model to solutions of the original model as the parameter α tends to zero.