Stability analysis for a recovered fracturing fluid model in the wellbore of shale gas reservoir

This paper is concerned with the study of stability analysis to a complicated recovered fracturing fluid model (RFFM, for short), which consists of a stationary incompressible Stokes equation involving multivalued and nonmonotone boundary conditions, and a reaction-diffusion equation with Neumann boundary conditions. Firstly, we introduce a family of perturbated problems corresponding to (RFFM) and deliver the variational formulation of perturbated problem which is a hemivariational inequality coupled with a variational equation. Then, we prove that the existence of solutions to perturbated problems and the solution sequence to perturbated problems are uniformly bounded. Finally, via employing Mosco convergent approach and the theory of nonsmooth, a stability result to (RFFM) is established.