Study of the Norton–Hoff operator coupled with the evolutive heat equation

We are interested in the study of quasistatic visco-plastic flows with thermal effects. The fluid motion is governed by the incompressible Norton–Hoff model coupled with the time-dependent heat equation where the dissipated mechanical power is the source term. The viscosity of the fluid is modeled by the non-linear Arrhenius law. The well-posedness of each decoupled system is given. The optimal regularities of the heat solution and of the scale factor are supplied. A non-linear operator describing the stand coupling is provided. The existence of a solution to the considered problem is established. We prove the compactness result of the set solutions.