We are interested in a barotropic motion of the non-Newtonian bipolar fluids . We consider a special case where the stress tensor is expressed in the form of potentials depending on eii and $\left(\frac{\partial e_{ij}}{\partial x_k}\right)$. We prove the asymptotic stability of the rest state under the assumption of the regularity of the potential forces.

The paper describes the special situation of barotropic nonnewtonian fluid, where stress tensor can be written in the form of potentials which depend on $e_{ij}$ and $\left(\frac{\partial e_{ij}}{\partial x_k}\right)$. For this case, we prove the existence and uniqueness of weak solution.