Existence and stopping time for solutions of a class of non newtonian viscous fluids with thixotropic or shear thinning flows

The aim of this paper is to study the existence of weak solutions and of a finite stopping time for a large class of generalized newtonian fluids with thixotropic and shear-thinning flows. In a first time, the existence of the weak solutions is established using variational inequalities. In order to prove the existence of solutions we regularize the non-linear term and then we apply a Galerkin method for finally passing to the limit with respect to both regularization and Galerkin discretization parameters. In a second time, we prove the existence of a finite stopping time for a class of fluids with threshold flows in dimension \(N \in \{2, 3\}\). More exactly, we show that if the extra non-linear term is of the form \(F(s) = s^{-\alpha}\) with \(\alpha \in (0, \frac{4}{N+2}]\) when \(s\) is large, then there exists a finite stopping time. This result extends the existing results concerning the existence of a finite stopping time for Bingham fluids in dimension two