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The aim of this paper is to study the existence of variational inequalityweak solutions and of a finite stopping time for a large class of generalizedNewtonian fluids shear-thinning flows. The existence of dissipative solutionsfor such flows is known since \cite{abbatiello-feireisl-20}. We submit here analternative approach using variational inequality solutions as presented in\cite{duvaut-lions} in the two-dimensional Bingham flow. In order to prove theexistence of such solutions we regularize the non-linear term and then we applya Galerkin method for finally passing to the limit with respect to bothregularization and Galerkin discretization parameters. In a second time, weprove the existence of a finite stopping time for Ostwald-De Waele and Binghamflows in dimension \(N \in \{2, 3\}\).

Variational inequality solutions and finite stopping time for a class of shear-thinning flows