On the existence of unique global-in-time solutions and temporal decay rates of solutions to some non-Newtonian incompressible fluids

Abstract

In this paper, we deal with a class of incompressible non-Newtonian fluids. We first give some conditions to the viscous part of the stress tensor to set our model. We then show that there exists a unique regular solution globally in time if $$u_{0}\in L^{2}\cap \dot{B}^{1}_{\infty ,1}$$ and is sufficiently small in $$\dot{B}^{1}_{\infty ,1}$$ . We finally derive temporal decay rates of the solution which are consistent with the decay rates of the linear part of our model.