Abstract: We studied the initial-boundary value problem of a class of non-Newtonian fluids with initial vacuum and linear damping. Under the condition that the initial energy was small and the initial value met certain compatibility, we used the technique of weighted energy estimation and Zlotnik inequality to overcome the nonlinearity of the viscous term of the momentum equation, and proved that the density function had a uniform upper bound of time. Furthermore, we adopted the continuity method to obtain the existence and uniqueness conclusion of the global strong solution of the initial-boundary value problem of non-Newtonian fluid.

Key words: global strong solution, non-Newtonian fluid, initial vacuum, weighted estimation