On the existence of strong solutions for unsteady motions of incompressible chemically reacting generalized Newtonian fluids

Abstract: We consider a system of nonlinear partial differential equations modeling the unsteady motion of an incompressible generalized Newtonian fluid with chemical reactions. The system consists of the generalized Navier-Stokes equations with power-law type viscosity with a power-law index depending on the concentration, and the convection-diffusion equation which describes chemical concentration. This system of partial differential equations arises in the mathematical models describing the synovial fluid which can be found in the cavities of movable joints. We prove the existence of a global strong solution for the two and three-dimensional spatially periodic domain, provided that the power-law index is greater than or equal to $(d+2)/2$ where $d$ is the dimension of the spatial domain. Moreover, we also prove that such a solution is unique under the further assumption that $p^+ < \frac{3}{2} p^-$ for the two-dimensional case and $p^+ < \frac{7}{6}p^-$ for the three-dimensional case, where $p^-$ and $p^+$ are the lower and upper bounds of the power-law index $p(\cdot)$ respectively.