On the existence of weak solutions for steady flows of generalized viscous fluids

In this work we investigate the existence of weak solutions for steady flows of generalized incompressible and homogeneous viscous fluids. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the exponent $q$ that characterizes the flow depends on the space variable: $q=q(\mathbf{x})$. For the associated boundary-value problem we prove the existence of weak solutions for any variable exponent $q\geq\alpha>\frac{2N}{N+2}$, where $\alpha=\mathrm{ess}\inf q$. This work improves all the known existence results in the sense that the lowest possible bound of $q$ is attained and no other assumption on the regularity of $q$ is required.