Abstract:
In this talk, a finite element approximation of the steady $p(\cdot)$-Navier--Stokes equations ($p(\cdot)$ is variable dependent) is examined for orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Numerical experiments confirm the quasi-optimality of the  \textit{a priori} error estimates (for the velocity).

The steady $p(\cdot)$-Navier--Stokes equations are a prototypical example of a non-linear system with variable growth conditions. They appear naturally in physical models for so-called \textit{smart fluids}, \textit{e.g.}, electro-rheological fluids, micro-polar electro-rheological fluids, magneto-rheological fluids,  chemically reacting fluids, and thermo-rheological fluids, and have the potential for an application in numerous areas,  \textit{e.g.}, in electronic, automobile, heavy machinery, military, and biomedical industry.