In this talk, a finite element approximation of the unsteady p(t,x)-Navier-Stokes equations (p(t,x) is time-space-dependent) is examined for weak convergence and orders of convergence (with respect to natural fractional regularity assumptions on the velocity vector field and the kinematic pressure). Numerical experiments confirm the optimality of the error decay rates. Due to the time-space-dependency of the power-law index, extracting compactness properties needed for the passage to the limit of a fully-discrete finite element approximation is not possible via standard results. We will see that all the necessary compactness is already included in the fully-discrete finite element approximation.
The unsteady p(t,x)-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 ‘smart fluids’, e.g., electro-rheological, magneto-rheological, thermo-rheological, and chemically-reacting fluids. In general, a fluid is called 'smart', when it can change its rheological properties through various external stimuli (e.g. electric or magnetic fields, temperature, light, pH value, concentrations of chemical molecules). This opens the way for a variety of technical applications in aerospace, automotive, heavy machinery, electronic, and biomedical industry.