On the global regularity of shear thinning flows in smooth domains

In some recent papers we have been pursuing regularity results up to the boundary, in W 2 , l ( Ω ) spaces for the velocity, and in W 1 , l ( Ω ) spaces for the pressure, for fluid flows with shear dependent viscosity. To fix ideas, we assume the classical non-slip boundary condition. From the mathematical point of view it is appropriate to distinguish between the shear thickening case, p > 2 , and the shear thinning case, p < 2 , and between flat-boundaries and smooth, arbitrary, boundaries. The p < 2 non-flat boundary case is still open. The aim of this work is to extend to smooth boundaries the results proved in reference [H. Beirão da Veiga, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (1) (2008) 175–185]. This is done here by appealing to a quite general method, introduced in reference [H. Beirão da Veiga, On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., in press], suitable for considering non-flat boundaries.