On the global regularity of shear thinning flows in smooth domains

Abstract

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 LadyzhenskayaSmagorinsky 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.

Solenoidal difference quotients and their application to the regularity theory of the p-Stokes system

“…Therefore, m h,1 ij is a standard kernel with respect to Q with a constant independent of h. Analogously, using (3.11), (4.16) and (4.17), we show that m h,2 ij is a standard kernel with respect to Q with a constant depending only on n and ̺ 3,∞ . Hence, M h ij is a standard kernel with respect to Q with a constant independent of h. By Theorem 3.4 there exists a positive nondecreasing function c p,2 such that for all h ∈ 0, 1 4 and all f ∈ C ∞ 0,0 (Q) we have…”

Boundary Regularity of Shear Thickening Flows

“…Concerning the shear thinning case, strongly related W 2, q regularity results up to the boundary, under the boundary condition (1.2), are proved, for flat boundaries in [4,5,10], for cylindrical domains in [20,21], and for smooth arbitrary boundaries in [7]. Appeal to Troisi's anisotropic embedding theorems (instead of classical, isotropic, Sobolev embedding theorems), also used below, was introduced in [10].…”

Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities

“…The literature on this subject is very large and we focus on the papers that are mostly connected with the results we are going to prove. In particular, for the steady problem, there are several results proving existence of weak solutions [32], [26], interior regularity [1], [33] and very recently regularity up-to-the boundary for the Dirichlet problem [53], [56], [7], [8], [9], [10], [11], [12], [13], [14], [16], [21], [22]. Concerning the time-evolution Dirichlet problem in a three-dimensional domain we have recent advances on the existence of weak solutions in [58] for p > 8 5 and in [31] for p > 6 5 .…”