On the regularity of shear thickening viscous fluids

Abstract

The aim of this note is to improve the regularity results obtained by H. Beirão da Veiga in 2008 for a class of p-fluid flows in a cubic domain. The key idea is exploiting the better regularity of solutions in the tangential directions with respect to the normal one, by appealing to anisotropic Sobolev embeddings.

On the global regularity for nonlinear -systems of the -Laplacian type in space variables

“…We restrict our proofs to the "cubic domain case" (see the next section), where the interesting boundary condition (Dirichlet) is imposed on two opposite sides, and periodicity in the other two directions. This choice, introduced in reference [5] and used in a series of other papers (see for instance [4,6,10,11]), is convenient in order to work with a flat boundary and, at the same time, with a bounded domain. The main reason is that, in proving the regularity theorem for p > 2 (see Theorem 2.1), we apply the difference quotients method: we appeal to translations parallel to the flat boundary, and then retrieve the normal derivatives from the equations.…”

“…11), side by side, and by settingV = u − w, we get    −∆V − (p − 2) ∇u • ∇∇V • ∇u (µ + |∇u|) |∇u| = 0, in Ω , V = 0 , on ∂Ω .…”

On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem

“…Concerning the flat boundary case, very recently (after this paper was accepted by the editors), the results stated in [4] have been improved in two forthcoming papers, namely, [11] and [6]. In this last reference we show that u ∈ W 2,l ∩ W 1,p+4 , where l = 3(p + 4)/(p + 1).…”

On theC1,γ(Ω¯)∩W2,2Crispo1, Grisanti2 2009Journal of Mathematical Analysis and ApplicationsSelf Cite23010View full textAdd to dashboardCite

“…We also recall the W 2,l -regularity results in papers [5,7,20], for flat boundaries, and in [6] for non-flat boundaries. Here l = l(p) < 2.…”