Boundary Regularity of Shear Thickening Flows

Abstract

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p ≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W 1,(np+2−p)/(n−2) (Ω). We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier-Stokes problem are also formulated. (2000). 35Q35, 35J65, 76D03. Mathematics Subject Classification

Steady flow for shear thickening fluids with arbitrary fluxes

“…The regularity part, i.e. (u T , P) ∈ W 2,l (Ω t ) × W 1,l (Ω t ), for any t ∈ (0, T ), is a corollary of the proof of Theorem 1.2 in [4]. Notice that if (u T , P) is a weak solution with u T in D 1,p 0 then v = u T + a is a weak solution in W 1,p (Ω T ) of (4.2)…”

Regularity theorems, up to the boundary, for shear thickening flows

“…We just mention here the forthcoming paper [7], where the authors prove, for small data f , that the solutions belong to C 1,α (Ω) ∩ W 2,2 (Ω), up to the boundary. Finally we claim that the method followed in [4] may lead to an improvement of the result in [9] for all p 2.…”

“…For some references we refer the reader to [4]. We just mention here the forthcoming paper [7], where the authors prove, for small data f , that the solutions belong to C 1,α (Ω) ∩ W 2,2 (Ω), up to the boundary.…”

“…Since in the following and in Ref. [4] we insert into S and F only symmetric tensors, we can drop the superscripts " sym " and restrict the admitted tensors to symmetric ones.…”

“…We refer the reader to [13,8,5] for a more detailed discussion leading to Assumption 1. We also define a function F : R n×n → R n×n sym , closely related to the extra stress tensor S, by…”