On the Existence, Uniqueness and $$C^{1,\gamma} (\bar{\Omega}) \cap W^{2,2}(\Omega)$$ Regularity for a Class of Shear-Thinning Fluids

Abstract

We consider a stationary Navier-Stokes system with shear dependent viscosity, under Dirichlet boundary conditions. We prove Hölder continuity, up to the boundary, for the gradient of the velocity field together with the L 2 -summability of the weak second derivatives. The results hold under suitable smallness assumptions on the force term and without any restriction on the range of p ∈ (1, 2).

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

“…Up to now, these are the stronger results known in literature for shear-thickening fluids without a smallness assumption on f . In the present paper we extend to fluids with non-standard growth conditions the regularity results obtained in [21] for shear-thinning fluids. Indeed our treatment is based on the same technique developed in the previous investigation [21] for generalized Newtonian fluids.…”

“…In the present paper we extend to fluids with non-standard growth conditions the regularity results obtained in [21] for shear-thinning fluids. Indeed our treatment is based on the same technique developed in the previous investigation [21] for generalized Newtonian fluids. More precisely we prove the existence and uniqueness of a solution u, for any n 2, such that the velocity field u belongs to C 1,γ (Ω) (the pressure field π is in C 0,γ (Ω)).…”

“…However in these papers the minimal regularity assumption on the extra-stress tensor S(Du) is the C 1 -regularity, which we replace here by a Lipschitz continuity assumption. In [21], in a smallness assumption on the size of f , we prove the existence, uniqueness and C 1,γ (Ω) ∩ W 2,2 (Ω) regularity for the velocity field, for any p ∈ (1, 2) and n 2. In the three-dimensional case, the most significant regularity result for shearthinning fluids (i.e.…”

“…As in our previous work, concerning shear-thinning fluids, a special attention is needed managing the constants in the estimates contained in [27]. The control in the growth of such constants is obtained adapting the computations made in the Appendix of [21] to the case where p is not a constant. We want to observe that, in this case too, no further pointwise regularity is expected, due to the non-differentiability of the modulus of the symmetric gradient in the equations.…”

Boundary Regularity of Shear Thickening Flows

“…Appeal to Troisi's anisotropic embedding theorems (instead of classical, isotropic, Sobolev embedding theorems), also used below, was introduced in [10]. In the forthcoming paper [23] the authors prove that, under a suitable smallness assumption on f , the solution u to the system (1.1), (1.3), under the boundary condition (1.2), belongs to C 1, α ∩ W 2,2 , up to the boundary. Still in the shear-thinning case, W 2, q − regularity results up to a flat or a polyhedral boundary, under no-stick boundary conditions, are proved in [28].…”