On the regularity of flows with Ladyzhenskaya Shear‐dependent viscosity and slip or non‐slip boundary conditions

Abstract

Navier-Stokes equations with shear dependent viscosity under the classical nonslip boundary condition were introduced and studied in the 1960s by O. A. Ladyzhenskaya and, in the case of gradient dependent viscosity, by J.-L. Lions. A particular case is the well-known Smagorinsky turbulence model. This is nowadays a central subject of investigation. On the other hand, boundary conditions of slip type seems to be more realistic in some situations, in particular in numerical applications. They are a main research subject. The existence of weak solutions u to the above problems, with slip-(or nonslip-) type boundary conditions, is well-known in many cases. However, regularity up to the boundary still presents many open questions. In what follows we present some regularity results, in the stationary case, for weak solutions to this kind of problem; see Theorem 3.1 and Theorem 3.2. The evolution problem is studied in a forthcoming paper [6]. A cornerstone in our proof is the classical Nirenberg translation method; see [38].

On the vanishing viscosity limit for the 3D Navier‐Stokes equations with a slip boundary condition

“…Such boundary conditions can be induced by effects of free capillary boundaries (see [3]), or a rough boundary as in [1,20], or a perforated boundary, which is then called Beavers-Joseph's law (see [4,19,32,35]), or an exterior electric field as in [9]. This type of boundary condition was first introduced by Navier in [30], which was followed by many applications and numerical studies and much analysis for various fluid mechanical problems; see, for instance, [1,2,3,4,5,8,11,15,17,18,19,20,21,25,26,27,28,29,32,33,35,36,37,46] and the references therein. In particular, we mention that such slip boundary conditions are used in the large eddy simulations of turbulent flows, which seek to compute the large eddies of a turbulent flow accurately while neglecting small flow structure.…”

“…For a rigorous mathematical analysis of the Navier-Stokes equations with Navier-type slip boundary conditions, the first pioneering paper is due to Solonnikov and Ščadilov [39] for the stationary linearized Navier-Stokes system under the boundary conditions (1.6) u · n = 0, (D(u)n) τ = 0, on ∂ , and the existence of weak solutions and regularity for the stationary Navier-Stokes equations with the Navier slip boundary condition (1.5) has been obtained by Beirão da Veiga [5] for half-space. In the case of two-dimensional, simply connected, bounded domains, the vanishing viscosity problem has been rigorously justified by Yodovich [50]; see also [26,27] for the free boundary condition…”

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

“…Thus, if we set 8) we obtain at once that these V and G solve (1.6) for any choice of Φ, and that Φ = S V dS. Thus, in Problem 1, the solution to (1.6) is given by V = G ϕ, while in Problem 2 it is just furnished by (1.8).…”

“…Concerning regularity issues (a topic that is still unsettled, in many respects, and currently under deep investigation) we refer to the papers of C. Ebmeyer [30] and to the more recent ones of H. Beirão da Veiga [8,9,10,11].…”

“…We shall also point out some basic open questions. (8) Let Ω be a semi-infinite straight pipe of constant cross-section S, defined as follows…”