On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions

• Published in: Communications on pure and applied mathematics Vol. 58; no. 4; pp. 552 - 577
• Main Author: BEIRAO DA VEIGA, H
• Format: Journal Article
• Language: English
• Published: New York, NY Wiley 01.04.2005; John Wiley and Sons, Limited
• Subjects: Computational methods in fluid dynamics; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematics; Partial differential equations; Physics; Sciences and techniques of general use; Viscosity; Smagorinsky turbulence model; Shear viscosity; Gradient; Turbulence; Weak solution; Navier Stokes equation; Existence of solution; Slip; Boundary condition; Regularity
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20036

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 mu 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]. [PUBLICATION ABSTRACT]