A mathematical model of the motion of a nonlinear viscous fluid with the condition of slip on the boundary

Abstract

INTRODUCTION Many mathematical papers are dedicated to various questions of resolvability of the Navier–Stokes equations. The number of papers devoted to mathematical models of non-Newtonian fluids, whose rheological correlations obey the principle of objectivity of the materials behavior, is essentially less (see [1]). Moreover, several authors (see, e. g., [2]) note the importance of the investigation of problems with boundary conditions which differ from the classic sticking conditions. This paper is devoted to the proof of the existence theorem for weak solutions to the boundary value problem which describes the stationary motion of a nonlinear viscous fluid with the condition of slip on the boundary. Note that various mathematical models of nonlinear viscous fluids were investigated earlier by V. G. Litvinov; the results obtained in this paper essentially improve some results of monograph [3]. Assume that the fluid under consideration entirely fills a container with hard walls represented by a bounded domainΩ ⊂ Rn, n ∈ {2, 3}, with the Lipschitz boundary S. It is well-known that the stationary motion of any fluid obeys the motion equation in the Cauchy form