A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

Abstract

We study a new class of elliptic variational-hemivariational inequalities in a reflexive Banach space. Based on a surjectivity result for an operator inclusion of Clarke's subdifferential type, we prove existence of solution. Then, we apply this result to a mathematical analysis of the steady Oseen model for a generalized Newtonian incompressible fluid. A variational-hemivariational inequality for the flow problem is derived and sufficient conditions for existence of weak solutions are obtained. The mixed boundary conditions involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and the threshold slip and leak condition of frictional type. K E Y W O R D Sfrictional type boundary condition, generalized Newtonian fluid, leak, operator inclusion, Oseen model, slip, unilateral boundary condition 2 0 1 0 M A In this paper we introduce and analyse a new class of operator inclusions of subdifferential type and a corresponding class of elliptic variational-hemivariational inequalities in a reflexive Banach space. Our main goal is to provide existence results to these classes of problems, and apply them to examine the stationary flow Oseen model of generalized Newtonian fluid described by equations − Div + ( ⋅ ∇) + ∇ = in Ω, div = 0 in Ω

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

“…The results will find important applications to model the semipermeable media, contact problems in solid and fluid mechanics, etc. Moreover, it would be desirable to extend the results with nonconvex constraints sets to second order evolution problems motivated by dynamic contact models in viscoelasticity, thermoviscoelasticty, see [17,19], and nonstationary fluid models, see [20,21].…”

“…can be found in classical monographs [8,9,16], and in two recent books [17,18], and the references therein. A unified method, based on the hemivariational inequality formulation, to study contact problems of viscoelasticity is given in [19], the abstract elliptic variational-hemivariational inequalities in reflexive Banach spaces with applications can be found in [1], and the variational-hemivariational inqualities which model fluid flow in mechanics were treated in [20,21] and very recently in [22,23]. Other recent developments on variational methods in the study of existence and multiplicity of solutions, see [24,25].…”

A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

“…They play an important role in a description of diverse mechanical problems arising in solid and fluid mechanics. We refer to [2][3][4]12,19,20,26,28,34,38,39] and the references therein for the recent results on the mathematical theory of contact mechanics and related issues.…”

A Quasi Variational-Hemivariational Inequality for Incompressible Navier-Stokes System with Bingham Fluid

“…The existence of solution to (1.1) is demanding and, as far as we know, this abstract problem has not been studied up to now. In a particluar case, if B = 0, K(u) = K, C = K, and φ(u, z) = Φ(N z), then (1.1) reduces to a variational-hemivariational inequality treated in [42]. In addition, if j = 0, then from (1.1) we arrive to an elliptic variational inequality of the second kind, see, e.g., [30].…”