Evolutionary Oseen Model for Generalized Newtonian Fluid with Multivalued Nonmonotone Friction Law

Abstract

The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system.

Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control

“…The necessity to understand time-dependent applied models involving nonsmooth and nonconvex energy principles gives rise to evolutionary hemivariational inequalities. A prototypical example is the study of nonstationary fluid flow problems modeled by nonmonotone and set-valued frictional laws, see [34][35][36].…”

Optimal control of an evolution hemivariational inequality involving history-dependent operators

“…The literature on hemivariational inequalities has been significantly enlarged in the last forty years, mainly because of their multiple relevant applications to various fields, see monographs [33,37,44] for analysis of various classes of such inequalities, see e.g. [2][3][4][5][11][12][13][14]21,22,24,[26][27][28]30,32,34,43,47,49,51] .…”

Hemivariational Inequality for Navier–Stokes Equations: Existence, Dependence, and Optimal Control

“…In Orcliz spaces, hemivariational inequalities for Newtonian and Non-newtonian Navier-Stokes equations has been recently studied in [25], [24]. Hemivariational inequalities for generalized Newtonian fluids are recently extensively studied see [13] and references therein, see also [23] for evolutionary Oseen model for generalized Newtonian fluid. For an equilibrium problem approach to hemivariational inequalities for Navier-Stokes equations we refer to [1] and [4].…”

“…For an equilibrium problem approach to hemivariational inequalities for Navier-Stokes equations we refer to [1] and [4]. For different aspects about nonsmooth optimization in the context of Navier-Stokes system we refer to [16,17,19,20,33,31,32,33,34,50] The goal of this paper is threefold. We aim to (1) show the existence of weak solutions to the hemivariational inequality corresponding to the problem (1.1)-(1.3), (2) prove a dependence result of solutions with respect to the hemivariational part and to the density of the external forces, (3) formulate and study the distributed parameter optimal control where the control is represented by the density of the external forces.…”