TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID

In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued,nonmonotone friction law.First,a variational formulation of the model is obtained;that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field.Then,an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated.Under mild assumptions,the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven.Furthermore,a uniqueness theorem for the abstract inequality is established by using a monotonicity argument.Finally,we employ the theoretical results to examine the nonstationary Oseen model.

MIXED VARIATIONAL INEQUALITIES DRIVEN BY FRACTIONAL EVOLUTIONARY EQUATIONS

The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation in the framework of Banach spaces. Using discrete approximation approach, an existence theorem of solutions for the inequality is established under some suitable assumptions.

A new class of generalized quasi-variational inequalities with applications to Oseen problems under nonsmooth boundary conditions

In this paper, we study a generalized quasi-variational inequality (GQVI for short) with twomultivalued operators and two bifunctions in a Banach space setting. A coupling of the Tychonov fixedpoint principle and the Katutani-Ky Fan theorem for multivalued maps is employed to prove a new existencetheorem for the GQVI. We also study a nonlinear optimal control problem driven by the GQVI and givesufficient conditions ensuring the existence of an optimal control. Finally, we illustrate the applicability of thetheoretical results in the study of a complicated Oseen problem for non-Newtonian fluids with a nonmonotone andmultivalued slip boundary condition (i.e., a generalized friction constitutive law), a generalized leak boundarycondition, a unilateral contact condition of Signorini’s type and an implicit obstacle effect, in which themultivalued slip boundary condition is described by the generalized Clarke subgradient, and the leak boundarycondition is formulated by the convex subdifferential operator for a convex superpotential.

Optimal control of differential quasivariational-hemivariational inequalities with applications

In this paper,we study a new class of differential quasivariational-hemivariational inequalities of the elliptic type.The problem consists of a system coupling the Cauchy problem for an ordinary differential equation with the variational-hemivariational inequalities,unilateral constraints,and history-dependent operators.First,based on the Minty formulation and the continuity of the solution map of a parametrized quasivariational-hemivariational inequality,and a fixed point theorem for a history-dependent operator,we prove a result on the well-posedness.Next,we examine optimal control problems for differential quasivariational-hemivariational inequalities,including a time-optimal control problem and a maximum stay control problem,for which we show the existence of solutions.In all the optimal control problems,the system is controlled through a distributed and boundary control,a control in initial conditions,and a control that appears in history-dependent operators.Finally,we illustrate the results by considering a nonlinear controlled system for a time-dependent elliptic equation with unilateral constraints.