Abstract
This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. The control parameter is the external force. We discuss both strong and weak solutions. Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. Moreover, a necessary condition for the existence of strong solutions is derived. Simultaneously, we introduce the concept of the marginal function and study its properties. In particular, it is shown that the marginal function of this control system is lower semicontinuous with respect to the directed Hausdorff distance.
Highlights
It is well known that the behavior of many real fluids cannot be satisfactorily described by the classical Navier–Stokes equations
An important example of non-Newtonian models is given by the second-grade fluids model, which forms a subclass of differential type fluids of complexity 2 and is one of basic constitutive models for viscoelastic fluid [1,2]
We initiated the mathematical study of nonlocal models of fluid dynamics by using methods of nonlinear functional analysis
Summary
It is well known that the behavior of many real fluids cannot be satisfactorily described by the classical Navier–Stokes equations. We shall focus on the case when the nonlinear terms αD (y)W (y) and αW (y) D (y), which contain products of the first derivatives of the velocity field y with respect to the space variables, are small compared to other terms in the third equality of (1) and can be discarded. We study an optimal control problem for the motion equations of viscous fluids with polymer additives in a bounded domain Ω ⊂ Rn with boundary Γ, assuming the flow is time-independent, that is, ∂t y = 0, ∂t p = 0, and ∂t u = 0.
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