Approximating-topological methods in some problems of hydrodynamics

Abstract

We describe a new approach to the study of initial-boundary value problems for evolutionary equations of hydrodynamics, which is based on approximation of the problems and subsequent application of topological degree theory for investigation of weak solvability for approximating problems. Use of this method turned out to be especially effective in problems of nonNewtonian hydrodynamics. We demonstrate its application to the study of weak solvability and attractors of the initial-boundary value problem for the Oldroyd model with regularized objective Jaumann derivative. (2000). Primary 35Q35; Secondary 47H11. Mathematics Subject Classification

Steady flows of an Oldroyd fluid with threshold slip

“…Here Ω is the domain occupied by the fluid, Γ denotes the boundary of Ω, v is the velocity, S is the extra-stress tensor, p is the pressure, f denotes the body force, E is the elastic part of the extra-stress tensor, D(v) is the strain-rate tensor, D(v) = 1 2 ∇v + (∇v) T , the operator D ρ /Dt is the regularized Jaumann derivative [42]. In the stationary case, this operator is defined by the formula…”

“…In contrast to (11), the constitutive law (4) is frame-indifferent (see [42]), i.e., the form of (4) does not change after a change of spatial variables. This means that the model considered here does not violate the principle of material objectivity [38].…”

“…We shall construct a weak solution of the original boundary value problem by passing to the limit n → ∞. In view of the estimates (41) and (42), we can assume without loss of generality that, for some v * ∈ X(Ω,…”

О Стационарном Движении Вязкоупругой Жидкости Типа Олдройда

“…[6]). Результаты о разрешимости начально-краевой задачи (с однородными краевыми условиями) для уравнений модели Олдройда с объективной регуляризованной производной Яуманна получены в работе [7]. Теорема о существовании и единственности локального по времени решения начально-краевой задачи для уравнений модели с объективной производной Олдройда доказана Гильопе и Со (см.…”

Uniform attractors for non-autonomous motion equations of viscoelastic medium

“…Existence of weak solutions for problem (0.1)-(0.4) for n = 2, 3 is proved in [16] using the approximating-topological method [17]. The problem of uniqueness of these solutions is open even at n = 2.…”