On the existence of weak solutions for the initial‐boundary value problem in the Jeffreys model of motion of a viscoelastic medium

Abstract

This paper deals with the initial-boundary value problem for the system of motion equations of an incompressible viscoelastic medium with Jeffreys constitutive law in an arbitrary domain of two-dimensional or three-dimensional space. The existence of weak solutions of this problem is obtained.

Trajectory and Global Attractors of the Boundary Value Problem for Autonomous Motion Equations of Viscoelastic Medium

“…From the results of [15] it follows that the estimate (3.19) together with Lemma 3.2 ensures convergence of all terms in identities (3.10), (3.11) with (u m,k , τ m,k ) substituted there to the corresponding terms in (3.5), (3.6) at least in the sense of scalar distributions on (0, T ). Therefore the pair (u, τ ) satisfies identities (3.5), (3.6) for all ϕ ∈ V and Φ ∈ C ∞ 0 .…”

“…Lemma 3.1 (the proof may be found in [15]). For u ∈ V , τ ∈ H 1 0 the following identities take place (τ, ∇u) + (u, Div τ ) = 0,…”

“…Now we are ready to turn to the proof of Theorem 3.1. In [15] (Theorem 3.2) it was proved that there exist solutions (u m,k , τ m,k ) (where m, k are natural parameters in (3.10), (3.11)) of problem (3.10)-(3.11), (3.8) Applying the formula of derivative of a product and multiplying by e 2γt we obtain …”

“…Therefore the pair (u, τ ) satisfies identities (3.5), (3.6) for all ϕ ∈ V and Φ ∈ C ∞ 0 . From the results of [15] and Lemma 3.2 it follows that the pair (u, τ ) belongs to class (3.7) and satisfies condition (3.8).…”

“…Existence of weak solutions for problem (0.1)-(0.4) for n = 2, 3 is proved in [15] using the approximating-topological method [16]. The problem of uniqueness of these solutions is open even at n = 2.…”