Báo Cáo Báo cáo hóa học ON WEAK SOLUTIONS OF THE EQUATIONS OF MOTION OF A VISCOELASTIC MEDIUM WITH VARIABLE

ON WEAK SOLUTIONS OF THE EQUATIONS OF MOTION OF A VISCOELASTIC MEDIUM WITH VARIABLE BOUNDARY V. G. ZVYAGIN AND V. P. ORLOV Received 2 September 2005 The regularized system of equations for one model of a viscoelastic medium with memory along trajectories of the field of velocities is under consideration. The case of a changing domain is studied. We investigate the weak solvability of an initial boundary value problem for this system. 1. Introduction The purpose of the present paper is an extension of the result of [21] on the case of a changing domain. Let Ωt ∈ Rn , 2 ≤ n ≤ 4 be a family of the bounded domains with boundary Γt , Q = {(t,x) : t ∈ [0,T], x ∈ Ωt }, Γ = {(t,x) : t ∈ [0,T], x ∈ Γt }. The following initial boundary value problem is under consideration: ρ vt + vi ∂v/∂xi ư µ1 Div = ư grad p + ρϕ, t 0 exp ư tưs λ (v) s,z(s;t,x) ds ư µ0 Div (v) Ωt div v = 0, x ∈ Ω0 , (t,x) ∈ Q; p dx = 0, (t,x) ∈ Γ. t ∈ [0,T];