On weak solutions of the equations of motion of a viscoelastic medium with variable boundary
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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 − µ1 Div
t 0
= − grad p + ρϕ,
v(0,x) = v0 (x),
exp −
t−s Ᏹ(v) s,z(s;t,x) ds − µ0 Div Ᏹ(v) λ
div v = 0, x ∈ Ω0 ,
(t,x) ∈ Q;
Ωt
v(t,x) = v1 (t,x),
p dx = 0,
t ∈ [0,T];
(t,x) ∈ Γ. (1.1)
Here v(t,x) = (v1 , ...,vn ) is a velocity of the medium at location x at time t, p(t,x) is a pressure, ρ, µ0 , µ1 , λ are positive constants, Div means a divergence of a matrix, the matrix Ᏹ(v) has coefficients Ᏹi j (v)(t,x) = (1/2)(∂vi (t,x)/∂x j + ∂v j (t,x)/∂xi ). In (1.1) and in the sequel repeating indexes in products assume their summation. The function z(τ;t,x) is defined as a solution to the Cauchy problem (in the integral form) z(τ;t,x) = x +
τ t
v s,z(s;t,x) ds,
τ ∈ [0,T], (t,x) ∈ Q.
(1.2)
The substantiation of model (1.1) is given in [21]. One can find the details in [12, Chapter 4]. We assume that a domain Q ⊂ Rn+1 is defined as an evolution Ωt , t ≥ 0 of the volume Ω0 along the field of velocities of some sufficiently smooth solenoidal vector field v˜(t,x) Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 215–245 DOI: 10.1155/BVP.2005.215
216
On weak solutions of the equations of motion
0 }, so that 0 = {(t,x) : t ∈ [0,T], x ∈ Ω which is defined in some cylindrical domain Q 0 . This means that Ωt = z˜(t;0,Ω0 ), where z˜(τ;t,x) is a solution to the Cauchy Ωt ⊂ Ω problem
z˜(τ;t,x) = x +
τ t
v˜ s, z˜(s;t,x) ds,
τ ∈ [0,T], (t,x) ∈ Q.
(1.3)
Thus, it is clear that the lateral surface Γ of a domain Q and the trace of the function v˜(t,x) on Γ will be smooth enough, if v˜(t,x) is smooth enough. We will assume sufficient smoothness of v˜(t,x), providing validity of embedding theorems for domains Ωt used below with the common for all t constant. Let us mention some works which concern the study of the Navier-Stokes equations ((1.1) for µ1 = 0) in a time-dependent domain (see [2, 5, 8, 13] etc.), by this, different methods are used and various results on existence and uniqueness of both strong and weak solutions are obtained. In the present work, the existence of weak solutions to a regularized initial boundary value problem (1.1) in a domain with a time-dependent boundary Γt is established. The approximation-topological methods suggested and advanced in [3, 4] are used in the paper. It assumes replacement of the problem under consideration by an operator equation, approximation of the equation in a weak se
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