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 prob- lem 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 Ω t ∈ R n ,2≤ n ≤ 4 be a family of the bounded domains with boundary Γ t , Q ={(t,x):t ∈ [0,T], x ∈ Ω t }, Γ ={(t,x):t ∈ [0,T], x ∈ Γ t }.Thefollow- ing initial boundary value problem is under consideration: ρ v t + v i ∂v/∂x i − µ 1 Div t 0 exp − t − s λ Ᏹ(v) s,z(s;t,x) ds − µ 0 DivᏱ(v) =−grad p + ρϕ,divv = 0, (t,x) ∈ Q; Ω t pdx= 0, t ∈ [0,T]; v(0,x) = v 0 (x), x ∈ Ω 0 , v(t,x) = v 1 (t,x), (t,x) ∈ Γ. (1.1) Here v(t,x) = ( v 1 , ,v n )isavelocityofthemediumatlocationx at time t, p(t,x)isa pressure, ρ, µ 0 , µ 1 , λ are positive constants, Div means a divergence of a matrix, the matrix Ᏹ(v)hascoefficients Ᏹ ij (v)(t,x) = (1/2)(∂v i (t,x)/∂x j + ∂v j (t,x)/∂x i ). 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)isgivenin[21]. One can find the details in [12,Chapter 4]. We assume that a domain Q ⊂ R n+1 is defined as an evolution Ω t , t ≥ 0ofthevolume Ω 0 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 which is defined in some cylindrical domain Q 0 ={(t,x):t ∈ [0,T], x ∈ Ω 0 },sothat Ω t ⊂ Ω 0 . This means that Ω t = ˜ z(t;0,Ω 0 ), where ˜ z(τ;t,x) is a solution to the Cauchy 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 ofadomainQ and the trace of the function ˜ v(t,x)onΓ 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 Ω t 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)forµ 1 = 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 reg- ularized initial boundary value problem (1.1) in a domain with a time-dependent bound- ary Γ t 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 sense and application of the topological theory of a degree that allows to establish the existence of solutions on the basis of a priori estimates and statements about passage to the limit. Note that in the case of a not cylindrical domain (with respect to t) the necessary spaces of differentiable functions cannot be regarded as spaces of functions of t with values in some fixed func- tional space. Consequently, the direct application of the method of [21] is not possible. The history of the motion equation from (1.1) is given in details in [21]. On the basis of the rheological relation of Jeffreys-Oldroyd type the existence theorem for weak solutions in a domain with a constant boundary was proved. The purpose of the present paper is to prove a similar result for a domain with changing boundary. The article is organized as follows. We need a number of auxiliary results about func- tional spaces for the formulation of the basic results. They are presented in Section 2. We also need some results about the linear problem in a non-cylindrical domain which are given in Section 3. By this the proofs of the part of the results (which require the rather long proofs) are given in Section 8.InSection 4, the main results are formulated, in Sections 5–7 the proofs of the main results are carried out. We will denote constants in inequalities and chains of inequalities by the same M if their values are not important. 2. Auxiliary results 2.1. Functional spaces. Let us introduce necessary functional spaces. Denote norms in L 2 (Ω t )andW k 2 (Ω t )by|·| 0,t and |·| k,t accordingly. Denote by · 0 anorminL 2 (Q) or in L 2 (Q 0 )(Q 0 = [0,T] × Ω 0 ), depending on a context. We will denote by D 0,t the set of functions, which are smooth, solenoidal and finite on domain Ω t . We will designate through H t and V t acompletionofD 0,t in the norms L 2 (Ω t )andW 1 2 (Ω t )accordingly. We denote by V ∗ t the conjugate space to V t and by |·| −1,t the norm in V ∗ t . We denote by v,h t an action of the functional v ∈ V ∗ t upon an element h ∈ V t .Thus,thescalar product (·,·) t in H t generates (see, e.g., [7, Chapter 1, page 29]) the dense continuous V. G. Zvyagin and V. P. Orlov 217 embeddings V t ⊂ H t ⊂ V ∗ t at every t ∈ [0,T]. It is clear that u,v t ≤|u| 1,t |v| −1,t , u ∈ V t , v ∈ V ∗ t . (2.1) Let D be the set of smooth vector functions on Q, solenoidal and finite on a domain Ω t for every t. It is easy to show that scalar functions ϕ(t) =|v(t,x)| 1,t , ψ(t) =|v(t,x)| −1,t , g(t) =|v t (t,x)| −1,t ,wherev t (t,x)isaderivativewithrespecttot of function v(t,x), are determined and continuous on [0,T]foreveryv ∈ D. We denote by E, E ∗ , E ∗ 1 , W, W 1 , CH, EC, L 2,σ (Q) the completion of D accordingly in norms v E = T 0 v(t,x) 2 1,t dt 1/2 , v E ∗ = T 0 v(t,x) 2 −1,t dt 1/2 , v E ∗ 1 = T 0 v(t,x) −1,t dt, v W =v E + v t E ∗ , v W 1 =v E + v t E ∗ 1 , v CH = max t∈[0,T] v(t,x) 0,t , v EC =v E + v CH , v 0 = T 0 v(t,x) 2 0,t dt 1/2 . (2.2) Let a sequence v n ∈ D, n = 1,2, be fundamental on Q in the norm · 0 : T 0 |v n (t,x) − v m (t,x)| 2 0,t dt → 0, n,m → +∞. Then (see [19, page 224]) there exists a subsequence v n k (t,x) which is fundamental at a.e. t in the norm |·| 0,t .Letv(t,x) ∈ L 2 (Ω t ) be the limit of v n k (t,x). Solenoidality of functions from D implies v(t,x) ∈ H t at a.e. t. It implies the possibility to get the completion of D in the norm · 0 as a subspace of usual functions from L 2 (Q). It is similarly shown that an element v ∈ E is a function v(t,x)ata.e.t, v(t,x) ∈ V t , v 2 E = T 0 |v(t,x)| 2 1,t dt,andv ∈ E ∗ is a function v(t,x) ∈ V ∗ t at a.e. t, v 2 E ∗ = T 0 |v(t, x)| 2 −1,t dt.Forv ∈ E ∗ 1 we have at a.e. tv(t,x) ∈ V ∗ t and v E ∗ 1 = T 0 |v(t,x)| −1,t dt. Lemma 2.1. Let v ∈ E, h ∈ E ∗ . The scalar funct ion v(t,x),h(t,x) t = ψ(t) is summable and T 0 v(t,x), h(t,x) t dt =v,h. (2.3) Proof. Choose such a sequences v n ,h n ⊂ D that v n − v E → 0, h n − h E ∗ → 0and|v n (t, x) − v(t,x) | 1,t → 0, |h n (t,x) − h(t,x)| −1,t → 0ata.e.t. Then the sequence of continuous functions ψ n (t) = (v n (t,x), h n (t,x)) t converges to ψ(t)ata.e.t and, hence, ψ(t)ismea- surable. As T 0 ψ n (t) dt v n E h n E ∗ M, (2.4) where M does not depend on n, then the Fatou’s theorem implies summability of ψ(t). From convergence v n → v in E, h n → h in E ∗ and the equality ψ(t) =v,h (2.3) easily follows. The lemma is proved. 218 On weak solutions of the equations of motion The scalar product (v,h) = T 0 (v(t,x), h(t,x)) t dt in L 2,σ (Q) generates the continuous embeddings E ⊂ L 2,σ (Q) ⊂ (E) ∗ .Here(E) ∗ is adjoint to E. Denote by v,h an action of the functional v ∈ (E) ∗ upon h. It turns out that (E) ∗ = E ∗ .Really,E ∗ is a subspace in (E) ∗ .IfE ∗ does not coincide with (E) ∗ , we can find an element v 0 = 0inE for which h,v 0 =0forallh ∈ E ∗ . Choosing elements h from the set D dense in E ∗ ,wegetthat h,v 0 =(h,v 0 ) = 0forallh ∈ D. This implies v 0 = 0. Therefore, E ∗ = (E) ∗ and the scalar product (v,h)inL 2,σ (Q) generate the continuous embeddings E ⊂ L 2,σ (Q) ⊂ E ∗ .Note that |v,h| v E h E ∗ . In the Banach spaces introduced above it is convenient for us to define equivalent norms by the rule v k,F = ¯ v F , ¯ v = exp(−kt)v, k>0. Here F is any Banach space of functions defined on Q. The space E ∗ is continuously embedded in E ∗ 1 .Belowv,u denotes an action of a functional v ∈ E ∗ upon a function u ∈ E. Besides, we need the set CG of functions z(τ;t,x)definedon[0,T] × Q which are continuous with respect to all variables and con- tinuously differentiable with respect to x. Moreover, these functions are diffeomorphisms of Ω t on Ω τ with the determinants equal to 1. We will consider CG as a metric space with the metrics ρ(z 1 ,z 2 ) =z 1 − z 2 CG where z CG = max τ max t z(τ;t,x) C( ¯ Ω t ) . We denote by W l,m 2 (Q) the usual Sobolev spaces of functions f (t,x)onQ,having generalized derivatives up to order l with respect to t and up to order m with respect to x which are square summable. · l,m stands for their norms. 2.2. Regularization operator. Problem (1.1) involves the integral which is calculated along the trajectory z(τ;t,x)ofaparticlex in the field of velocities v(t, x)wherebyz(τ;t,x) is a solution to the Cauchy problem (1.2). However, even strong solutions v(t,x)ofprob- lem (1.2), having a derivative w ith respect to t and the second derivatives with respect to x, square summable on Q, do not provide uniquely solvability of problem (1.1). As an exit from this situation in [21](following[9]) the regularization of the field of velocity with the help of introduction of a linear bounded operator S δ,t : H t → C 1 ( ¯ Ω t ) ∩ V t for δ>0 such that S δ,t (v) → v in H t at δ → 0andfixedt was offered. As far as the boundar y Γ t of a domain Ω t is concerned, it was assumed to be sufficiently smooth. In the construction of this operator a smooth decomposition of the unit for Ω t , some homothety transforma- tions in R n and the operator P t of orthogonal projection in L 2 (Ω t )onH t were used. Let v(t,x) ∈ L 2,σ (Q). We define on L 2,σ (Q)theoperator S δ (v) = v where v(t,x) = S δ,t (v(t,x)) at t ≥ 0. As the operator P t is constructed by means of solutions of the Dirichlet and Neumann problems for Ω t (see [18, page 20]) and Ω t in our case smoothly depends on t,fromthe structure of the operator S δ,t at fixed t it follows that the operator S δ is a linear bounded operator from L 2,σ (Q)inE ∩ L 2 (0,T;C 1 ), and S δ (v) → v in E at δ → 0. Here L 2 (0,T;C 1 ) is a Banach space which is a completion of the set of smooth functions on Q in the norm v L 2 (0,T;C 1 ) = ( T 0 v(t,x) 2 C 1 ( ¯ Ω t ) dt) 1/2 .Letnowu(t,x) ∈ L 2,σ (Q), v(t,x) = u(t,x)+ ˜ v(t,x). Let us define the regularization operator S δ : L 2,σ (Q) → L 2 (0,T;C 1 ) ∩ E by the formula S δ (v) = S δ (v − ˜ v)+ ˜ v = S δ (u)+ ˜ v. It is clear that S δ (v) → v in L 2,σ (Q)atδ → 0. Consider problem (1.2)forv(t,x) ⊂ L 2 (0,T;C 1 ). The solvability of problem (1.2)for the case of a cylindrical domain Q was established in [10]forv ∈ L 2 (0,T;C 1 ) vanishing V. G. Zvyagin and V. P. Orlov 219 on Γ. In the same place, the estimate was obtained: z 1 (τ;t,x) − z 2 (τ;t,x) C( ¯ Ω t ) ≤ M τ t v 1 (s,x) − v 2 (s,x) C 1 ( ¯ Ω s ) ds , (2.5) where v 1 ,v 2 ∈ L 2 (0,T;C 1 ), t,τ ∈ [0,T]. The same f acts are fair and for the case of a non cylindrical Q and for functions coinciding with ˜ v on Γ. The proofs are similar to ones resulted in [10] with minor alterations. Inequality (2.5)isrequiredtousinwhatfollows below. We replace (1.2)for(1.1) by the equation z(τ;t,x) = x + τ t S δ v s,z(s;t,x) ds, τ,t ∈ [0,T], x ∈ Ω t . (2.6) For every v(t, x) ∈ E the function S δ (v) ∈ L 2 (0,T;C 1 ), and, hence, problem (2.6)is uniquely solvable. we designate by ˜ Z δ (v) the solution to problem (2.6). Note that S δ (v) coincides with ˜ v on Γ. In particular it means that all trajectories z(τ;t,x)ofproblems(2.6) lay in Q. 3. Linear parabolic operator on noncylindrical domain Consider a linear operator L : E ∗ → E ∗ defined on the set D(L) of smooth solenoidal functions v(t,x) vanishing on Γ and at t = 0bytheformula Lv,h= T 0 v t − ∆v,h t dt. (3.1) Here h(t,x) ∈ E.Obviously,D(L) is dense in E ∗ . Let us show that the operator L admits a closure and study its properties. First we establish auxiliary results. The following result is known (see [11, page 8]). Lemma 3.1. Let F(t,x) be a smooth scalar function. Then d dt Ω t F(t,x)dx = Ω t F t (t,x)dx + Γ t F(t,x) ˜ v n (t,x) dx. (3.2) Here ˜ v n (t,x)istheprojectionof ˜ v(t,x) on the direction of the external normal n(x)at apointx ∈ Γ t . Corollary 3.2. If F(t,x) = 0 for (t,x) ∈ Γ then (d/dt) Ω t F(t,x)dx = Ω t F t (t,x)dx. Lemma 3.3. Afunctionv ∈ D(L) satisfies the inequality: v t E ∗ +sup 0≤t≤T v(t,x) 0,t + v E ≤ MLv E ∗ . (3.3) 220 On weak solutions of the equations of motion Proof. Integration by parts and use of Corollary 3.2 yields t 0 (Lv,v) s ds = t 0 v t (s,x),v(s,x) s − v(s,x), v(s,x) s ds = t 0 1 2 d ds v(s,x) 2 0,s + ∇ v(s,x), ∇v(s,x) s ds = t 0 1 2 d ds v(s,x) 2 0,s + v(s,x) 2 1,s ds = 1 2 v(t,x) 2 0,t + t 0 v(s,x) 2 1,s ds. (3.4) From this it follows that Lv,v=1/2|v(T, x)| 2 0,T + v 2 E .As|Lv,v|≤Lv E ∗ v E we get from (3.4) the inequality sup t |v(t,x)| 0,t + v E ≤ MLv E ∗ . On the other hand, v t E ∗ = sup h∈E,h E =1 T 0 v t ,h t dt ≤ sup h∈E,h E =1 T 0 v t −v,h t dt +sup h∈E,h E =1 T 0 (v,h) t dt ≤Lv E ∗ + v E . (3.5) Thus, v t E ∗ ≤ MLv E ∗ . The last inequalities imply (3.3). The lemma is proved. Lemma 3.4. Let v ∈ W. Then |v(t,x)| 0,t is absolutely continuous in t on [0,T],differentiable at a.e. t ∈ [0,T] and 1 2 d dt v(t,x) 2 0,t = v t (t,x), v(t, x) t , (3.6) v(t,x) 0,t ≤ ε v t E ∗ + M(ε)v E , ε>0, t ∈ [0,T]. (3.7) Proof. Let v(t,x) be smooth. Then (3.6)followsfromCorollary 3.2.Letusprove(3.7). It follows from (3.6)and(2.1)thatfor0≤ τ ≤ T v(t,x) 2 0,t ≤ v(τ,x) 2 0,τ +2 t τ v s (s,x) −1,s v(s,x) 0,s ds ≤ v(τ,x) 2 0,τ +2 v t E ∗ v E (3.8) is valid. Supposing t ≥ T/4 and integrating over τ on [0, T/4], we have v(t,x) 2 0,t ≤ 4/T T/4 0 v(s,x) 2 0,s ds+2 v t E ∗ v E ≤ M v 2 L 2 (Q) + v t E ∗ v E . (3.9) The same inequality for t ≤ 3T/4 is established by means of integ ration over τ on [3T/4,T]. Using inequality v L 2 (Q) ≤v E and standard arguments we obtain from this inequality (3.7). The lemma is proved. V. G. Zvyagin and V. P. Orlov 221 In the cylindrical case this fact is proved in [18, Lemma 1.2, page 209]. Lemma 3.5. The space W is embedded in EC and v EC ≤ Mv W . The proof of the lemma follows from Lemma 3.4. Let W 0 ={v : v ∈ W, v(0,x) = 0}. It is not hard to show that W 0 can be easily obtained by means of the closure in the W-norm of the set of smooth on Q functions which are solenoidal on Ω t at every t and vanish on ∂Ω t and Ω 0 .Infact,letv ∈ W 0 . By the defini- tion of W there exists a sequence of functions v n smooth on Q and solenoidal at every t such that v − v n W → 0byn →∞.Letϕ n (t) be a smooth nondecreasing on [0,T] func- tion such that ϕ n (t) ≡ 0whent ∈ [0,T/n]andϕ n (t) ≡ 1whent ∈ [2T/n]. The function u n (t,x) = ϕ n (t)v n (t,x) vanishes at t = 0andon∂Ω t at every t.Obviously,u n converges to v by n →∞in the W-norm. Theorem 3.6. The operator L admits a closure ¯ L : E ∗ → E ∗ with D( ¯ L) = W 0 ,itsrangeR( ¯ L) is closed and ¯ L is invertible on R( ¯ L). Proof. Fro m (3.3) it follows that L admits a closure ¯ L.ItsdomainD( ¯ L) consists of those v ∈ E ∗ for which there exists such a sequence v n ∈ D(L)thatv n → v and Lv n → u in E ∗ . Then by definition ¯ Lv = u. Let us show that D( ¯ L) ⊆ W 0 .Letv ∈ D( ¯ L). Then there exists such a sequence v n ∈ D(L)thatv n → v in E and Lv n → u in E ∗ . Then by means of passing to the limit we have from (3.3)forv n that v ∈ W 0 and the inequality holds: v t E ∗ +sup t v(t,x) 0,t + v E ≤ M ¯ Lv E ∗ . (3.10) Thus, D( ¯ L) ⊆ W 0 . Let us show that D( ¯ L) ⊇ W 0 .Letv ∈ W 0 and v n → v, v n ∈ D,intheW-norm that v ∈ W 0 .From(3.1) it follows that for h ∈ E ¯ Lv n ,h=Lv,h= T 0 (∇v n (t,x), ∇h(t,x)) t dt + v n t ,h takes place. The passage to the limit gives the validity of ¯ Lv,h=v t ,h + T 0 (∇v(t,x), ∇h(t,x)) t dt for v ∈ D( ¯ L). From the obtained above it follows that the right- hand side part defines an element u ∈ E ∗ for any v ∈ W 0 . By this v ∈ D( ¯ L)and ¯ Lv = u. Thus, W 0 ⊆ D( ¯ L) and consequently W 0 = D( ¯ L). From (3.10) it follows that R( ¯ L)isclosedand ¯ L is invertible on R( ¯ L). The theorem is proved. Remark 3.7. From (3.4) established for smooth v by means of the passage to the limit and the differentiation with respect to t it is easy to show that the scalar function ( ¯ Lv,v) t for v ∈ D(L) and a.e. t satisfies the relation ( ¯ Lv,v) t = 1 2 d dt v(t,x) 2 0,t + ∇v(t,x) 2 0,t . (3.11) Theorem 3.8. The range R( ¯ L) of the operator ¯ L is dense in E ∗ . We give the proof of this theorem in Section 8. From Theorems 3.6 and 3.8 the next result follows. 222 On weak solutions of the equations of motion Theorem 3.9. For every f ∈ E ∗ the equation ¯ Lv = f has a unique solution v and the esti- mate holds: v t E ∗ +sup t v(t,x) 0,t + v E ≤ M f E ∗ . (3.12) Let k>0. Ever ywhere below we set ¯ v = exp(−kt)v. It is easy to show that exp(−kt) ¯ L(v) = ¯ L( ¯ v)+k ¯ v. From here it follows that if L(v) = f then ¯ L( ¯ v)+k ¯ v = ¯ f . Corollary 3.10. For the solution v of the equation ¯ Lv = f by any k>0 the estimate holds: v t E ∗ ,k + v EC,k ≤ M f E ∗ ,k . (3.13) To prove it is enough to make the change ¯ v = exp(−kt)v and take advantage of Theorem 3.6.FromCorollary 3.10 and from Theorem 3.9 there follows the following the- orem. Theorem 3.11. For every ¯ f ∈ E ∗ the equation ¯ L( ¯ v)+k ¯ v = ¯ f has a unique solution ¯ v and theestimateholds: ¯ v t E ∗ +sup t ¯ v(t,x) 0,t + ¯ v E + k ¯ v 0 ≤ M ¯ f E ∗ . (3.14) 4. Formulation of the main results We are interested in the solvability of the regularized problem (1.1). By this we suppose without loss of generality µ 0 = µ 1 = ρ = 1, replace z(s;t,x)byZ δ (v) and restrict ourselves with the case v 0 (x) = ˜ v(0,x), x ∈ Ω 0 , v 1 (t,x) = ˜ v(t,x), (t, x) ∈ Γ. Thus, we get the prob- lem v t + v k ∂v/∂x k − DivᏱ(v) − Div t 0 exp(s − t)Ᏹ(v) s, ˜ Z δ (v) (s;t,x)ds =−∇p + Φ, divv(t,x) = 0, (t,x) ∈ Q; Ω t p(t,x)dx = 0, t ∈ [0,T]; v(0,x) = ˜ v(0,x), x ∈ Ω 0 , v(t,x) = ˜ v(t,x), (t,x) ∈ Γ. (4.1) One should mark that the case of smooth and satisfying accordance conditions func- tions v 0 (x)andv 1 (t,x)in(1.1) can be reduced to the conditions in (4.1). Let Φ(t,x) ∈ V ∗ t at a.e. t.Let w(t, y) = w(t,z(t;0, y)) for an arbitrary function w(t,x)definedonQ. Definit ion 4.1. A function v(t,x) = ˜ v(t,x)+w(t,x), w(t,x) ∈ E, w ∈ L 2 0,T;W 1 2 Ω 0 W 1 1 0,T;W −1 2 Ω 0 (4.2) V. G. Zvyagin and V. P. Orlov 223 is called a weak solution of problem (4.1)ifforanyh(t,x) ∈ D, h(T,x) = 0 the identity holds: − T 0 v(t,x), h t (t,x) t dt + T 0 v i (t,x)v j (t,x), ∂h i (t,x)/∂x j t dt + T 0 Ᏹ ij v(t,x) ,Ᏹ ij h(t,x) t dt + T 0 t 0 exp(s − t)Ᏹ ij (v) s, Z δ (v)(s;t,x) ds,Ᏹ ij h(t,x) t dt = Φ(t,x),h(t, x) − ˜ v(0,x), h(0,x) 0 . (4.3) The following main result takes place. Theorem 4.2. Let Φ = f 1 + f 2 , f 1 ∈ E ∗ 1 , f 2 ∈ E ∗ . Then the problem (4.1) has at least one weak solution. The proof of the theorem is organized as follows. Following [21], we need to consider a family of approximating operator e quations with a more weak nonlinearity. Alongside with the operator ¯ L : W 0 → E ∗ introduced above we will consider the operators K i t : V t −→ V ∗ t , i = 1,2,3, K 3 t (w),h t = w i w j ,∂h i /∂x j t , w,h ∈ V t ; K 1 t (w),h t = w i ˜ v j ,∂h i /∂x j t , w,h ∈ V t ; K 2 t (w),h t = ˜ v i w j ,∂h i /∂x j t , w,h ∈ V t ; (4.4) the functional ˜ g ∈ E ∗ : ˜ g, h= T 0 ˜ v i ˜ v j ,∂h/∂x j t dt, h ∈ E; (4.5) the functional ˜ V ∈ E ∗ : ˜ V, h= T 0 ˜ v t − ˜ v,h t dt, h ∈ E; (4.6) the operator A t : E → E ∗ : A t (v),h t = (Ᏹ ij (v),Ᏹ ij (h)) t , v,h ∈ V t ;theoperatorC t : E × CG → V ∗ t C t (v,z), h t = t 0 exp(s − t)Ᏹ ij (v + ˜ v) s,z(s;t,x)ds,Ᏹ ij (h) t , v ∈ E, z ∈ CG. (4.7) 224 On weak solutions of the equations of motion The operators A t and C t naturally generate the operators A : E → E ∗ ,andC : E × CG → E ∗ : A(w),h = T 0 A t (w),h t dt h ∈ E ∗ , w ∈ E; C(v,z),h = T 0 C t (v,z), h t dt, v ∈ E, z ∈ CG, h ∈ E. (4.8) Alongside with the operators K i t we will consider for ε>0theoperatorsK i t,ε : V t → V ∗ t and the operators K i ε : E → E ∗ generated by them: K 1 t,ε (u),h t = u i ˜ v j 1+ε|u| 2 , ∂h i ∂x j t , K 2 t,ε (u),h t = ˜ v i u j 1+ε|u| 2 , ∂h i ∂x j t , K 3 t,ε (u),h t = u i u j 1+ε|u| 2 , ∂h i ∂x j t , u,h ∈ V t ; K i ε (u),h = T 0 K i t,ε (u),h(t,x) t dt, u,h ∈ E. (4.9) Let K ε (v) = 3 i=1 K i ε (v). Let Z δ (w) = ˜ Z δ ( ˜ v + w)forw ∈ E. Consider for ε>0theoperator equation ¯ Lw − K ε (w)+C w,Z δ (w) = f ε . (4.10) Theorem 4.3. For any f ε ∈ E ∗ (4.10) has at least one solution w ε ∈ W 0 . Let us approximate a function f 1 ∈ E ∗ 1 by f 1,ε → f 1 at ε → 0, f 1,ε ∈ L 2,σ (Q). Let v ε = w ε + ˜ v,wherew ε is a solution to (4.10). Using the passage to the limit at ε → 0 we establish that functions v ε converge to the function v which is a weak solution to problem (4.1), that is, we get the assertion of Theorem 4.2. The proof of Theorem 4.3 is carried out in Section 6.Theoperatortermsinvolvedin (4.10) are investigated in Section 5. 5. Investigation of propert ies of operators To investigate the operator terms of (4.10) we need the additional properties of functional spaces. Let ˜ z be a solution to the Cauchy problem (1.3)andu(t, y) = ˜ z(t;0,y), y ∈ Ω 0 .Itis clear that u maps Q 0 in Q, Q 0 = [0, T] × Ω 0 and u(t,Ω 0 ) = Ω t .LetU(t,x) = ˜ z(0; t,x). The maps u(t, y)andU(t,x)atfixedt are mutually inverse and U t,u(t, y) = y, y ∈ Ω 0 , u t,U(t,x) = x, x ∈ Ω t . (5.1) [...]... 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(8.15) is uniquely solvable in W2 (Ω) (see [15]) −1 −1 Its solution has the form Ψ = N γn ∇Φ Here N is a linear bounded operator acting 1/2 2 from W2 (Γ0 ) into W2 (Ω0 ) (see [17]), and γn is the operator of taking of the normal component of a trace on the boundary of a function defined on Ω0 The operator γn 236 On weak solutions of the equations of motion 1/2 1 is bounded as an operator from W2 (Ω0 )... problem of the motion of a viscous incompressible fluid that is bounded by a free surface, Izv Akad Nauk SSSR Ser Mat 41 (1977), no 6, 1388–1424, 1448 (Russian) R Temam, Navier-Stokes Equations Theory and Numerical Analysis, Mir, Moscow, 1987 B Z Vulikh, Short Course of Theory of Functions of Real Variable, Nauka, Moscow, 1978 V G Zvyagin, On the Theory of Generalized Condensing Perturbations of Continuous... Operators, “Nauka” Sibirsk Otdel., Novosibirsk, 1986 K Amedodji, G Bayada, and M Chambat, On the unsteady Navier-Stokes equations in a timemoving domain with velocity-pressure boundary conditions, Nonlinear Anal Ser A: Theory Methods 49 (2002), no 4, 565–587 V T Dmitrienko and V G Zvyagin, The topological degree method for equations of the NavierStokes type, Abstr Appl Anal 2 (1997), no 1-2, 1–45 , Approximation-topological... estimate easily E proved inequality | (w)|−1,t ≤ M |w|1,t , the boundedness of the map C on E and estimate (7.1) admit to get the required estimate (7.2) from (7.11) The theorem is proved 232 On weak solutions of the equations of motion ˜ Let us go over to the proof of Theorem 4.2 Let v = w + v It is easy to show that v is a weak solution to problem (4.1) if and only if w satisfies the integral identity... Kopachevski˘, S G Kre˘n, and H C Ngo, Operator Methods in Linear hydrodynamics, ı ı “Nauka”, Moscow, 1989 O A Ladyˇ enskaja, An initial -boundary value problem for the Navier-Stokes equations in doz mains with boundary changing in time, Zap Nauˇ n Sem Leningrad Otdel Mat Inst c Steklov (LOMI) 11 (1968), 97–128 (Russian) W G Litvinov, Regualr model and nonstationary problem for the nonlinear viscoelastic. .. fluid, Siberian J Differential Equations 1 (1997), no 4, 351–382 V P Orlov and P E Sobolevskii, On mathematical models of a viscoelasticity with a memory, Differential Integral Equations 4 (1991), no 1, 103–115 V V Pukhnachev, Motion of a Viscous Fluid with Free Boundaries, Novosibirskii gos universitet, Novosibirsk, 1989 O Rozanova (ed.), Analytical Approaches to Multidimensional Balance Laws, Nova Science,... cylindrical case ([3, Lemma 2.1 and Theorem 2.2]) For i = 1,2 the proof is easier because of the smoothness ˜ of v Lemma 5.6 For any v ∈ E, z ∈ CG the inclusion C(v,z) ∈ E∗ is valid and the map C : E × CG → E∗ is continuous and bounded The proof repeats the proof of Lemma 2.2 in [21] for the cylindrical case which is fit for the non cylindrical case as well Lemma 5.7 The map Zδ : W1 → CG is continuous and . 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. 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 sense and application of