On one problem of viscoelastic fluid dynamics with memory on an infinite time interval

In the present paper we establish the existence of weak solutions of one boundary value problem for one model of a viscoelastic fluid with memory along the trajectories of the velocity field on an infinite time interval. We use solvability of related approximating initial-boundary value problems on finite time intervals and responding pass to the limit.

The problem (1)-(4) describes the motion of a fluid corresponding to the Jeffreys-Oldroyd constitutive law (see [12]) (1 + λ D)σ = 2ν(1 + κν −1 D)E, λ, κ, ν > 0, (5) where D means the total derivative with respect to time, σ is the deviator of the stress tensor, E is the strain rate tensor, µ 0 = 2κ/λ and µ 1 = 2(ν − κ)/λ ≥ 0. Details can be found in [19]. The presence of the integral term in (1) means that the memory on the stresses is taken into account. Various models with memory was appeared and studied in many papers (see e.g. [1], [3]- [6], [13]- [14], [18]). But as usual, mathematical statements of problems took into account the contribution of memory on a finite time interval but not along trajectories of particles (see e.g. [1], [13]). In this regard, the system (1)-(4) turned out to be without the equation (3) and became much more easier. In the case of a memory along trajectories of particles on a finite time interval problem (1)-(4) is considered in [20], where due to regularization of the velocity field v in (3) its weak solvability is established. A regularization of v provided good properties of the solution z to equation (3) and allowed to prove the solvability of the Cauchy problem (3) in the classical sense. Recent results on the solvability of the Cauchy problem in the class of regular Lagrangian flows (briefly RLF) (see e.g. [2], [7]- [8]) allow to avoid the regularization of (3) and establish the weak solvability of (1)-(4) on a finite time interval (see [21]).
Problems with memory along trajectories on the half-line (−∞, T ] occur for many models of viscoelastic fluids motion (see e.g. [3]- [4], [14]), but theorems on the existence of solutions of these problems for infinite intervals (−∞, T ] with memory along the trajectories of velocity field are unknown to us. The goal of the present paper is to establish a weak solvability of problem with a memory along the trajectories (1)-(4) without the requirement of a regularization of the velocity field.
Let us note, that we consider the case of exponential memory kernels, that are typically for Oldroyd type models (see [19], Ch.1). But there are various models with other memory kernels of another types of decay (see e.g. [6]). However, the investigation of such models requires particular analysis.
The paper is organized as follows. In Section 2 we recall some auxiliary results and formulate the main result. Section 3 is devoted to the proof of the main result and it is divided into several Subsections. In Subsection 3.1 we construct a series of approximative problems. In Subsection 3.2 uniform estimates of approximating solutions on the half-line are given. In Subsection 3.3 we find the limit of approximating solutions. In Subsections 3.4 and 3.5 we investigate the properties of the limit function. Finally, Subsection 3.6 completes the proof of the main result.
Constants in inequalities and chains of inequalities which do not depend on significant parameters are denoted by M .

2.
Preliminaries and main result. We will need Hilbert spaces H and V (see [16], Chapter I, Section 1.4) of divergence free vector functions. Here V = {v ∈ W 1 2 (Ω) n : v| Γ = 0, div v = 0} and H is the closure of V with respect to the norm of the space L 2 (Ω) n . Let V −1 be the space conjugate to V .
Norms in the spaces H and L 2 (Ω) n we denote by | · | 0 , in V by | · | 1 and in the space W β 2 (Ω) for β ∈ R 1 by | · | β , respectively. The norms in L 2 (−∞, T ; H) and L 2 (−∞, T ; L 2 (Ω) n ) are defined by the following formulae v L2(−∞,T ;H) = ( The sign (·, ·) stands for the scalar product in Hilbert spaces L 2 (Ω), H, L 2 (Ω) n , L 2 (Ω) n×n . It is clear from the context in which of them the scalar product is taken. The action of a functional f ∈ V −1 on an element ϕ ∈ V is denoted by f, ϕ .
If v ∈ L 1 (0, T ; C 1 (Ω)) and v| ∂Ω = 0 the problem (3) is nonlocal uniquely solvable in the classical sense (see [13]). However, in the case of only integrable with respect to x derivatives of v the situation is much more complicated, and a more general concept of solution to the problem (3) is required, namely, the concept of regular Lagrangian flows.
• for a.e. x and any t ∈ [0, T ] the function γ(τ ) = z(τ ; t, x) is absolutely continuous and satisfies the equation • m(z(τ ; t, B)) = m(B) for any t, τ ∈ [0, T ] and every Borel set B ⊂ Ω; • for t i ∈ [0, T ], i = 1, 2, 3 and a.e. x ∈ Ω the following relation is valid: Here m is the Lebesgue measure in R n . Above the definition of RLF is given in the particular case of a bounded domain Ω and a divergence free field v.
In a more general form this result is given in [7], Corollaries 3.6, 3.7 and 3.9.
Same results hold if we replace [0, T ] by any interval [r, T ] ⊂ (−∞, T ). We introduce the following functional spaces Let W = W 1 for n = 2 and W = W 2 for n = 3.
for any ϕ ∈ V and a.e. t ∈ (−∞, T ]. Here z is the RLF associated to v. The scalar products in the third and fourth terms of (8) are taken in L 2 (Ω) n×n . Let us formulate the main result.
3. Proof of theorem 2.5. The proof of Theorem 2.5 we will carry out in several steps.
3.1. Approximative problems. Consider the family of regularized initial-boundary value problems Here m = 1, 2, . . . , f m is the restriction of f on [−m, T ] × Ω, The motivation of introduction of the regularization operator K m in (9) is caused by the reason that K m (v) ∈ L 2 (0, T ; V −1 ) while K(v) = n i=1 v i ∂v/∂x i belongs, generally speaking, only to L 1 (0, T ; V −1 ) for v ∈ W . The change of K(v) by K m (v) ON ONE PROBLEM OF VISCOELASTIC FLUID DYNAMICS WITH MEMORY 3859 in (1) provides the existence of solutuion to problem (9)- (11) in the class W , together with the a regularization (smoothing)ṽ m of the field v m in (10).
The regularization of the velocity field in (10) is caused by the fact that the study of Cauchy problem (10) for v m ∈ W runs into difficulty because in this case the velocity field, generally speaking, doesn't determine the trajectory of fluid particles. One possible way to avoid this situation (see [19], section 7.1) is regularizatioñ v m = S 1/m v m of the velocity field v m ∈ W .
Introduce the functional space holds for all ϕ ∈ V and a.e. t ∈ [−m, T ] and the initial condition in (11) holds.
Here z m is RLF associated toṽ m . Theorem 3.3 (see [20]). Given f m ∈ L 2 (−m, T ; V −1 ), the problem (9)-(11) has at least one weak solution v m ∈ W (m) and inequalities hold: Remark 1. Note that in [20] the problem (9)- (11) has been considered on the interval [0, T ]. It is obvious, however, that the change of variable Below we will show that v m (with up to a subsequence) converges to v weakly in L 2 (−∞, T ; V ), * -weakly in L ∞ (−∞, T ; H) and strongly in L 2 (−∞, T ; H). Using these facts we prove that the limit function v is a weak solution of the problem (1)-(4).
Uniform on m estimates similar to (17)-(18) play an important role for our proof.

3.2.
Uniform on m estimates of approximating solutions on the half-line.
. Then for extended solutions v m of problems (9)-(11) the following uniform on m estimate holds: Proof of Lemma 3.4. In the same way as in [20] we obtain from (9) Here we have in mind that v m = 0 for t ≤ −m.
Integrating (20) with respect to time over (−∞, t], we get For the right hand side of (21) we have for arbitrary ε > 0 Denote the third term in the left hand side of (21) by (23) Let us transform K 3 (t). Using the change of variables x = z m (τ ; T, y) we have for the integrand I in K 3 (t) Here A : B for matrices A = {a ij } and B = {b ij } means A : B = n i,j=1 a ij b ij . Note that in (24) it was used the fact that the Jacobian J(τ ; T, y) = det z m y (τ ; T, y) of the change x = z m (τ ; T, y) is equal to one due to divergence free v m . In fact, it is well known that ∂ τ J(τ ; T, y) = (div v m )(τ, z m (τ ; T, y))J(τ ; T, y). Consequently, J(τ ; T, y) = exp((div v m )(τ, z m (τ ; T, y)))J(T ; T, y). Since div v m = 0 and J(T ; T, y) = det z m y (T ; T, y) = 1 we obtain the identity J(τ ; T, y) = 1. It follows from (23) and (24) that Using (22), (25) and Korn's inequality (see e.g. [11]) we get from (21) 3862 From (26) choosing ε > 0 small enough we obtain the inequality (19).
holds with independent on m but dependent on k constant C(k).
x) is continuous with respect to t as function with values in V −1 , it can be considered on [−k, T ] as a weak solution to the initial-boundary value problem From [20] there follows an estimate for u(t, x) The assertion of Lemma 3.5 follows from this estimate, equality u(t, x) = v m (t, x) and (19). 3.3. Passage to the limit. Estimate (19) means that the sequence v m is bounded in L 2 (−∞, T ; V ) and L ∞ (−∞, T ; H). This implies the existence of function v ∈ L 2 (−∞, T ; V ) ∩ L ∞ (−∞, T ; H) such that v m (with up to a subsequence) converges to v weakly in L 2 (−∞, T ; V ) and * -weakly in L ∞ (−∞, T ; H). Using properties of lower limits of weakly and * -weakly converging sequences (see e.g. [17], Theorems 1 and 9, Chapter V, section 1) we get from estimates (19) and (27) the following inequality sup In addition, estimates (19) and (27) entail (see e.g. [16], Chapter III, proof of Theorem 3.2) the convergence of v m to v (with up to a subsequence) a.e. on [−k, T ] × Ω for any k > 0, and, therefore, on (−∞, T ] × Ω. Let us show that the limit function v is a weak solution to problem (1)-(4). Defined on (−∞, T ] function v m satisfies, obviously, the identity for any ϕ ∈ V and ψ ∈ C ∞ 0 (−∞, T ). It is easy to see that the sequence f m converges to f in L 2 (−∞, T ; H). From estimates (18) and (19) it follows (see e.g. [20]) that the sequence v m converges (with up to subsequence) to v weakly in L 2 (−∞, T ; V ), *-weakly in L ∞ (−∞, T ; H), strongly in L 2 (Q) n , a.e. on Q = (−∞, T ]×Ω, and the sequence of derivatives dv m /dt is bounded in the norm of the space L 1 (−k, T ; V −1 ) and converges to dv/dt in the sense of distributions on [k, T ] for any −∞ < k < T .
Next, we will need several Lemmas.
But in virtue of (13) we have for integrand in (30) In addition, on the strength of the first inequality in (12) function g 2 (t) satisfies the inequality g 2 (t) ≤ M |v(t, ·)| 2 0 . Hence, in virtue of the Lebesgue Theorem I 2 (m) 0 → 0 as m → +∞. The assertion of Lemma 3.6 follows from the fact that I k (m) 0 → 0 as m → +∞ for k = 1, 2.
for any ϕ ∈ V and ψ ∈ C ∞ 0 (−∞, T ). Proof of Lemma 3.8. First, let ϕ ∈ V be smooth. Introduce the notation for terms in the left hand side of (29): Let the corresponding terms in the left hand side of (31) be Here z is the RLF associated to v.
(see [9]). Since the integration in J m 2 and J 2 is performed over the finite interval [k, T ] ⊃ supp ψ, then in virtue of Lemma 2.2 from [9] there follows the convergence of J m 2 to J 2 as m → +∞. Proposition 1 is proved.
It follows that h n (z(s; t, x)) = h n (y) → h(y) = h(z(s; t, x)) on the dense set Ω 1 . Thus, h(z(s; t, x)) is measurable. From (35) and the convergence of h n (x) to h(x) in L 1 (Ω) it follows that the sequence h n (z(s; t, x)) L1(Ω) is uniformly boundeded. Using the Fatou theorem for |h n (z(s; t, x))| we get the summability of h(z(s; t, x)).
The relation (34) for a bounded function h(x) follows from (35). In the case of arbitrary h(x) it suffices to take a sequence of truncation functions [h(z(s; t, x))] k for h(z(s; t, x)) and using the equality [h(z(s; t, x))] k = [h] k (z(s; t, x)), to obtain (34) by means of the passage to the limit as k → +∞.
Proposition 2 is proved.
The proof of the summability of v x (s, z(s; t, x)) as a function of three variables is similar to the proof of Proposition 2 but it consists of some tedious arguments and we skip it.
Proof of Proposition 3. It is easy to see that where We will show that lim m→∞ Z m 1 = 0. Denote the integral over Ω in Z m 1 by Let's make in I the change of variable  s, y)) : E(ϕ)(z m (t; s, y)) dy ψ(t)dt ds y)) : (E(ϕ)(z m (t; s, y)) − E(ϕ)(z(t; s, y))) dy ψ(t) dt ds y)) : E(ϕ)(z(t; s, y)) dy ψ(t) dt ds Using the boundedness of functions ψ and exp ((s − t)/λ) and applying the Cauchy-Schwartz and Hölder inequalities we get Let us show that Z m 121 → 0 as m → +∞. Denote the last factor in (41) by Represent it in the form Let us establish the convergence g m (s) → 0 as m → +∞ for all s ∈ [−k, T ]. It is easy to see that Let ε > 0 be a small number. The continuity of function ϕ x in Ω implies the existence of δ 1 (ε) > 0 such that if |x − x | ≤ δ 1 (ε), then Since the sequence z m (t; s, y) converges to z(t; s, y) in (t, y) Lebesgue measure on [s, T ] × Ω for s ∈ [−k, T ], then for δ 1 (ε) there exists such N = N (δ 1 (ε)) that as m ≥ N the following inequality holds m({(t, y) : |z(t, s, y) − z m (t, s, y)| ≥ δ 1 (ε)}) ≤ ε.
Thus, (41) and (42)  Using the boundedness of ϕ x and ψ, we get It is clear that for a sufficiently large |R| we can make |Z m 111 | small enough. The relation |Z m 112 | → 0 as m → +∞ is established similarly to the case of |Z m 12 |. Thus, we have proved that Show that if m → +∞ then |Z m 2 | → 0.
From inequalities (55) and (56) it follows that Sinceṽ is compactly supported, then Since z m (s; t, x) converges a.e. to z m (s; t, x) on [k 1 , 0] × Ω uniformly with respect to t and the functionṽ x (t, x) is smooth and bounded, we get the convergence (53) as m → +∞ by the Lebesgue theorem. The assertion of Proposition 3 follows from (52) and (53).
Proposition 3 is proved.
Thus, by virtue of Propositions 1 and 3 it is possible to pass to the limit in each term in (29), that yields the identity (31) for any smooth ϕ.
Proposition 4. Let ϕ be smooth. Then Proof of Proposition 4. We denote Here (v i (t, ·)v(t, ·), ∂ϕ(·)/∂x i ) , Let us estimate R i . It is easy to see that for any −∞ < m 1 < m 2 ≤ T the following inequality holds
Proposition 4 is proved.
Since the set of smooth functions is dense in V , for ϕ ∈ V there exists a sequence of smooth functions ϕ l ∈ V such that |ϕ l − ϕ| 1 → 0 as l → +∞. By (58) for ϕ = ϕ l we have This equality and (61) entail Taking into account inequality (67) and passing to the limit when l → +∞ in (31) for ϕ = ϕ l we get (58), or that is the same, the identity (31) for any ϕ ∈ V .
In order to prove that v is a weak solution to problem (1)-(4) it is now enough to prove that v ∈ W and satisfies the identity (8).
First, let us prove the second assertion.
Lemma 3.9. The limit function v satisfies the identity (8).
The proof of Lemma 3.9 consists in the deduction of (8) from (31) and is a rather standard application of the following fact, formulated in Lemma 1.1, Chapter III from [16].
Then the following statements are equivalent: 2) for each η ∈ X (the space X is adjoint to X) d u, η /dt = g, η takes place in the sense of distributions on (a, b); 3) u(t) is differentiable by a.e. t as a function with values in X and du(t)/dt = g(t).
To complete the proof of Theorem 2.5 it remains to show that v ∈ W .
Lemma 3.10. The limit function v belongs to the space W .
Hence from (72) using the change of variable τ = s − t we obtain Proof of Proposition 7. Use the statement of Proposition 5. From Lemma 3.9 it follows that v satisfies by a.e. t the identity (8). We rewrite it in the form d(v, ϕ)/dt = ĝ, ϕ , Find out summability properties of the summands in the right hand side of (78). From inequality (68) and estimate (28) it follows that From inequality (73) and estimate (28) it follows that