WEAK SOLVABILITY OF FRACTIONAL VOIGT MODEL OF VISCOELASTICITY

. In the present paper we establish the existence of weak solutions to one fractional Voigt type model of viscoelastic ﬂuid. This model takes into account a memory along the motion trajectories. The investigation is based on the theory of regular Lagrangean ﬂows, approximation of the problem under consideration by a sequence of regularized Navier-Stokes systems and the following passage to the limit.


To Professor Rafael de la Llave
Abstract. In the present paper we establish the existence of weak solutions to one fractional Voigt type model of viscoelastic fluid. This model takes into account a memory along the motion trajectories. The investigation is based on the theory of regular Lagrangean flows, approximation of the problem under consideration by a sequence of regularized Navier-Stokes systems and the following passage to the limit.
1. Introduction. It is well known the Cauchy momentum equation of a fluid which occupies a bounded domain Ω in R N , N = 2, 3, ∂Ω ∈ C 2 , (see [9]) has the form: ρ(∂v/∂t + v i ∂v/∂x i ) = −∇ p + Div σ + ρf (t, x) (t, x) ∈ Q T = [0, T ] × Ω. (1) Here v(t, x) = (v 1 (t, x), . . . , v N (t, x)) is the velocity vector of a particle at the point x of Ω at time t, ρ(t, x) is the fluid density (which is supposed to be equal to 1), p = p(t, x) is the pressure of the fluid at the point x at time t, σ(t, x) is the deviator of the stress tensor, f (t, x) is the density of external forces acting on the fluid; Div σ(t, x) is the vector, coordinates of which are divergences with respect to x of the rows of matrix σ(t, x).
The rheological relation determines the type of a continuum (fluid) (see eg. [4], [22], [23] and the references therein). A wide range of continua is determined by rheological relation of the form where D α 0t denotes some fractional derivative and ε is the strain tensor.
Description of a wide range of polymers caused introduction of models with fractional derivatives. Such a models reflect the influence of creep and relaxation effects.
Scott-Blair, Zener, Burgers, generalized Maxwell and Kelvin-Voigt fractional models describe above mentioned polymers. In [12] there are given a mechanical interpretations of these models and a good bibliographical review.
It has been noted in [10] that many authors used in (2) various types of fractional derivatives, for example the fractional derivatives of Grunwald-Letnikov, Liouville, Caputto-Liouville, Riemann-Liouville etc.
Though there exists a lot of fractional models, as far as we know there are no nonlocal existence theorems of weak solutions to the corresponding initial-boundary value problems. This is caused by the presence of singularities in integral representations of the fractional derivatives and integrals, in contrast to the integer models (see [15]).
The experiments demonstrate that the usage of full derivatives instead of time ones in (3) give a more precise description of nonlinear effects in fluids (see [23], [22]).
In some models it allows to express explicitly σ in terms of v along trajectories of the vector field v (see eg. [15]). This implies the appearance of an integral term along the trajectories of the field v in the motion equation. This means that the Cauchy problem (in integral form) determining the trajectories z = z(τ ; t, x) of the field v has to be added to the momentum equation. Let us mark that the presence of the integral along trajectories in the momentum equation means the presence of a memory in the fluid.
The presence of z in the momentum equation requires the unique solvability of (4). However, the existence of solutions to (4) for fixed v is known only in the case of v ∈ L 1 (0, T ; C(Ω) N ) and this is the unique solution if v ∈ L 1 (0, T ; C 1 (Ω) N ), v| (0,T )×∂Ω = 0 (see e.g. [14]). But even for strong solutions (v ∈ L 2 (0, T ; W 2,2 (Ω) N ) equation (4) is generally speaking not unique solvable and consequently the trajectories z are not determined uniquely.
One possible way out of this situation is a regularization of the velocity field (see [20]).
In the study of weak solvability of equations of the form (1) it is usual that v ∈ L 2 (0, T ; W 1,2 (Ω) N ). But this is unsufficient for classical solvability of irregularizied Cauchy problem (4). Recently (see eg. [6]- [7]) the unique solvability of the Cauchy problem (4) in the case of v belonging to a Sobolev space was established in the class of Regular Lagrangian Flows, a generalization of the concept of classical solutions.
In the present work, this allowed to proof the existence of weak solutions without a regularization of v in equation (4).
Below we consider the special case of model (2) which is some fractional equivalent to the Voigt model, a rheological relation of which has the form (see [13]) σ = µ 0ε + µ 1 D α 0tε , 0 < α < 1. The structure of the work is as follows. In section 2 auxiliary assertions are given. In section 3 the fractional model under consideration is discussed. In section 4 we formulate the main results. Section 5 is devoted to the study of regularized problems and consists of 4 subsections. In subsection 5.1 we consider ε-regularization of the original problem. In subsection 5.2 we investigate regularized Navier-Stokes system. In subsections 5.3-5.4 we construct a sequence of approximations for ε-regularization and establish their solvability and estimates of solutions for small T . In section 6 using passage to the limit in the approximating problems we establish the solvability of ε-regularization for small T . In section 7 we prove a priori estimates for εregularization which are necessary for the proof of the solvability of ε-regularization for arbitrary T . In section 8 we prove the solvability of ε-regularization for arbitrary T . In section 9 the solvability of the main problem is obtained via passage to the limit as n → +∞ in 1/n-regularization.
Constants in inequalities and chains of inequalities which do not depend on significant parameters are denoted by a single letter M .
2. Basic definitions and auxiliary results. Functional spaces. Let C ∞ 0 (Ω) N be the set of infinitely differentiable compactly supported R N -valued functions on Denote by H and V the closures of V w.r.t. norms of L 2 (Ω) N and W 1,2 (Ω) N , respectively. Let V −1 denote the conjugate to V space.
Denote by E(v) the matrix with components E ij (v) = 1 2 (∂v i /∂x j + ∂v j /∂x i ). The space V is a Hilbert space with the scalar product (v, u) ) and the corresponding norm. This norm in the space V is equivalent to the norm of W 1,2 (Ω) N . Denote by f, v the action functional f from the adjoint to V space V −1 on a function v from V .
The sign (·, ·) stands for the scalar product in Hilbert spaces L 2 (Ω) N , H, L 2 (Ω) and L 2 (Ω) N ×N . From a context it is clear what the space is meant.
The identification of the Hilbert space H with its conjugate space H −1 and the theorem of Riesz lead to the continuous embedding V ⊂ H = H −1 ⊂ V −1 . In addition, for u, w ∈ V the relation u, w = (u, w) is valid with the scalar product in H.
Fractional Riemann-Liouville integrals and derivatives. Recall some facts about fractional derivatives and integrals (see [12], [16]). The fractional integrals of where Γ(α) =  In particular, if 0 < α < 1 then D α 0t = 1 (t − s) −α y(s) ds, and when α = n > 0 is integer, then D n 0t = d n dt n y(t) is the usual derivative of order n. The fractional differential operator D α 0t is inverse to the left-side operator of fractional integration: D α 0t I α 0t y(t) = y(t). Regular Lagrangian Flows. Consider the Cauchy problem (in integral form) In the case of v ∈ L 1 (0, T ; C 1 (Ω)) with the zero condition on the boundary the problem (6) has a unique solution in the classical sense (see [14]). However, in the case of only summable with respect to x vector-function v, the situation is much more complicated and one has to use a more general concept of the solution to (6). 3) for all t i ∈ [0, T ], i = 1, 2, 3 and a.a. x ∈ Ω z(t 3 ; t 1 , x) = z(t 3 ; t 2 , z(t 2 ; t 1 , x)).
The definition of RLF can be found, for example, in [1], [5], [7]. Here the definition of RLF is given in the particular case of a bounded domain Ω and divergence free function v.
Let us recall some results on RLF . Let D = [0, T ] × [0, T ] and L be the set of measurable on D functions which is considered as a metric space with the metric Let v x be the Jacobian matrix of a vector function v.
are valid. Let v m converges to v in L 1 (Q T ) N as m → +∞. Let z m (τ ; t, x) and z(τ ; t, x) be RLF associated to v n and v, respectively. Then the sequence z m converges (up to a subsequences) to z w.r.t. Lebesgue measure on the set In a more general formulation this result is proved in [5], Corollaries 3.6, 3.7, 3.9.
3. Fractional Voigt model. This model has a mechanical interpretation in the form of the parallel connection of Newton and Scott-Blair elements (see [12]). Indeed, a Newton element N is determined by the rheological relation σ 1 = ν 1ε1 and a Scott-Blair element SB is determined by the rheological relation σ 2 = ν 2 D α 0t ε 2 . For the parallel connection N ||SB of elements N and SB the relations σ = σ 1 + σ 2 and ε = ε 1 = ε 2 are valid where σ is the deviator of the stress tensor and ε is the strain tensor of the element N ||SB.
It follows that Here E(v) is the strain rate tensor and Model takes into account the history of the fluid motion along the spatial variable x. However, more realistic are models which take into account the history of the fluid motion along the trajectories of fluid motion. In the case of rheological relation (11) such a model has the form Here z(τ ; t, x) is the solution to the Cauchy problem (6). Substituting (12) in (1) we obtain the initial-boundary value problem

VICTOR ZVYAGIN AND VLADIMIR ORLOV
and is denoted as u . By this the action of functional u(t) ∈ V −1 on ϕ ∈ V is determined as follows u(t), ϕ = (u(t, ), ϕ).
It will be useful for us to consider a solution v(t, x) to problem (13) Introduce functional space (16) and the identity   5. Regularized problems.
5.1. ε-regularization of problem (13)- (16). In order to establish the solvability of (13)- (16) we consider the dependent on ε > 0 auxiliary regularized problem Here z(τ ; t, x) is the solution to the Cauchy problem (20) and K ε is determined by the formula (21) is a function v ∈ W 1 (0, T ) satisfying the initial condition (21) and the identity To prove the solvability of problem (18)-(21) let us consider the successive approximations v n , n = 1, 2, 3, ... defined as the solution of the auxiliary regularized problems Here and v 1 is chosen as v 1 = 0. Below we show that weak solutions v n of problem (23)-(27) converge to a weak solution of problem (18)-(21) as n → +∞.
Let us first recall some facts on the regularized Navier-Stokes system.

5.2.
Regularized Navier-Stokes system and properties of its solutions. Problem (23)-(25) is a regularizaton for the Navier-Stokes system of the form for any ϕ ∈ V .
In [19], Theorem 3.1, Chapter III, the weak solvability of problem (28) was established for ε = 0 (the Navier-Stokes system) and for any F ∈ L 2 (0, T ; V −1 ), v 0 ∈ H in the class W 1 (0, T ). It is obvious that the regularized problem (28) (ε > 0) also has a solution in W 1 (0, T ). We show that for ε > 0 the solution possesses better properties, namely it belongs to W (0, T ). The space W (0, T ) is defined by the formula hold true with independent on ε constant M 0 .
Proof of Theorem 5.3. Using the terms of (29) we introduce functional on V and hence the map We introduce the operator K ε : For a function v ∈ W 1 (0, T ) the relation is valid (see [19], Chapter III, Then the problem (28) can be rewritten in the operator form (see [19], Section Let v be a weak solution to problem (28) from W 1 (0, T ). Then from (32) it follows that In [20] it is established that for any ε > 0 It follows that all summands in the right-hand side part of equation (32) belong to L 2 (0, T ; V −1 ) and therefore v ∈ L 2 (0, T ; V −1 ).
Thus, v ∈ W (0, T ). Let us apply both sides of (32) (which belong to On the strength of Lemma 1.2, Chapter III in [19] for v ∈ W (0, T ) the following relation is valid Using (35), definition of operator A and having in mind that K ε (u), u = 0 for u ∈ V (see [20], [23], p. 208) we have From Korn's inequality it follows that |E(u)| 0 ≥ m|u| 1 , m > 0 for u ∈ V . Using this fact and elementary transformations we find that for arbitrary δ > 0 the following inequality holds d dt Choosing δ > 0 small enough, shifting the last term in (36) to the left side and integrating by t, by simple arguments we obtain the inequality (30). Let us establish estimate (31). From equation (35), the second estimate (34), inequality (30) and monotonicity of the L p norms w.
Theorem 5.3 is proved.
Remark 5.1. Note that the statement of Theorem 5.3 is obviously true for problem (28) on Q T for any T ≤ T with the change of T by T in (30) and (31).
Properties of solutions of the regularized Navier-Stokes system will be used below. First, note that for v n−1 ∈ L 2 (0, T ; V ) the Cauchy problem (26) defines a unique RLF z n−1 (τ ; t, x) due to Theorem 4.2.
Establish the following fact.
Lemma 5.4. Let v n−1 ∈ L 2 (0, T ; V ) and z n−1 (τ ; t, x) be RLF assotiated to the Cauchy problem (26). Then for Proof. Let A : Ω → R N ×N be a matrix function and A ∈ L 2 (Ω) N ×N . It is easy to show (see e.g. [11], [18]) that Let It is obviously that Let's make the change of variable x = z n−1 (t; s, y) in integral J ij (the inverse change is y = z n−1 (s; t, x)). Since RLF z n−1 (s; t, x) is associated to divergence free v n−1 then for Jacobian matrix z n−1 In virtue of (38) and (39) and easily checking inequality (see [16], Theorem 2.6), then it follows that Lemma 5.4 is proved.
Establish estimates for functions v n . (23)

Estimates of weak solutions of regularized problem
Here Let T be such that q < 1. Using estimate (46) step by step we obtain is valid. Here M 4 doest depend on n but depend on T . From (48) it follows the validity of (44). Next, w n ∈ L 2 (0, T ; V −1 ) due to Lemma 5.5. It follows then from Theorem 5.3 that v n ∈ W (0, T ).
Let us prove estimate (45). It is obviously that v n satisfies equation where w n is defined by (27). On the strength of estimate (31) it follows from here that Using estimates (43) and (44) we easily get Estimates (49) and (50) imply inequality (44). Theorem 5.6 is proved.
Let T be small. Then problem (18)-(21) has at least one weak solution v ∈ W (0, T ) satisfying the uniform on ε estimates sup Here M 6 doesn't depend on ε.
We will assume that v n converges (with up to a subsequence) to some v weakly in L 2 (0, T ; V ) and strongly in L 2 (0, T ; H).
From estimates (30), it follows that due to the boundedness in L ∞ (0, T ; H) of the sequence v n it converges (with up to a subsequence) to v * -weakly in L ∞ (0, T ; H).
We will show that v is a weak solution of problem (13)- (16).
To do this, pass to the limit in problem (23) , E(ϕ)(·)) ds, We rewrite the identity (53) in the form and pass to the limit in (53) Weak convergence of v n to v in L 2 (0, T ; V ) and strong in L 2 (0, T ; H) suggests (see [20]) that Consider I 4 (n). Let us recall that Making the change of variable y = z n (s, t, x) ( x = z n (t, s, y) is the inverse change), using (7) and the fact that det z n−1 x (s; t, x) = 1 for the Jacobian matrix z n−1 x (s; t, x) since v n−1 is divergence free, we find that y) : E(ϕ)(z n−1 (t; s, y)) ds dy.

VICTOR ZVYAGIN AND VLADIMIR ORLOV
Using this relation and changing the integration order we have 1 (t, s, y)) dt.
Consider ψ n . Since the sequence z n (t, s, y) converges to z(t, s, y) w.r.t. (t, y) measure we will assume that z n converges to z(t, s, y) a.e. (up to a subsequence). Due to the smoothness of ϕ the function E(ϕ)(z n (t, s, y)) is bounded and E(ϕ)(z n (t, s, y)) converges a.e. on Q T to the bounded function E(ϕ)(z(t, s, y)). In virtue of the Lebesgue Theorem the uniformly bounded sequence ψ n (s, y) converges a.e. on Q T to the bounded function ψ(s, y) = T s (t − s) −α E(ϕ)(z(s, t, y)) dt. Thus, in the integrand of 1 (s, y)) : ψ n (s, y) dy ds (57) the first factor converges weakly in L 2 (Q T ) N ×N while the second one a.e. in Q T . This implies that in (57) one can pass to the limit as n → +∞ and get , E(ϕ)(z(t, s, y))) dt ds.
Changing the integration order and making the change of variable y = z(s, t, x), we get From the established convergence of the summands I i (n) it follows that the function v(t, x) satisfies for any smooth ϕ.
Let ϕ ∈ V be arbitrary. Choose a sequence of smooth ϕ m ∈ V , m = 1, 2, . . . such that ϕ m converges in V to ϕ as m → +∞. Taking ϕ = ϕ m in (59) and passing to the limit as m → +∞ we get (59) for arbitrary ϕ ∈ V . The passage to the limit is possible, since the convergence of ϕ m to ϕ in V implies the convergence of E(ϕ m ) to E(ϕ) in L 2 (Ω) N ×N and, in addition, the scalar products in (59) are continuous w.r.t. to its factors.
It is easy to show that (59) is true for any t ∈ (0, T ) instead of T . Hence, using differentiation w.r.t. t we obtain that v satisfies the identity (22). Let us show that v ∈ W (0, T ). Similarly to the case of problem (23)-(27) rewrite problem (18)- (21) in the operator form where z(s; t, x)) ds.
Let us establish estimates (51) and (52). Since v is a weak solution to regularized Navier-Stokes problem for known w, then due to Theorem 5.3 the following estimates are valid: with independent on ε constant. In the same way as in the proof of (43) it is shown that From estimates (61) and (63) it follows that for small T inequality (51) holds true.
Establish the solvability of problem (18)-(21) for arbitrary T . For this we need a priori estimates of weak solutions to problem (32).

7.
A priori estimates of weak solutions for ε-regularization. The following fact is valid.
Then if the solution of the problem (32) v ∈ W (0, T ), then it satisfies the estimates with independent on ε constant M 7 .
Proof of Theorem 7.1. For a function v ∈ W (0, T ) relation (35) is valid. Applying both sides of equation (60) to v ∈ V , using the obtained relation and K ε (u), u = 0 for u ∈ V (see [20], [23], p. 208) by means of standard transformations with usage of Lemma 1.2 (see [19]) we obtain the identity Changing t by s in (66), integrating w.r.t. s over interval [0, t] ⊂ [0, T ] we have For the first term on the right-hand side in (67) we get for sufficiently small η > 0 Next, it is easy to see that due to Korn's inequality Consider the last term in (67) and rewrite it in the form Making elementary transformations we have for arbitrary η > 0 Let Making in I the change of variables x = z(t; s, y), y = z(s; t, x).
and having in mind that det z x (s; t, x) = 1 in the case of divergence free v we get Thus, from (72) and (74) it follows that Using (68), (75), (69) and choosing η small enough, we obtain from (67) Denoting the last summand via Z 2 (t), we have Then using the change of variable ξ = s − τ we get Using the Minkowski integral inequality in (77) and the invariance of L 2 (−∞, +∞) norms w.r.t. shift we have From the relations (76) and (79) it follows that for 0 < t ≤ t 0 where t 0 > 0 is small enough, the inequality is valid. Estimate (64) in the case 0 < T ≤ t 0 follows from (80). Now consider the case of arbitrary T > t 0 . Let t > t 0 . Represent Z 2 (t) for t > t 0 in the form where It is clear that Z 21 = Z 2 (t 0 ) and from (79), it follows that Consider Z 22 and rewrite it in the form For Z 221 we have Let us estimate Z 222 . Using the change of variable ξ = s − τ we get In the same way as in the derivation of estimates (80), using the Minkowski integral inequality one has From estimates (82), (84) and (86) it follows that for t > t 0 Using the inequality (87) for estimation of the last term in (76) and supposing t 0 small enough, simple transformations yields the inequality Dropping in (88) the first summand, we get the Gronwall inequality for ϕ(t) = t 0 |v(τ, ·)| 2 1 dτ , from which it follows that From (88) Supposing the solution of problem (18)- (21) to be known on [0, T k ], extend it on [T k , T k+1 ]. Letv(t, x) ∈ W (0, T k ) be a solution to problem (90)-(93) on [0, T k ]. Letz be RLF associated tov. Construct a continuation ofv(t, x) on [T k , T k+1 ].
By Theorem 5.6v(t, x) ∈ W (0, T k ) and satisfies the estimates Consider on [T k , T k+1 ] the problem A weak solution to problem (96)-(99) is a function v ∈ W (T k , T k+1 ) satisfying the appropriate identity and the initial condition.
Lemma 8.2. The function f 1 belongs to L 2 (T k , T k+1 ; V −1 ) and estimate is valid.
Here the justification of the passage to the limit is the same as in the proof of Theorem 6.1, with the exception of term I 2 (n). However, weak convergence of v n to v in L 2 (0, T ; V ) and strong in L 2 (0, T ; H) suggests (see [20]) that Hence, in the same way as in the proof of Theorem 6.1 it is deduced that v is a weak solution to (13)- (16).
Theorem 4.2 is proved.