Keywords: nonlinear systems, stabilization, quantum ensembles, Lyapunov stability, LaSalle invariance, infinite dimensional systems, continuous spectrum

Feedback stabilization of an ensemble of non interacting half spins described by Bloch equations is considered. This system may be seen as a prototype for infinite dimensional systems with continuous spectrum. We propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin +1/2 or -1/2. The closed-loop stability analysis is done locally around the equilibrium. The local convergence is shown to be a weak asymptotic convergence for the H 1 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the equilibrium

Time-periodic feedback stabilization for an ensemble of half-spin systems

Karine Beauchard * Paulo Sérgio Pereira da Silva * * ,2 Pierre Rouchon * * * ,1

1. INTRODUCTION

Infinite dimensional systems with continuous spectra

Most controllability results available for infinite dimensional bilinear systems are related to systems with discrete spectra (see for instance, [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] for exact controllability results and [START_REF] Chambrion | Controllability of the discrete-spectrum schrödinger equation driven by an external field[END_REF], [START_REF] Nersesyan | Growth of Sobolev norms and controllability of Schrödinger equations[END_REF] for approximate controllability results).

As far as we know, very few controllability studies consider systems admitting a continuous part in their spectra.

In [START_REF] Mirrahimi | Lyapunov control of a quantum particle in a decaying potential[END_REF] an approximate controllability result is given for a system with mixed discrete/continuous spectrum: the Schrödinger partial differential equation of a quantum particle in an N-dimensional decaying potential is shown to be approximately controllable (in infinite time) to the ground bounded state when the initial state is a linear superposition of bounded states.

In Li andKhaneja [2006, 2009] a controllability notion, called ensemble controllability, is introduced and discussed for quantum systems described by a family of ordinary differential equations (Bloch equations) depending continuously on a finite number of scalar parameters and with a finite number of control inputs. Ensemble controllability means that it is possible to find open-loop controls that compensate for the dispersion in these scalar parameters: the goal is to simultaneously steer a continuum of systems between states of interest with the same control input. Such continuous family of ordinary differential systems sharing the same control inputs can be seen as the prototype of infinite dimensional systems with purely continuous spectra.

The article [START_REF] Li | Control of inhomogeneous quantum ensembles[END_REF] highlights, the role of Lie algebras and non-commutativity in the design of a compensating control sequence and consequently in the characterization of ensemble controllability. In [START_REF] Beauchard | Controllability issues for continuous spectrum systems and ensemble controllability of bloch equations[END_REF], this analysis is completed by functional analysis methods developed for infinite dimensional systems governed by partial differential equations (see, e.g., [START_REF] Coron | Control and Nonlinearity[END_REF] for samples of these methods). In [START_REF] Beauchard | Controllability issues for continuous spectrum systems and ensemble controllability of bloch equations[END_REF], several mathematical answers are given, with dicrimination between approximate and exact (simultaneous) controllability, and finite time and infinite time (simultaneous) controllability, for the Bloch equation.

The goal of this article is to investigate feedback stabilization of such specific infinite dimensional systems with continuous spectra. As in [START_REF] Mirrahimi | Lyapunov control of a quantum particle in a decaying potential[END_REF], the feedback design is based on a Lyapunov function closely related to the norm of the state space, a Banach space.

The studied model

We consider here an ensemble of non interacting half-spins in a static field

0 0 B0
in R 3 , subject to a transverse radio frequency field ṽ(t) -ũ(t) 0 in R 3 (the control input). The ensemble of half-spins is described by the magnetization vector M ∈ R 3 depending on time t but also on the Larmor frequency ω = -γB 0 (γ is the gyromagnetic ratio). It obeys to the Bloch equation:

Ṁ (t, ω) = (ũ(t)e 1 + ṽ(t)e 2 + ωe 3 ) × M (t, ω), (1)
where -∞ < ω * < ω * < +∞, ω ∈ (ω * , ω * ), (e 1 , e 2 , e 3 ) is the canonical basis of R 3 , × denotes the wedge product on R 3 and Ṁ (t, ω) = ∂M (t,ω)

∂t

. The equation ( 1) is a bilinear control system in which, at time t,

• the state is (M (t, ω)) ω∈(ω * ,ω * ) ; for each ω, M (t, ω) ∈ S 2 , the unit sphere of R 3 , • the two control inputs ũ(t) and ṽ(t) are real.

It must be stressed that ũ(t) and ṽ(t) are common controls for all the members of the ensemble, and they cannot depend on ω.

The state M (t, ω) = (x(t, ω), y(t, ω), z(t, ω)) of this ensemble of dynamic systems depends on t and ω and its initial condition is a map

M 0 : (ω * , ω * ) → S 2 ⊂ R 3 , M (0, ω) = M 0 (ω).
In coordinates, one may write ẋ = -ωy + ṽz, ẏ = ωx -ũz, ż = -ṽx + ũy.

(2)

Formally, the spectrum of the operator A defined by ω) where δ ω0 is the Dirac distribution in ω 0 : φ(ω)δ ω0 (ω)dω = φ(ω 0 ) for any continuous function φ.

AM | ω := ωe 3 × M (ω) for every M : (ω * , ω * ) → S 2 , is -ı(ω * , ω * ) ∪ ı(ω * , ω * ): for every ω 0 ∈ (ω * , ω * ), the eigenvector associated to ±ıω 0 is 1 ∓ı 0 δω 0 (

Outline

The goal of this article it to propose a first answer to the following question.

Local Stabilization Problem. Define an explicit control law (ũ, ṽ) = (ũ(t, M ), ṽ(t, M )) and a neighborhood U of -e 3 (in some space of functions (ω * , ω * ) → S 2 to be determined) such that, given any initial condition M 0 ∈ U , the solution of the closed loop system is defined for every t ∈ [0, +∞), is unique and converges to -e 3 , when t → +∞, uniformly with respect to ω ∈ (ω * , ω * ).

Section 2 is devoted to control design and closed-loop simulations: the feedback law is the sum of a Dirac comb and of a time-periodic feedback law based on a Lyapunov function; proposition 1 proved in appendix guaranties that the closed-loop initial value problem is always well defined; simulations illustrate the convergence rates observed for an initial state quite far from goal south pole M ≡ -e 3 .

In section 3 we state and prove the main convergence result, theorem 1: the closed-loop convergence towards the constant profile M (ω) = -e 3 is shown to be local and weak for the H 1 topology on M (ω). Some concluding remarks are gathered in section 4.

LYAPUNOV H 1 APPROACH

The impulse-train control

It is proved in [START_REF] Beauchard | Controllability issues for continuous spectrum systems and ensemble controllability of bloch equations[END_REF] that controls containing sums of Dirac masses are crucial to achieve the controllability of the Bloch equation. In view of the controls used in this reference, it is natural to consider a control that admits the following "impulse-train" structure

ũ = u + +∞ k=1 π δ(t -kT ), ṽ = (-1) E( t T ) v (3)
for some period T > 0 (E (γ) denotes the integer part of the real number γ). The new controls u and v are bounded and measurable time functions. Then, after each impulse that is applied at time t equal to kT , x remains unchanged, but y and z are moved to their opposites (see [START_REF] Li | Control of inhomogeneous quantum ensembles[END_REF] for an explanation of this fact), that is

x(kT -) = x(kT + ), y(kT -) = -y(kT + ), z(kT -) = -z(kT + ).
The resulting state diffeomorphism

(x, y, z) → (X = x, Y = -y, Z = -z) (4) transforms (2) into Ẋ = ωY -ṽZ, Ẏ = -ωX -ũZ, Ż = ṽX + ũY. Let ς = (-1) E( t T ) .
Considering the identification (4), one gets the following dynamics ẋ = -ςωy + vz, ẏ = ςωx -uz, ż = -vx + uy, (5) with the new control (u, v). It is as if, between [kT, (k + 1)T ] and [(k + 1)T, (k + 2)T ], one is changing the sign of ω, but the solution, after the identification (4), remains continuous (but not differentiable in time at the instants t = kT, k ∈ N). In other words, the application of the impulses at t = kT changes the sense of rotation of the null input solution. One would expect that this impulse-train control is reducing the average dispersion of the solution. Roughly speaking, the dispersion observed for the openloop system (2) with (ũ, ṽ) as input is strongly reduced and almost canceled for the open-loop system (5) with (u, v) as input.

Heuristics of the Lyapunov-like control

Now let Z(t, ω) and Ω(t) be two complex numbers defined by Z = x + ıy, Ω = v -ıu where x, y, z refer to the transformed dynamics (5). Then one may write (5) in the form

Ż(t, ω) = ıς(t)ωZ(t, ω) + Ω(t)z(t, ω), ż(t, ω) = -Ω(t)Z(t, ω) ,
where (ξ) (resp. ξ) denotes the real part (resp. the complex conjugate) of a complex number ξ ∈ C. It is easy to see that the following transformation

Z(t, ω) = Z(t, ω)e -ıω t 0 ς(τ )dτ converts the system into the driftless form      Ż(t, ω) = Ω(t)z(t, ω)e -ıω t 0 ς ż(t, ω) = -Ω(t)Z(t, ω)e -ıω t 0 ς (6)
where, for notation simplicity, one lets Z(t, ω) stand for Z(t, ω), and one lets t 0 ς stand for t 0 ς(τ )dτ .

For the moment one shall assume that the input Ω(t) will be chosen in such a way that the solution (Z(t, ω), z(t, ω)) of ( 6) does exist, it is unique and it is regular enough in a way that one may consider that the derivatives Z (t, ω) = ∂Z ∂ω (t, ω) and z (t, ω) = ∂z ∂ω (t, ω) exists almost everywhere, and they are solutions of the differential equation3 that is obtained by differentiation of ( 6) with respect to ω, namely

       Ż = Ω z -ı t 0 ς z e -ıω t 0 ς , ż = - Ω Z -ı t 0 ς Z e -ıω t 0 ς , (7)
where Ż stands for ∂ ∂t Z , and ż stands for ∂ ∂t z . Now consider the following Lyapunov-like functional:

L = 1 2 ω * ω * G |Z | 2 + (z ) 2 + |Z| 2 dω (8)
where G is a positive real number and Z(t, ω), Z (t, ω) and z (t, ω) refer to the solutions respectively of ( 6) and ( 7). One may write

d dt L(t) =   ω * ω * G Z Ż + z ż + Z Ż dω   (9)
and so, taking into account ( 6) and ( 7), the fact that Ω(t) does not depend on ω, one gets

d dt L(t) = [Ω(t)H(t)] (10)
where

H(t) = ω * ω * -ıG   t 0 ς   Z z -Zz + Zz e -ıω t 0 ς dω.
Hence one may take Ω(t) = -K p H(t), where K p is a positive real number, obtaining

Ω(t) = -Kp ω * ω * ıG   t 0 ς   Z z -Zz + Zz e ıω t 0 ς dω. (11) It follows that d dt L(t) = -1 Kp |Ω(t)| 2 .
Recall that M = ( (Z), (Z), z) = (x, y, z). Consider the system (6) in closed loop with the control law (11), thereby called by closed loop system. The state space of this system is H 1 ((ω * , ω * ), R 3 ), which is the set of functions f ∈ L 2 (ω * , ω * ) such that the distributional derivative f belongs to L 2 (ω * , ω * ). This space, equipped with the norm

f H 1 :=   ω * ω * |f (ω)| 2 + |f (ω)| 2 dω   1/2 is a Banach space.
In other words, this system may be considered to be a differential equation of the form

Ṁ (t) = F (t, M (t)) M (0) = M 0 ∈ H 1 ((ω * , ω * ), S 2 ) where F (t, M ) is a continuous map F : R × H 1 ((ω * , ω * ), R 3 ) → H 1 ((ω * , ω * ), R 3 ).
Moreover, F is periodic in t and locally Lipschitz in M . Using the same ideas as in the proof of the Cauchy-Lipschitz (Picard-Lindelöf) theorem, we get the following result, whose proof is detailed in Appendix. Proposition 1. For every initial condition M 0 belonging to H 1 ((ω * , ω * ), S 2 ), the closed loop system (6), ( 11) admits a unique solution

M in C 1 [0, ∞), H 1 (ω * , ω * ), S 2 such that M (0) = M 0 .

Closed-loop simulations

We assume here ω * = 0, ω * = 1 and we solve numerically the T -periodic system (5) with the T -periodic feedback law (11) where Z = x + ıy and Ω = v -ıu. The parameters are

T = 2π/(ω * -ω * ), K p = 1, G = 1 2T 2 . The simulation is for t ∈ [0, T f ], T f = 50T . The pro- file [ω * , ω * ] ω → (x(t, ω), y(t, ω), z(t, ω)) is discretized {0, . . . , N } k → (x k (t), y k (t), z k (t)) with a regular mesh of step N = ω * -ω * N with N = 100: (x k (t), y k (t), z k (t)) is then an approximation of (x(t, k N ), y(t, k N ), z(t, k N )).
We have checked that the closed-loop simulations are almost identical for N = 100 and N = 200. In the feedback law (11), the integral versus ω is computed assuming that (x, y, z) and (x , y , z ) are constant over

](k -1 2 ) N , (k + 1 2 ) N [, their values being (x k , y k , z k ) and x k+1 -x k-1 2 N , y k+1 -y k-1 2 N , z k+1 -z k-1 2 N
. The obtained differential system is of dimension 3(N + 1). It is integrated via an explicit Euler scheme with a step size h = T /50. We have tested that h = T /100 yields to almost the same numerical solution at t = T f = 50T . After each time-step the new values of (x k , y k , z k ) are normalized to remain in S 2 . Figures 1 and2 summarize the main convergence issues when the initial ω-profiles of (x, y, z) ∈ S 2 are z(0, ω) = 0.8 -0.1 sin(4πω), x(0, ω) = cos(πω) 1 -z 2 (0, ω) and y(0, ω) = sin(πω) 1 -z 2 (0, ω). The convergence speed is rapid at the beginning and tends to decrease at the end. The control problem is quite hard due to the fact that one has a continuous spectrum, that is, an infinite ensemble of systems with a common control input Ω(t). Hence, as time increases, the control must fight against the dispersion of the solutions M (t, ω) for different values of ω. Simulations (not presented here) on much longer times until 10 4 T and with the same initial conditions and parameters always yield to smaller final value for the Lyapunov function (for example we get L(10 4 T ) = 0.0395). This is a strong indication of asymptotic converge of the profile M (t, ω) toward -e 3 even if the convergence speed seems to be very slow. This numerically observed convergence is confirmed by theorem 1 here below.

MAIN RESULT

The main result of this paper shows that the control law (11) is a solution of the local stabilization problem stated at the end of the introduction. Theorem 1. There exists δ > 0 such that, for every

M 0 ∈ H 1 ((ω * , ω * ), S 2 ) with M 0 + e 3 H 1 ≤ δ , M (
t) → -e 3 weakly in H 1 when t → +∞. In particular, as the injection of H 1 in C 0 is compact, M (t, ω) converges to -e 3 when t → +∞ uniformly with respect ω ∈ (ω * , ω * ) (convergence in the sup norm of C 0 ).

The proof of Theorem 1 relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. The first step of the proof consists in checking that, locally, the invariant set is reduced to {-e 3 }. Proposition 2. There exists δ > 0 such that, for every

M 0 ∈ H 1 ((ω * , ω * ), S 2 ) with M 0 + e 3 H 1 < δ, the map t → L(t) is constant on [0, +∞) if and only if M 0 = -e 3 .
Proof: We can absorb the positive parameter G via a scaling on ω. Thus we will assume during all the following proofs that G = 1. Let us assume that L(t) is constant. Then, Ω(t) = 0, Z(t, ω) = Z 0 (ω), z(t, ω) = z 0 (ω) and we have

ω * ω * ıt(Z 0 z 0 -Z 0 z 0 ) + Z 0 z 0 e ıωt dω = 0, ∀t ∈ [0, T ].
Considering the power series expansion versus t of the left hand side, we get, for any polynomials

P ∈ C[ω] ω * ω * P Z 0 z 0 -Z 0 z 0 + P Z 0 z 0 dω = 0. ( 12
)
Polynomials are dense in H 1 , thus, the previous equality holds for every P ∈ H 1 (ω * , ω * ). In particular, with P (ω) = Z 0 (ω), we get

ω * ω * Z 0 Z 0 z 0 -Z 0 z 0 + Z 0 Z 0 z 0 dω = 0.
We deduce that

ω * ω * |Z 0 | 2 + |Z 0 | 2 dω = ω * ω * (1 + z 0 ) |Z 0 | 2 + |Z 0 | 2 -Z 0 Z 0 z 0 dω.
This relation provides an inequality of the type

Z 0 2 H 1 ≤ C(M 0 ) Z 0 2 H 1
where the constant C(M 0 ) tends to 0 when M 0 tends to -e 3 in H 1 . There exists δ > 0 such that, for every

M 0 ∈ H 1 ((ω * , ω * ), S 2 ) with M 0 + e 3 H 1 < δ, we have C(M 0 ) < 1.
If L is constant along the trajectory associated to such an initial condition M 0 , then, the previous argument shows that Z 0 = 0, thus M 0 = -e 3 . 2 Remark 1. The relation ( 12) shows that the invariant set contains at least -e 3 , e 3 and any function M 0 taking values in the equator (i.e. for which z 0 = 0). Thus, with these feedback laws, global stabilization (i.e. for every M 0 ∈ H 1 ((ω * , ω * ), S 2 )) cannot be expected. In order to get global stabilization results, one needs other tools.

For the proof of Theorem 1, we need the continuity with respect to initial conditions, of the solutions of the closed loop system (6), ( 11), for the H 1 2 (ω * , ω * )-topology. This space is defined by interpolation between L 2 (ω * , ω * ) and H 1 (ω * , ω * ) and we have a compact injection H 1 (ω * , ω * ) → H 1 2 (ω * , ω * ). First, let us recall the following Lemma. Lemma 1. Let T > 0. There exists c 1 > 0 such that, for every ϕ ∈ H 1 2 (ω * , ω * ) and for every α ∈ [0, T ], the map ω → ϕ(ω)e ıαω belongs to

H 1 2 (ω * , ω * ) and satisfies ϕ(ω)e ıαω H 1 2 ≤ c 1 ϕ H 1 2 . Proof: We have ϕ(ω)e ıαω L 2 = ϕ L 2 , ∀ϕ ∈ L 2 (ω * , ω * ), and, for every ϕ ∈ H 1 (ω * , ω * ), ϕ(ω)e ıαω H 1 =   ω * ω * |ϕ (ω) + iαϕ(ω)| 2 + |ϕ(ω)| 2 dω   1/2 ≤   ω * ω * 2|ϕ (ω)| 2 + (2α 2 + 1)|ϕ(ω)| 2 dω   1/2
thus we get the conclusion with, for example, c 1 := (2T 2 + 2) 1/4 by interpolation.

Proposition 3. There exists δ > 0 such that, for every

(M 0 n ) n∈N ∈ H 1 ((ω * , ω * ), S 2 ) N , M 0 ∞ ∈ H 1 ((ω * , ω * ), S 2 ) such that • M 0 n + e 3 H 1 < δ , ∀n ∈ N, • M 0 n M 0 ∞ weakly in H 1 when n → +∞, • M 0 n → M 0 ∞ strongly in H 1 2
when n → +∞, the solutions M n (t, ω), M ∞ (t, ω) of the closed loop system associated to these initial conditions satisfy the following convergences, when n → +∞, for every t ∈ [0, +∞),

M n (t) → M ∞ (t) strongly in H 1 2 , Ω n (t) → Ω ∞ (t).
Proof: First let us emphasize that L(M ) and M + e 3 H 1 are equivalent norms on a small enough H 1 ((ω * , ω * ), S 2 )neighborhood of -e 3 : there exists η, c * , c * > 0 such that, for every

M ∈ H 1 ((ω * , ω * ), S 2 ) with M 0 + e 3 H 1 < η, we have c * L(M ) ≤ M + e 3 H 1 ≤ c * L(M ).
(13) Indeed, we have

M + e 3 2 H 1 = ω * ω * |Z | 2 + (z ) 2 + |Z| 2 + (1 + z) 2 dω. When M ∈ H 1 ((ω * , ω * ), S 2 ) is close enough to -e 3 in H 1 , we have Z L ∞ < 1/ √ 2 (because the injection H 1 → L ∞ is continuous), which implies 1 2 √ 2 |Z| 2 ≤ 1 + z = 1 -1 -|Z| 2 ≤ 1 2 |Z| 2 .
Now, let δ := min{δc * /c * , η}, where δ is as in Proposition 2. Thanks to the monotonicity of L, we have, for every t ∈ [0, +∞),

M n (t) + e 3 H 1 ≤ c * L(M n (t)) ≤ c * L(M 0 n ) ≤ c * c * M 0 n + e 3 H 1 < c * δ c * ≤ δ.
We have

M n (t) -M ∞ (t) H 1 2 ≤ M 0 n -M 0 ∞ H 1 2 + t 0 F (s, M n (s)) -F (s, M ∞ (s)) H 1 2 ds. Let us prove the existence of C > 0 such that, for every M, M ∈ H 1 (ω * , ω * ) satisfying M + e 3 H 1 < δ, we have F (s, M ) -F (s, M ) H 1 2 ≤ C M -M H 1 2 , ∀s ∈ R.
Then, the proof may be concluded thanks to the Gronwall Lemma. Let us work, for example, on the third component of F :

F 3 (t, M ) = -Re Ω(t)Ze -ıω t 0 ζ ,
where Ω is defined by (11). We have

F 3 (t, M ) -F 3 (t, M ) H 1 2 ≤ |Ω(t) -Ω(t)| Ze -ıω t 0 ζ H 1 2 + | Ω(t)| (Z -Z)e -ıω t 0 ζ H 1 2 ≤ |Ω(t) -Ω(t)|c 1 Z H 1 2 + Kc 1 Z -Z H 1 2
where c 1 is as in the previous Lemma and K = K(δ). It is sufficient to prove the existence of a constant C > 0 such that, for every M, M ∈ H 1 * , ω * ), S 2 ) satisfying M + e 3 H 1 , M + e 3 H 1 < δ, we have

|Ω(t) -Ω(t)| ≤ C M -M H 1 2 , ∀t ∈ [0, +∞).
Let us prove it only on one of the terms that compose Ω (the other terms may be treated as well):

ω * ω * Z z -Z z e ı|ω| t 0 ζ dω ≤ ω * ω * Z -Z ze ı|ω| t 0 ζ dω + ω * ω * Z z -z e ı|ω| t 0 ζ dω ≤ Z -Z H -1 2 ze ı|ω| t 0 ζ H 1 2 + Z H -1 2 (z -z)e ı|ω| t 0 ζ H 1 2 ≤ c 1 K M -M H 1 2
. 2

Proof of Theorem 1: Let δ be as in the previous proof.

Let M 0 ∈ H 1 ((ω * , ω * ), S 2 ) be such that M 0 + e 3 H 1 < δ and M ∈ C 1 ([0, +∞), H 1 ((ω * , ω * ), S 2
)) be the solution of the closed loop system such that M (0) = M 0 .

First step: Let us prove that Ω(t) → 0 when t → +∞.

Thanks to the choice of the feedback law, M (t) is bounded in H 1 , uniformly with respect to t ∈ [0, +∞). Computing explicitly dΩ dt (t), we see that dΩ dt (t) is bounded in C uniformly with respect to t ∈ [0, +∞) -NT . Thus, Ω is uniformly continuous on [0, +∞). Since Ω ∈ L 2 (0, +∞), it has to satisfy Ω(t) → 0 (Barbalat's lemma).

Second step: Let us prove that -e 3 is the only possible weak H 1 limit. Let M 0 ∞ be a weak H 1 limit of the trajectory starting from M 0 . There exists a sequence (t n ) n∈N of [0, +∞) such that t n → +∞,

M (t n ) M 0 ∞ weakly in H 1 when n → +∞, M (t n ) → M 0
∞ strongly in H 1/2 when n → +∞. Working as in the previous proof, one may prove that

M (t n ) + e 3 H 1 < δ, ∀n ∈ N, (14) There exists t ∞ ∈ [0, T ) such that t n mod T → t ∞ .
Let M ∞ (t, ω) be the solution of the closed loop system associated to the initial condition M ∞ (t ∞ ) = M 0 ∞ . Let us prove that L is constant along this trajectory, by proving that the associated control Ω ∞ vanishes. In order to simplify, we assume that t ∞ = 0 (otherwise, consider an additional shift). For every t > 0, M (t n + t) → M ∞ (t) strongly in H 1/2 when n → +∞, thanks to the previous proposition. This allows to pass to the limit in the feedback law: Ω(t n + t) → Ω ∞ (t) when n → +∞, for every t > 0. Thanks to the first step, we get Ω ∞ = 0.

In order to apply Proposition 2, we only need to check that M 0

∞ + e 3 H 1 < δ, which is a consequence of ( 14). 2

CONCLUSION

We have investigated here the stabilization of an infinite dimensional system admitting in open-loop a continuous spectrum. We have designed a Lyapunov based feedback. Closed-loop simulations illustrate the asymptotic convergence towards the goal steady-state. We have provided a local and weak convergence result for the H 1 topology.

Simulations indicate that the domain of attraction is far from being local and thus we can expect a large attraction domain for this feedback law.

More generally, this feedback and convergence analysis open the way to asymptotic stabilization of neutrally stable systems of infinite dimension with continuous spectra. For example, it will be interesting to see if the following system (1D Maxwell-Lorentz model for the propagation of an electro-magnetic wave in a non-homogenous dispersive material)

∂ 2 E ∂t 2 + ∂ 2 P ∂t 2 = ∂ 2 E ∂x 2 , ∂ 2 P ∂t 2 = p 2 (x)(E-P ), x ∈ (0, 1) E(0, t) = u(t), E(1, t) = v(t)
with two controls u and v, can also be stabilized to zero. Notice that when p(x) is a smooth strictly increasing positive functions, this system admits as continuous spectrum ±ı]p(0), p(1)[. F (s, M (s)) H 1 ds.

By definition, we have

F (s, M (s)) 2 H 1 = Ω(s)z(s)e -ıω s 0 ς 2 H 1 + Ω(s)Z(s)e -ıω s 0 ς 2 H 1 ≤ |Ω(s)| 2 C 2 1 ( z(s) 2 H 1 + Z(s) 2 H 1 ) = C 2 1 |Ω(s)| 2 M (s) 2 H 1 .
Moreover, the Cauchy-Schwarz inequality gives |Ω(s)| ≤ K p (2G + 1) M (s) 2 H 1 , thus, Θ(M ) L ∞ ((0,T * ),H 1 ) ≤ M 0 H 1 + T * C 1 Kp(2G + 1)R 3 ≤ R thanks to (A.4).

Second step: Let is prove that Θ is a contraction. For M 1 , M 2 ∈ E and t ∈ [0, T * ], using (A.3), we get

Θ(M 1 )(t) -Θ(M 2 )(t) H 1 ≤ t 0 F (s, M 1 (s)) -F (s, M 2 (s)) H 1 ≤ tC 2 M 1 -M 2 L ∞ ((0,T * ),H 1 ) ,
thus Θ is a contraction, thanks to (A.4).

Third step: Let us prove the existence and uniqueness of strong solutions, defined on [0, +∞). Thanks to the Banach fixed point theorem, the map Θ has a unique fixed point.

We have proved that, for every R > 0, there exists T * = T * (R) > 0 such that, for every M 0 ∈ B R [H 1 ((ω * , ω * ), S 2 )], there exists a unique weak solution M ∈ C 0 ([0, T * ], H 1 ((ω * , ω * ), R 3 ) in the sense M (t, ω) = M 0 (ω) + t 0 F (s, M (s, ω))ds in H 1 (ω * , ω * ), ∀t ∈ [0, T * ]. We deduce that M ∈ C 1 ([0, T * ], H 1 ((ω * , ω * ), R 3 )) and dM dt (t, ω) = F (t, M (t, ω)) in H 1 ((ω * , ω * ), R 3 ), ∀t ∈ [0, T * ]. Since the embedding H 1 → L ∞ is continuous, we also have dM dt (t, ω) = F (t, M (t, ω)), ∀(t, ω) ∈ [0, T * ] × (ω * , ω * ). This has 2 consequences. Firstly, M (t, .) takes values in S 2 for every t ∈ [0, T * ], indeed, M 0 does and the following computation is licit d dt M (t, ω) 2 = 2 M (t, ω), F (t, M (t, ω)) = 0. Secondly, the computations (9), (10) are licit, thus L(t) is not increasing. Therefore, we have

Fig. 1 .Fig. 2 .

 12 Fig. 1. Lyapunov function L(t) defined by (8) and the closed-loop control Ω(t) defined by (11) .

  | 2 + z (T ) 2 + |Z(T )| 2 + z(T ) 2 dω ≤ max {1/G, 1} 2L(T ) + ω * -ω * ≤ max {1/G, 1} 2L(0) + ω * -ω * ≤ R 2thanks to (A.1). Thus, we can apply the previous result with M 0 replaced by M (T * ): it provides a solution on [0, 2T * ]. Iterating this again, we get a solution defined for every t ∈ [0, +∞).2

KB and PR were partially supported by the "Agence Nationale de la Recherche" (ANR), Projet Blanc C-QUID number BLAN-3-139579.

2 Partially supported by CNPq -Conselho Nacional de Desenvolvimento Cientifico e Tecnologico -Brazil, under

grant 308465/2006-7.

Or at least they are solutions of the corresponding integral equations.

Appendix A. PROOF OF PROPOSITION 1

Let M 0 ∈ H 1 ((ω * , ω * ), S 2 ) and R > 0 be such that

where B R [F ] denote the closed ball centered at 0 with radius R, of the space F . Let T * = T * (R) > 0 be small enough so that

4) Let us consider the map Θ, defined on the space

for every (t, ω) ∈ [0, T * ] × (ω * , ω * ).

First


Arnaud Heibig

Liviu Iulian Palade
email: liviu-iulian.palade@insa-lyon.fr

Well posedness of a linearized fractional derivative fluid model

Keywords: objective fractional derivative constitutive equation, viscoelasticity, rest state stability analysis, Hadamard stability analysis, solution existence, uniqueness, smoothness

The one-dimensional fractional derivative Maxwell model (e.g. Palade et al. Rheol. Acta 35, 265, 1996), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade et al. Int. J. Non-Linear Mech. 37, 315, 1999, to objective threedimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade J. Math. Phys. 49, 043101, 2008, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results.

Introduction

Fractional derivative constitutive equations (CEs) have been found to accurately predict, among others, the stress relaxation of viscoelastic fluids in the glass transition and glassy (high frequency) states. The experimental behavior of storage G ′ and loss G ′′ moduli (obtained upon using the time -temperature superposition principle -see [START_REF] Palade | Time-temperature superposition and linear viscoelasticity of polybutadienes[END_REF], [START_REF] Robertson | Unified application of the coupling model to segmental, Rouse, and terminal dynamics of entangled polymers[END_REF]) of a linear, narrow molecular weight series of polybutadienes is exceptionally well predicted by linearized fractional derivative models as can be reckoned from [START_REF] Palade | A modified fractional model to describe the entire viscoelastic behavior of polybutadienes from flow to glassy regime[END_REF]. Polybutadienes are of utter importance for the tire industry, for manufacturing certain solid propergols, etc. Similar excellent agreements between frequency sweep experimental data obtained on other polymers (e.g. polystyrenes) and theoretical predictions of linear fractional derivative models are reported in [START_REF] Friedrich | Relaxation Functions of Rheological Constitutive Equations With Fractional Derivatives: Thermodynamical Constraints[END_REF][START_REF] Kausch | Matériaux Polymères: Propriétés Mécaniques et Physiques[END_REF][START_REF] Mainardi | Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models[END_REF].

The object of study is the below given objective, fractional derivative viscoelastic (incompressible) fluid constitutive equation (CE) (see [START_REF] Palade | Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models[END_REF])

S(t) + λ α F(t) t -∞ µ 1 (t -τ )F -1 (τ ) ▽ S (τ ) F -1 (τ ) T dτ F(t) T = Gλ β F(t) t -∞ µ 2 (t -τ )F -1 (τ )A 1 (τ ) F -1 (τ ) T dτ F(t) T (1.1)
Function S is the (objective) stress tensor and ▽ S its objective upper convected derivative defined by (with D/Dt denoting the material derivative and L the velocity gradient; see for example [START_REF] Huilgol | Fluid Mechanics of Viscoelasticity[END_REF], [START_REF] Morrison | Understanding Rheology[END_REF], [START_REF] Cleja-T ¸igoiu | Rheology and Thermodynamics, Part I -Rheology[END_REF]):

▽ S = DS Dt -LS -SL T , (1.2)
Function F is the strain gradient and A 1 = ∇u + (∇u) T = L + L T is the first Rivlin-Ericksen tensor. The model parameters are such that 0 < λ, 0 < α < β < 1. µ 1,2 (t) are two memory kernels given by:

µ 1 (t -τ ) = (t -τ ) -α Γ(1 -α) , µ 2 (t -τ ) = (t -τ ) -β Γ(1 -β) (1.
3)

The stability of the rest state is now investigated using the linearized theory. As shown in [START_REF] Palade | Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models[END_REF] and [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF], it is first assumed that the stress tensor S = O(ǫ) and the deformation gradient F(t) = 1 + ǫJ(t) + O(ǫ 2 ). Since L = ḞF -1 (see for ex. [START_REF] Huilgol | Fluid Mechanics of Viscoelasticity[END_REF], [START_REF] Joseph | Fluid dynamics of viscoelastic fluids[END_REF], [START_REF] Morrison | Understanding Rheology[END_REF]), L = O(ǫ). Hence the velocity u = O(ǫ), and the first Rivlin-Ericksen tensor A 1 = O(ǫ) as well. Therefore, keeping only terms of O(ǫ), within the linear response theory eq.(1.1) reduces to :

S(t) + λ α t -∞ µ 1 (t -τ ) ∂S(τ ) ∂τ dτ = Gλ β t -∞ µ 2 (t -τ )A 1 (τ )dτ (1.4)
Next, assume the fluid is contained in a bounded volume Ω ⊂ R 3 whose boundary ∂Ω is sufficiently smooth, and set in motion at t = 0. The CE in eq.(1.4) then takes the form:

S(t) + λ α t 0 µ 1 (t -τ ) ∂S(τ ) ∂τ dτ = Gλ β t 0 µ 2 (t -τ )A 1 (τ )dτ (1.5)
The above may be re-written in condensed form using the Caputo operators D α t and I 1-β t as:

S(t) + λ α D α t S = Gλ β I 1-β t A 1 (1.6)
where for an absolutely continuous function f : R + → C:

D α t f (t) = 1 Γ(1 -α) t 0 f ′ (τ ) (t -τ ) α dτ (1.7)
and for f ∈ L 1 loc (R + ),

I 1-β t f (t) = 1 Γ(1 -β) t 0 f (τ ) (t -τ ) β dτ (1.8)
As shown in [START_REF] Palade | Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models[END_REF][START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF], investigating the stability of the rest state is tantamount to studying the existence and the uniqueness of solutions to the following initial boundary value problem (IBVP):

∂u ∂t = -∇p + ∇ • S (1.9a) S + λ α D α t S = Gλ β I 1-β t A 1 , A 1 = ∇u + (∇u) T (1.9b) ∇ • u = 0, in [0, +∞[×Ω, Ω ⊂ R 3 (1.9c) u = 0, in [0, +∞[×∂Ω (1.9d) u(t = 0) = u 0 , S(t = 0) = S 0 (1.9e) (1.9f)
In the above system of equations we assume u :

[0, +∞[×Ω → R 3 , ∇ • u 0 = 0, p : [0, +∞[×Ω → R, S : [0, +∞[×Ω → M 3,3 (R), 0 < α < β < 1. Denote δ = β -α > 0.
A change of variables on (x, t) can be performed to eliminate the CE parameters λ and G (see [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]). This is carried out only for convenience; in no way the generality of this paper results is shrinked down. Therefore, from now on assume λ = G = 1.

At this stage recall that an existence result for the initial boundary value problem given in equations (1.9)was presented in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]. The present paper, which is a continuation of [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF], is organized as follows:

• Section 2 presents the weak formulation of the boundary value problem.

• Section 3 is devoted to proving the existence and uniqueness of the solutions. We further on use the existence theorem obtained in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] to state a general existence and uniqueness result.

• Section 4 deals with the functional framework within which the solution continuity at t = 0 is proved.

• Section 5 presents the proof of the solution continuity at t = 0.

• Section 6 contains results on the solution smoothness.

Weak formulation of the IBVP

All time-depending functions involved in the current stability analysis, save for when stated otherwise, are causal functions (i.e. set equal to zero on R -). Hence the convolution in time is simply (f * g)(t)

:= t 0 f (s)g(t -s)ds.
We first present the weak formulation of the boundary value problem eqs.(1.9): find

u ∈ C 0 ([0, +∞[, L 2 (Ω) 3 ) ∩L 1 loc (R + , H 1 0 (Ω) 3 ), ∇ • u = 0, [S] ij ∈ C 0 (]0, +∞[, L 2 (Ω) ∩ L 1 loc (R + , L 2 (Ω))), i, j = 1, 2, 3, such that for any test-functions ∀θ ∈ (H 1 0 (Ω)) 3 , ∇ • θ = 0, ∀a ∈ (D(Ω)) 3 , ∀ψ ∈ C ∞ 00 ([0, +∞[),
where C ∞ 00 ([0, +∞[) denotes the space of C ∞ class functions that vanish in a neighborhood of +∞, the following equations hold true:

ψ(0) Ω u 0 (x) • θ(x)dx + +∞ 0 Ω u(t, x) • θ(x)ψ ′ (t)dxdt = +∞ 0 Ω (S(t, x) : ∇θ(x))ψ(t)dxdt (2.1) - +∞ 0 Ω ψ(τ ) Γ(1 -α)τ α [S 0 ] ij (x)[a] j (x)dxdτ - +∞ 0 +∞ τ Ω ψ ′ (t) Γ(1 -α)τ α [S] ij (t -τ, x)[a] j (x)dxdtdτ + +∞ 0 Ω [S] ij (t, x)[a] j (x)ψ(t)dxdt = - 1 Γ(1 -β) +∞ 0 t 0 Ω ψ(t) (t -τ ) β {(∇ • a) [u] i + [u • ∇a] i } (τ, x)dxdτ dt (2.2)
Summation over repeated indices is understood in equations (2.1) and (2.2) above. We now detail the functional framework. Let V = {h ∈ H 1 0 (Ω) 3 s.t. ∇ • h = 0} be the Hilbert space endowed with the inner product:

f|g V = 3 i,j=1 Ω ∂f i ∂x j ∂g i ∂x j (x)dx (2.3)
and denote V the corresponding norm. The closure of V in (L 2 (Ω)) 3 is denoted by H, the later space being endowed with the inner product:

f|g H = 3 i=1 Ω f i g i (x)dx (2.4) with H being the corresponding norm. Let 0 < λ 1 ≤ λ 2 ≤ . . . λ n ≤ . . . -→ n→+∞ +∞ and w i ∈ V, i ∈ N * ,
be the eigenvalues and the corresponding eigenfunctions of the Stokes operator in H, i.e.:

∀φ ∈ V, w k | φ V = λ k w k | φ H , where w k H = 1 (2.5)
To prove the solution uniqueness, we first eliminate S from equations (2.1) and (2.2).

Denote

L := {f ∈ L 1 loc (R + ) s.t. ∃M > 0, so that f e -M t ∈ L 1 (R + )} ⊂ L 1 loc (R + ). Next, let f ∈ L 1 loc,R + (R).
For any a ∈ R and α ∈]0, 1[, define D α t,a f by:

D α t,a f, ϕ = 1 Γ(1 -α) -a +∞ 0 ϕ(τ ) τ α dτ - +∞ 0 f (t) +∞ 0 ϕ ′ (t + τ ) τ α dτ dt (3.1)
for any test function ϕ ∈ D(R).

Observe that D α t,a f ∈ D ′ (R). Moreover, for any f ∈ L , one easily sees that x ≥ M, e -xt D α t,a f ∈ S ′ (R). The hat ( ) notation to be used below stands for the usual Laplace transform. Proposition 3.1. Let f ∈ L . Then, for ∀s ∈ C, with Re(s) large enough, one has D α t,0 f (s) = s α f (s).

Proof. Let M > 0 such that e -M t f ∈ L 1 (R + ). For any s ∈ C, Re(s) ≥ M and ϕ(t) = e -st , Eq.(3.1) -still valid for this particular choice of ϕ(t) = e -st / ∈ D(R) -gives:

D α t,0 f, e -st = - 1 Γ(1 -α) +∞ 0 f (t) +∞ 0 -se s+τ τ α dτ dt (3.2)
As +∞ 0 e -sτ τ α dτ = Γ(1 -α) s 1-α , for Re(s) ≥ M > 0 one gets:

D α t,0 f (s) = D α t,0 f, e -st = - s Γ(1 -α) +∞ 0 e -st f (t)Γ(1 -α)s α-1 dt = s α f (s) (3.3)
The following classical result (see [START_REF] Carpinteri | Fractals and fractional calculus in continuum mechanics[END_REF]) is stated here within our functional framework. Recall first that (see also equations 14 and 15 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]):

W 0 (t) = sin(απ) π +∞ 0
e -rt r α-1 r 2α + 2r α cos(απ) + 1 dr, t ≥ 0 (3.4)

E α (t) = sin(απ) π ∞ 0
r α e -rt r 2α + 2r α cos(απ) + 1 dr, t > 0 (3.5) Proposition 3.2. Let F ∈ L . Then, for any a ∈ R and any α ∈]0, 1[, the equation

D α t,a f + f = F (3.6)
has a unique solution f ∈ L , given by f = E α * F + aW 0 .

Proof. Existence:

Assume F ∈ C 1 (R + ). Then, f = E α * F +aW 0 is a solution of Eq.(3.6) (cf [7]). Now, if one assumes that F ∈ L , then there exists (F n ) n∈N * , F n ∈ C 1 (R + ) such that F n -→ n→+∞ F in L 1 loc (R + ). Since E α ∈ L 1 loc (R + ), then f n = E α * F n + aW 0 -→ n→+∞ E α * (t) F + aW 0 in L 1 loc (R + ). Hence the equation D α t,a f n + f n = F n , for n → +∞, becomes D α t,a f + f = F . We must next show that f ∈ L . Notice first that aW 0 ∞ ≤ |a|W 0 (t). Therefore aW 0 ∈ L . Denote g = e -M t F ; choose M > 0 so that e -M t F ∈ L 1 (R). Then E α * (t) F = t 0 E α (t -s)e M s g(s)ds ≤ e M t E α * (t) g . Since g ∈ L 1 (R + ), and E α ∈ L 1 (R + ), then E α * g ∈ L 1 (R + ). Finally E α * F ∈ L and f = E α * F + aW 0 ∈ L .
Uniqueness: Let f, g ∈ L be two solutions of Eq.(3.6). Then D α t,0 (f -g) + (f -g) = 0, from which it follows that (s α + 1) f -g(s) = 0, for Re(s) large enough. Therefore f -g(s) = 0, thus f = g.

We shall use the following result to prove the uniqueness property:

Lemma 3.1. For any g ∈ L , I 1-β t g ∈ L .
Proof. Since g ∈ L , there exists G ∈ L 1 (R + ) and M > 0 such that g = Ge -M t . Therefore, for t ≥ 0 a.e., I 1-β

t g(t) ≤ K t 0 g(t -u) u β du ≤ K t 0 |G(t -u)| e -M (t-u) u β du ≤ Ke -M t |G| * u β e -M t . Now, G ∈ L 1 (R + ), u β e -M t ∈ L 1 (R + ) leads to |G| * u β e -M t ∈ L 1 (R + ). Therefore, I 1-β t g(t) ≤ e M t H(t), with H ∈ L 1 (R + ), which gives I 1-β t g ∈ L .
Making use of Proposition 3.2 and of Lemma 3.1, we get: Corollary 3.1 (Solution uniqueness). Let u 0 ∈ H and S 0 ∈ L 2 (Ω) 9 . The system of equations Eqs.(2.1)-(2.2) has at most one solution that belongs to the functional space

F := {(u, S) ∈ [C 0 (R + , H) ∩ L 1 loc (R + , V )]× C 0 (]0, +∞[, L 2 (Ω) 9 ), such that ∇u L 2 (Ω) 9 ∈ L , S L 2 (Ω) 9 ∈ L }.
Proof. Let (u, S) ∈ F be a solution to Eqs.(2.1)-(2.2). For any test function ϕ ∈ D(Ω) 9 , as a consequence of Eq.(2.2) and of the fact that u ∈ L 1 loc (R + , V ), one has

D α t, S 0 |ϕ L 2 (Ω) 9 S|ϕ L 2 (Ω) 9 + S|ϕ L 2 (Ω) 9 = I 1-β t A 1 |ϕ L 2 (Ω) 9 (3.7) However, S|ϕ L 2 (Ω) 9 ≤ S L 2 (Ω) 9 ϕ L 2 (Ω) 9 . Since S L 2 (Ω) 9 ∈ L and ∇u L 2 (Ω) 9 ∈ L , we infer that S|ϕ L 2 (Ω) 9 ∈ L , and A 1 |ϕ L 2 (Ω) 9 ∈ L . Now Lemma 3.1 implies I 1-β t A 1 |ϕ L 2 (Ω) 9 ∈ L ,

and Proposition 3.2 leads to

S|ϕ L 2 (Ω) 9 = E α * I 1-β t A 1 |ϕ L 2 (Ω) 9 + S 0 |ϕ L 2 (Ω) 9 W 0 = ρ * A 1 |ϕ L 2 (Ω) 9 + S 0 |ϕ L 2 (Ω) 9 W 0 (3.8)
Notice that Eq.(3.8) still holds true for ϕ ∈ L 2 (Ω) 9 . Let θ ∈ V , ψ ∈ C +∞ 00 ([0, +∞[). We deduce from (3.8) and (2.1) that:

ψ(0) Ω u 0 (x) • θ(x)dx + +∞ 0 Ω u(t, x) • θ(x)ψ ′ (t)dxdt = - +∞ 0 ρ * A 1 |∇θ L 2 (Ω) 9 + S 0 |∇θ L 2 (Ω) 9 W 0 (t)ψ(t)dt (3.9)
We search for u ∈ L 1 loc (R + , V ). In this case, for almost every t > 0, u can be expressed as

u(t) = +∞ q=1 α q (t)w q (3.10) the series being convergent in V . It follows, by taking θ = w k and ψ ∈ D(]0, +∞[) in equation (3.9), that α ′ k = -λ k (ρ * α k ) -b k √ λ k W 0 , with b k := Ω (S 0 : ∇w k ) (x)dx, the equality holding true in D ′ (]0, +∞[). Recall that α k = u|w k H ∈ C 0 (R + ). As W 0 ∈ C 0 (R + ), then necessarily α k ∈ C 1 (R + ).
However, (cf [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]) the Cauchy's initial value problem

α ′ k (t) = -λ k (ρ * α k ) (t) -b k λ k W 0 (t) (3.11a) α k (0) = α 0 k (3.11b)
has a unique solution in C 1 (R + ). The uniqueness of the solution u is thus proved, and that of S follows.

We now state an existence and uniqueness result. Denote

C 0 b (R * + , V ) := {u ∈ C 0 (R * + , V ) s.t. sup t≥1 u(t) V < +∞}. The functional space C 0 b (R * + , L 2 (Ω) 9
) is defined in a similar way.

Theorem 3.1 (First Existence and Uniqueness Theorem). Let u 0 ∈ H, S 0 ∈ L 2 (Ω) 9 . Then the boundary value problem given by the system of equations (2.1)-( 2.2) has a unique solution

(u, S) ∈ C 0 (R + , H) ∩ C 0 b (R * + , V ) ∩ L 1 loc (R + , V ) × C 0 b (R * + , L 2 (Ω) 9 ) ∩ L 1 loc (R + , L 2 (Ω) 9 ) (3.12) Moreover, u(0) = u 0 .
Proof. The existence of at least one solution 9 ) follows from Theorem 8.4 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]. It remains to be proved that the solution (u, S) obtained in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] satisfies

(u, S) ∈ C 0 (R + , H) ∩ C 0 (R * + , V ) ∩ L 1 loc (R + , V ) × C 0 (R * + , L 2 (Ω) 9 ) ∩ L 1 loc (R + , L 2 (Ω)
sup t≥1 u(t) V + sup t≥1 S(t) L 2 (Ω) 9 < +∞ (3.13)
This is essentially contained in the arguments presented in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]. Indeed, since u ∈ C 0 (R + , V ), we write

u = +∞ k=1 α k (t)w k , t > 0.
From equation 125 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] one gets

λ k |α k (t)| 2 ≤ M    1 t δ/2 + 1 t 1-δ/2 2 + tλ 1/(2-δ) k 2(1-δ/2) t 2(1-δ/2) e -2γtλ 1/(2-δ) k    |α 0 k | 2 + 1 t 2(1-δ) + (tλ 1/(2-δ) k ) 2(1-δ) t 2(1-δ) e -2atλ 1/(2-δ) k |b k | 2 (3.14)
In the above, δ = (α -β) ∈]0, 1[, a > 0, γ > 0, and

α 0 k = u 0 |w k H . Clearly u 0 ∈ L 2 (Ω) 3 , S 0 ∈ L 2 (Ω) 9 implies that +∞ k=1 |α 0 k | 2 < +∞ and +∞ k=1 |b k | 2 < +∞. Hence u 2 V = +∞ k=1 λ k |α k (t)| 2 ≤ M t inf(δ,2-2δ) -→ t→+∞ 0 (3.15) and u 0 ∈ C 0 b (R * + , V
). We use the equation that defines S given right below equation 137 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]. Then: 9 (3.16)

S L 2 (Ω) 9 ≤ M +∞ k=1 λ k |ρ * α k (t)| 2 + |W 0 (t)| S 0 L 2 (Ω)
From (3.4) we see that

|W 0 (t)| -→ t→+∞ 0 (3.17)
Moreover, (see equation 130 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF])

λ k |ρ * α k (t)| 2 ≤ M       1 t δ + tλ 1/(2-δ) k δ t δ exp -atλ 1/(2-δ) k    |α 0 k | 2 + 1 t 2ǫ + exp(-2at) |b k | 2    , ǫ > 0, a > 0 Hence +∞ k=1 λ k |α k (t)| 2 ≤ M t inf(δ,2ǫ) , for t ≥ 1 (3.18) Finally (see (3.16)-(3.18)): S L 2 (Ω) 9 -→ t→+∞ 0. It follows that S ∈ C 0 b (R * + , L 2 (Ω) 9
). The existence of at least one solution belonging to the functional space of (3.12) is thus proved.

The uniqueness of such a solution results from Corollary 3.1 and from the fact that

C 0 (R + , H) ∩ C 0 b (R * + , V ) ∩ L 1 loc (R + , V ) × C 0 b (R * + , L 2 (Ω) 9 ) ∩ L 1 loc (R + , L 2 (Ω) 9 ) ⊂ F .

Functional spaces

In order to prove the continuity of the (u, S) at t = 0 we recall several classical functional spaces (see also [START_REF] Lions | Problèmes aux Limites Non-Homogènes et Applications[END_REF] and [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF]).

Denote ǫ k = ∇w k √ λ k , k ∈ N * . Let Π : L 2 (Ω) 9 → L 2 (Ω) 9 be the orthogonal projection operator of L 2 (Ω) 9 onto [Vect(ǫ k ) k∈N * ] ⊥ , θ ≥ 0. For any f ∈ L 2 (Ω) 9 , denote f 2 D θ := +∞ q=1 λ θ q | f|ǫ q L 2 (Ω) 9 | 2 + Π(f) 2 L 2 (Ω) 9 . Let D θ := {f ∈ L 2 (Ω) 9 s.t. f D θ < +∞}. For any f, g ∈ D θ , denote f|g D θ := +∞ q=1 λ θ q f|ǫ q L 2 (Ω) 9 g|ǫ q L 2 (Ω) 9 + Π(f)|Π(g) L 2 (Ω) 9 . The functional space (D θ , | D θ ) is a Hilbert space. For any f ∈ H, let f 2 H θ := +∞ q=1 λ θ q | f|w q H | 2 . Next, let H θ := {f ∈ H, f H θ < +∞}. For any f, g ∈ H θ , f |g H θ := +∞ k=1 λ θ q f |w k H g|w k H . As the sequence (w k ) k∈N * is complete in L 2 (Ω) 9 , the functional space (H θ , | H θ ) is a Hilbert space. Remark that, for any 0 ≤ θ ≤ θ ′ ≤ 1 ≤ θ ′′ , one has: H = H 0 ←֓ H θ ←֓ H θ ′ ←֓ H 1 = V ←֓ H θ ′′ (4.1) L 2 (Ω) 9 = D 0 ←֓ D θ ←֓ D θ ′ ←֓ D 1 ←֓ D θ ′′ (4.2)
The above injections are dense; use of them will be often made from now on.

The following ∆ θ spaces are closely related to the D θ ones. Let P : L 2 (Ω) 9 → L 2 (Ω) 9 be the orthogonal projection operator from L 2 (Ω) 9 onto

+∞ k=1 ǫ k , ǫ T k ⊥ . Let θ ∈ R. For any element f ∈ L 2 (Ω) 9 , denote f 2 ∆ θ = +∞ q=1 λ θ q f|ǫ q L 2 (Ω) 9 2 + +∞ q=1 λ θ q f|ǫ T q L 2 (Ω) 9 2 + P (f) 2 L 2 (Ω) 9 (4.3)
For any θ ≥ 0, ∆ θ := {f ∈ L 2 (Ω) 9 s.t. f ∆ θ < +∞}. The functional space ∆ θ endowed with the inner product defined as:

∀f ∈ ∆ θ , ∀g ∈ ∆ θ , f|g ∆ θ = +∞ q=1 λ θ q f|ǫ q L 2 (Ω) 9 g|ǫ q L 2 (Ω) 9 + +∞ q=1 λ θ q f|ǫ T q L 2 (Ω) 9 g|ǫ T q L 2 (Ω) 9 + P (f)|P (g) L 2 (Ω) 9 (4.4)
is a Hilbert space. For any θ < 0, let ∆ θ denote the topological dual space of ∆ -θ , i.e. ∆ θ = (∆ -θ ) ′ . The space ∆ θ is the completion of (L 2 (Ω) 9 , ∆ θ ). Next, note that whenever 0 ≤ γ ≤ γ, the following injections 9 and (L 2 (Ω) 9 ) ′ are isomorphic to each other.

L 2 (Ω) 9 = ∆ 0 ←֓ ∆ γ ←֓ ∆ γ are dense. It results that ∆ -γ ←֓ ∆ -γ = (∆ γ ) ′ ←֓ (L 2 (Ω) 9 ) ′ ≃ L 2 (Ω) 9 = ∆ 0 ←֓ ∆ γ ←֓ ∆ γ , invoking the fact that L 2 (Ω)
Next, for any f ∈ H 1 0 (Ω) 9 , one has:

f 2 ∆ 1 = +∞ q=1 λ q f|ǫ q L 2 (Ω) 9 2 + +∞ q=1 λ q f|ǫ T q L 2 (Ω) 9 2 + P (f) 2 L 2 (Ω) 9 = +∞ q=1 ∇ • f|w q L 2 (Ω) 3 2 + ∇ • f T |w q L 2 (Ω) 3 2 + P (f) 2 L 2 (Ω) 3 ≤ K f 2 H 1 0 (Ω) 9 (4.5)
due to the Poincaré's inequality. Consequently H 1 0 (Ω) 9 ֒→ ∆ 1 and the restriction r :

∆ ′ 1 → H -1 (Ω) 9 , such that T r → T | H 1 0 (Ω) 9 is continuous. Lemma 4.1. Let θ ∈ R + . (a) The sequence (ǫ k ) k∈N * is orthogonal in D θ . Moreover, ǫ k D θ = λ θ/2 k . (b) The sequence (ǫ T k ) k∈N * is orthogonal in D θ . (c) The sequence (ǫ k + ǫ T k ) k∈N * is orthogonal in D θ , and ǫ k + ǫ T k D θ = (1 + λ θ k ) 1/2 .
(d) The sequence (w k ) k∈N * is orthogonal in H θ , and

w k H θ = λ θ/2 k . (e) Let f ∈ L 2 (Ω) 9 . Denote f = +∞ k=1 a k ǫ k + +∞ k=1 b k ǫ T k + P (f). Then: f 2 ∆ -θ = +∞ k=1 λ -θ k |a k | 2 + |b k | 2 + P (f) 2 L 2 (Ω) 9 .
Proof. Observe that (cf. eq.(2.5)) for any (k, q) ∈ N * 2 :

ǫ k |ǫ q L 2 = w k |w q L 2 = δ kq (4.6)
On the other hand, since ∇ • w k = 0,

ǫ k |ǫ T q L 2 = i,j Ω ∂(w k ) i ∂x j (x) ∂(w q ) j ∂x i (x)dx = Ω i ∂(w k ) i ∂x j (x) j ∂(w q ) j ∂x i (x) dx = 0 (4.7)
Hence:

ǫ k |ǫ T q L 2 = 0 (4.8)
The statements (a) to (e) result from Eqs.(4.6)-(4.8).

Except for the injection H 2k ֒→ H 2k (Ω) 3 (see below), the following description of the spaces will not be used in this paper.

Let first θ ∈ [0, 1].

Let Λ : V → H, such that Λ k∈N * a k w k = k∈N * a k λ k w k . Then, for any (u, v) ∈ V 2 , Λu|Λv H = u|v V , and H θ = D(Λ θ ) = [V, H] 1-θ
, where [V, H] 1-θ stands for the holomorphic interpolation of spaces V and H, and D(Λ θ ) for the domain of Λ θ . Denote H 0 0 (Ω) ≡ L 2 (Ω). Let the canonical injection 3 (the last continuous injection ֒→ boils down to an equality = whenever θ = 1/2). Let now γ n denote the normal-trace application. It is well known that H = {u ∈ H 0 (Ω) 3 , s.t. ∇ • u = 0, γ n (u) = 0}. From the preceding arguments it results that we have the continuous injection

H i ֒→ H 0 0 (Ω) 3 and V i| V ֒→ H 1 0 (Ω) 3 be its restriction. Then H θ = [V, H] 1-θ i| [V,H] 1-θ ֒→ [H 1 0 (Ω) 3 , H 0 0 (Ω) 3 ] 1-θ ֒→ H θ 0 (Ω)
H θ ֒→ H ∩ H θ 0 (Ω) 3 = {u ∈ H θ 0 (Ω) 3 , s.t. ∇ • u = 0, γ n (u) = 0}
, with the space H ∩ H θ 0 (Ω) 3 being endowed with the H θ 0 (Ω) 3 topology. Let now θ ∈ N * . As quoted on page 106 in [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF], 3 ∩V . Also, invoking Agmon -Douglis -Nirenberg's Theorem as stated on page 832 in [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF], leads to

H 2 = D(Λ 2 ) = H 2 (Ω)
H 2k = D(Λ 2k ) ֒→ H 2k (Ω) 3 ∩ V , k ∈ N * . Here H 2k (Ω) are classical Sobolev spaces.
5 The continuity of the solution (u, S) at t = 0

From now on (u, S) denotes the solution to equations (2.1)-(2.2), with initial data (u 0 , S 0 ) ∈ H × L 2 (Ω) 9 (see Theorem 3.1). In order to prove continuity results we recall several representation formulas for u and S. First, functions α k , k ∈ N * , are defined by equations (3.11a)-(3.11b). Equivalently, for x ∈ R + (see [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]),

α k (t) = 1 2π lim A→+∞ +A -A T λ k (x + iy)e (x+iy)t dyα 0 k -lim A→+∞ +A -A (T λ k w)(x + iy)e (x+iy)t dy λ k b k (5.1) with T µ (s) = s 1-β (s α + 1) s 2-β (s α + 1) + µ , w(s) = 1 s 1-α (s α + 1) , µ ∈ R + , s ∈ C -R -, and
α 0 k = u 0 |w k H , b k = Ω (S 0 : ∇w k )(x)dx.
Notice that eq.(5.1) is given in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] only for x ≥ M. The general result (x ∈ R + ) follows from a simple use of the Cauchy formula; details are omitted. Regarding function S, recall the following formula from [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]:

S = +∞ k=1 (ρ * α k ) ⊗ (∇w k + ∇ T w k ) + W 0 ⊗ S 0 (5.2) Notation h = f ⊗ g means h(x, y) = f (x)g(y)
. As quoted in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF], the series in (5.2) converges in

C 0 (R * + , L 2 (Ω) 9 ) and in L 2 loc (R + , L 2 (Ω) 9 ). Here ρ = E α * t -β Γ(1 -β)
, and 0 < ρ(t) ≤ kt -δ (see [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]),

δ = β -α.
The following estimate will give the continuity at t = 0 of (u, S).

Lemma 5.1. For any µ 0 > 0, ∃M > 0, such that ∀(x, t) ∈ (R + ) 3 × R + , and ∀µ ≥ µ 0 , we have:

lim A→∞ +A -A T µ w(x + iy)e (x+iy)t dy ≤ +∞ -∞ |(T µ w)(iy)| dy ≤ M µ 1/(2-δ) (5.3) Proof. Whenever y > 0, |(T µ w)(iy)| = 1 |y| δ 1 µ + (ye iπ/2 ) 2-δ + (ye iπ/2 ) 2-β
. Therefore, µ + ye iπ/2 2-δ + ye iπ/2 2-β ≥ Im µe iπ(β-2)/2 + y 2-δ e iπα/2 + y 2-β ≥ -µ sin(πβ/2) + y 2-δ sin(πα/2) ≥ µ sin(πβ/2) -y 2-δ sin(πα/2) ≥ Kµ, for y ≤ µ 1/(2-δ) (5.4)

The constant K = sin(πβ/2)sin(πα/2) > 0 is independent of µ. Moreover,

µ + ye iπ/2 2-δ + ye iπ/2 2-β ≥ Im µ + ye iπ/2 2-δ + ye iπ/2 2-β = y 2-δ sin(π -πδ/2) + y 2-β sin(π -πβ/2) = y 2-δ sin(πδ/2) + y 2-β sin(πβ/2) ≥ Ky 2-δ , for y ≥ µ 1/(2-δ) 0
(5.5)

From the above estimates we infer that:

+∞ 0 |(T µ w)(iy)|dy = µ 1/(2-δ) 0 dy y δ µ + (ye iπ/2 ) 2-δ + (ye iπ/2 ) 2-β + +∞ µ 1/(2-δ) dy y δ µ + (ye iπ/2 ) 2-δ + (ye iπ/2 ) 2-β ≤ µ 1/(2-δ) 0 K µy δ dy + +∞ µ 1/(2-δ) K y δ y 2-δ dy ≤ M µ 1/(2-δ)
(5.6)

A similar estimate can be obtained for Denote, δ = β -α, ω = δ/(2 -δ) and notice that 0 < ω < δ < 1. From now on we shall sometimes write α k (u 0 , S 0 ) instead of α k ; of course α k is linear w.r.t. initial data (u 0 , S 0 ). Most of the following estimates are already proved in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF], save for those derived from Lemma 5.1.

Proposition 5.1. Let u 0 ∈ H, S 0 ∈ L 2 (Ω) 9 . Then exists ∃M > 0, such that, ∀t ∈ R + and ∀k ∈ N * , (i) |α k (t)| 2 ≤ M |α 0 k | 2 + λ -ω k |b k | 2 . (ii) λ k |α k (t)| 2 ≤ M |α 0 k | 2 t 2-δ + |b k | 2 t 2-2δ .
(iii) for any µ ∈ [0, 1] and any τ ∈ [0, 1],

|α k (t)| 2 ≤ M |α 0 k | 2 λ µ k t µ(2-δ) + |b k | 2 λ τ +(1-τ )ω k t 2τ (1-δ)
(iv) for any µ ∈ [0, 1] and any τ ∈ [0, 1],

λ k |ρ * α k | 2 (t) ≤ M |α 0 k | 2 λ µ-1 k t µ(2-δ)+2(δ-1) + |b k | 2 λ -(1-τ )(1-ω) k t -2(1-τ )(1-δ) Proof. (i)
Since α k is linear with respect to (u 0 , S 0 ), we have, by eq.( 121) in Theorem 8.1 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] and eq.(5.1):

|α k (u 0 , S 0 )| 2 ≤ 2|α k (u 0 , 0)| 2 + 2|α k (0, S 0 )| 2 ≤ M |α 0 k | 2 + +∞ -∞ (T λ k w)(iy)e iyt dy λ k b k 2 (5.7)
Invoking Lemma 5.1 we get

|α k (u 0 , S 0 )| 2 ≤ M |α 0 k | 2 + |b k | 2 λ k λ 2(2-δ) k (5.8)
which gives (i).

(ii)

Estimate (ii) is obtainable right away from eq.( 122) in Theorem 8.1 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF], with M T instead of M. The proof that M can be chosen independently of T is deferred until Corollary 6.1 in Section 6. Hence we take here M independent of T and proceed further on.

(iii)

Notice first that (ii) above gives

|α k (u 0 , 0)| 2 (t) ≤ M|α 0 k | 2 λ k t 2-δ (5.9)
and

|α k (0, S 0 )| 2 (t) ≤ M|b k | 2 λ k t 2-2δ
(5.10)

Next, combining eq.(5.9) and eq.(5.8) with S 0 = 0 on one hand, and eq.(5.10) and eq.(5.8) with u 0 = 0 on the other, making further use of eq.(5.7) leads to estimate (iii).

(iv) Since 0 ≤ ρ(t) ≤ K/t δ , estimate (iii) gives

λ k |ρ * α k | (t) ≤ λ k (ρ * |α k |) (t) ≤ M √ λ k |α 0 k | λ µ/2 k t µ(2-δ)/2+δ-1 + √ λ k |b k | λ [τ +ω(1-τ )]/2 k t τ (1-δ)+δ-1 ,
from which (iv) is obtained.

In order to work on spaces H θ and D γ , we need to reformulate Proposition 5.1. Let [ ] + denote the positive part of a real number. Lemma 5.2. Let u 0 ∈ H, S 0 ∈ L 2 (Ω) 9 , and 0 ≤ γ ≤ θ ≤ γ + 1. Then ∃M > 0, so that ∀t ≥ 0, (i)

λ θ k |α k (t)| 2 ≤ M λ θ k |α 0 k | 2 + |λ k | γ |b k | 2 t 2(1-δ)[(θ-γ-ω)/(1-ω)] + (ii) λ γ k λ k |ρ * α k (t)| 2 (t) ≤ M λ θ k |α 0 k | 2 t δ(θ-γ-ω)/ω + λ γ k |b k | 2 Proof. (i) Assume 0 ≤ γ ≤ θ ≤ γ+ω. Multiplying (i) of Theorem 5.1 by λ θ k leads to λ θ k |α k (t)| 2 ≤ A λ θ k |α 0 k | 2 + λ θ-ω k |b k | 2 . Since min k≥1 λ k > 0 and θ -ω ≤ γ, one gets λ θ k |α k (t)| 2 ≤ M λ θ k |α 0 k | 2 + λ γ k |b k | 2 .
Assume now that 0 ≤ γ +ω ≤ θ ≤ γ +1. Use part (iii) of Theorem 5.1 with µ = 0 and τ

= θ -γ -ω 1 -ω ∈ [0, 1]. A simple calculation leads to τ + (1 -τ )ω = τ (1 -ω) + ω = (θ -γ -ω) + ω = θ -γ. Henceforth: |α k (t)| 2 ≤ M |α 0 k | 2 + |b k | 2 λ θ-γ k t 2(1-δ)(θ-γ-ω)/(1-ω) which leads to (i). (ii) Assume 0 ≤ γ ≤ θ ≤ γ + 1. Letting µ = γ -θ + 1 ∈ [0, 1] and τ = 1 in part (iv) of Lemma 5.1, gives µ(2 -δ) + 2(δ -1) = (γ -θ)(2 -δ) + δ = -δ(θ -γ -ω)/ω. Hence λ k |ρ * α k | 2 (t) ≤ M |α 0 k | 2 λ γ-θ k t -δ(θ-γ-ω)/ω + |b k | 2
which gives (ii).

As a consequence, we have: In both cases, for any t ≥ 0,

Corollary 5.1. Let 0 ≤ γ ≤ θ ≤ γ + 1, u 0 ∈ H θ , S 0 ∈ D γ . Then u ∈ C 0 (R * + ,
u(t) H θ ≤ M u 0 H θ + S 0 Dγ t (1-δ)[(θ-γ-ω)/(1-ω)] + (5.11) S(t) Dγ ≤ M t δ(θ-γ-ω)/(2ω) u 0 H θ + S 0 Dγ (5.12)
Proof. We first prove that u ∈ C 0 R * + , H γ+1 . From (d) of Lemma 4.1 and (ii) in Proposition 5.1, we reckon that, for any

N ≤ M and t ∈ [T 1 , T 2 ], where 0 < T 1 < T 2 , M k=N α k (t)w q 2 H γ+1 = M k=N |α k (t)| 2 λ γ+1 k ≤ M T +∞ k=N λ γ k |α 0 k | 2 T 2-δ 1 + λ γ k |b k | 2 T 2(1-δ) 1
(5.13) Since γ ≤ θ, we have that u 0 ∈ H θ ⊂ H γ . Also S 0 ∈ H γ . Hence Eq.(5.13) implies that sup

t∈[T 1 ,T 2 ] M k=N α k (t)w q H γ+1 -→ N →+∞ 0 (5.14)
Finally, as α k ∈ C 0 (R + ), from (5.14) above we deduce that u

= +∞ k=N α k ⊗ w q ∈ C 0 R * + , H γ+1 .
Next we proceed with the rest of the proof.

(ρ * α k ) (t) ∇w k + ∇ T w k 2 Dγ = M k=N |ρ * (t) α k | 2 (1 + λ γ k )λ k ≤ K +∞ k=N λ θ k |α 0 k | 2 t δ(θ-γ-ω)/ω + λ γ k |b k | 2 (5.15)
Since u 0 ∈ H θ , and

S 0 ∈ D γ , then +∞ k=1 (ρ * α k ) (t) ∇w k + ∇ T w k is uniformly convergent w.r.t. t on any compact subset [T 0 , T 1 ] ⊂ R * + , in D γ . Given that α k ∈ C 0 (R + )
, and that ρ ∈ L 1 loc (R + ) -and hence

(ρ * α k ) ∈ C 0 (R + ) -, we conclude that +∞ k=1 (ρ * α k ) ⊗ ∇w k + ∇ T w k ∈ C 0 ([T 0 , T 1 ], D γ ). As W 0 ∈ C 0 (R + ),
one gets that (see (5.2))

S = +∞ k=1 (ρ * α k ) ⊗ ∇w k + ∇ T w k + W 0 ⊗ S 0 ∈ C 0 (R + , D γ ) (5.16)
The inequality Eq.(5.12) results from eq.(5.15) by letting N = 1, M = +∞, and from the fact that

W 0 (t)S 0 Dγ ≤ W 0 ∞ S 0 Dγ for any t ∈ R + , since W 0 ∈ L ∞ (R + ) ∩ C 0 (R + ). In a similar way we prove that u ∈ C 0 (R + , H θ ).
The inequality eq.( 5.11) is a consequence of (i) in Lemma 5.2 and of the fact that u ∈ C 0 (R + , H θ ). Finally, as H θ ֒→ H and lim t→0 u(t) = u 0 in H (see Theorem 8.4 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]), lim t→0 u(t) = u 0 in H θ .

(b)

Whenever θ ≥ ω + γ, u 0 ∈ H γ+ω and S 0 ∈ D γ . The inequalities (i) and (ii) of Lemma 5.2, for θ ′ = ω + γ and γ ′ = γ, read

λ θ ′ k |α k (t)| 2 ≤ M λ θ ′ k |α 0 k | 2 + λ γ ′ k |b k | 2 λ γ ′ k λ k |ρ * α k | 2 (t) ≤ M λ θ ′ k |α 0 k | 2 + λ γ ′ k |b k | 2
The proof of (i) of Lemma 5.2 entails the uniform convergence (with respect to t on [0, +∞[) of

+∞ k=1 (ρ * α k ) (t) ∇w k + ∇ T w k in D γ . Therefore S ∈ C 0 (R + , D γ ). Moreover, as W 0 (0) = 1, we get (see (5.2)) S(0) = +∞ k=1 (ρ * α k ) (0) ∇w k + ∇ T w k W 0 (0)S 0 = S 0 .
Arguing as in (a) above one proves that: u ∈ C 0 (R + , H γ+ω ) and u(0) = u 0 .

In the case γ ≥ 0 and θ = γ + ω we get the continuity of (u, S) at t = 0: Theorem 5.1 (Existence Theorem). Let γ ≥ 0. Assume that u 0 ∈ H γ+ω and S 0 ∈ D γ . Then the system of Eqs.(2.1)-( 2.2) has at least one solution (u, S)

∈ C 0 (R + , H γ+ω ) ∩ C 0 (R * + , V ) ∩ L p loc (R + , V ) × C 0 (R + , D γ ), p ∈ [1, 2/(2 -δ)[. Moreover, ∃A > 0, such that ∀t ∈ R + , u(t) H γ+ω + S(t) Dγ ≤ A u 0 H γ+ω + S 0 Dγ
The statement (a) of Corollary 5.1 says that, for any 0 ≤ γ ≤ θ ≤ γ + ω, u 0 ∈ H θ , S 0 ∈ D γ , and S ∈ C 0 (R * + , H γ ). This does not ensure continuity at t = 0. Nevertheless, for 0 ≤ γ ≤ θ ≤ γ + ω, θ ≥ ω and still holding on the assumptions u 0 ∈ H θ and S 0 ∈ D γ , we have S 0 ∈ D θ-ω . Therefore (see Theorem 5.1) S ∈ C 0 (R + , D θ-ω ), and u ∈ C 0 (R + , D θ ).

From now on we shall focus on the case 0 ≤ γ ≤ θ ≤ ω < 1. Proceeding as previously we get:

Lemma 5.3. Assume that 0 ≤ γ ≤ θ ≤ ω ≤ 1, u 0 ∈ H θ and S 0 ∈ D γ .
Then ∃M > 0, such that for any t ≥ 0,

λ θ k |α k (t)| 2 ≤ M λ θ k α 0 k 2 + λ θ-ω k |b k | 2 (5.17)
and

λ θ-ω k (λ k |ρ * α k | 2 (t)) ≤ M λ θ k |α 0 k | 2 + λ θ-ω k |b k | 2 ) (5.18)
Proof. Eq.(5.17

-δ) + 2(δ -1) = [1 -δ/(2 -δ)](2 -δ) + 2δ -2 = Henceforth, λ k (|ρ * α k | 2 (t)) ≤ M λ ω k |α 0 k | 2 + |b k | 2 which ends the proof. Hence: Corollary 5.2. Assume 0 ≤ γ ≤ θ, u 0 ∈ H θ and S 0 ∈ D γ . (a) whenever θ ≥ ω, u ∈ C 0 (R + , H θ ), and S ∈ C 0 (R + , D θ-ω ). Moreover, ∃A > 0 s.t. ∀t ≥ 0: u(t) H θ + S(t) D θ-ω ≤ A u 0 H θ + S 0 D θ-ω (5.19) (b) whenever θ ≤ ω, u ∈ C 0 (R + , H θ ), and S ∈ C 0 (R + , ∆ θ-ω ). Moreover, ∃A > 0 s.t. ∀t ≥ 0, u(t) H θ + S(t) ∆ θ-ω ≤ A u 0 H θ + S 0 ∆ θ-ω (5.20) Proof. (a)
The proof is a direct consequence of the discussion preceding Lemma 5. 9 ֒→ ∆ θ-ω , we get by Eq.(5.21) and Eq.(5.2) that

(ρ * α k ) (t) ∇w k + ∇ T w k 2 ∆ θ-ω = 2 N k=M λ θ-ω k λ k |ρ * (t) α k | 2 ≤ M N k=M λ θ k |α 0 k | 2 + λ θ-ω k |b k | 2 (5.21) Since u 0 ∈ H θ , S 0 ∈ D γ ֒→ L 2 (Ω)
(S -W 0 ⊗ S 0 ) ∈ C 0 (R + , ∆ θ-ω ) (5.22)
Moreover, W 0 ∈ C 0 (R + ) and S 0 ∈ D γ ֒→ ∆ θ-ω . Therefore, by Eq. (5.22), S ∈ C 0 (R + , ∆ θ-ω ). Inequality Eq. (5.20) follows right away after invoking Eq.5.17, Eq.(5.2), Eq.(5.21) with M = 1 and N = +∞, and that W 0 ∈ L ∞ (R + ) and S 0 ∈ ∆ θ-ω .

Remark 5.1. Using part (b) in Corollary 5.2 and by a density argument one may prove that, for u 0 ∈ H and S 0 ∈ ∆ -ω , the system of equations (1.9) has a weak solution (u, S) ∈ C 0 (R + , H) × C 0 (R + , ∆ -ω ). Of course the integrals have to be replaced by inner product functionals.

Corollary 5.3. Assume 0 ≤ γ ≤ θ ≤ ω ≤ 1, u 0 ∈ H θ and S 0 ∈ D γ . Then u ∈ C 0 (R + , H θ ), and S ∈ C 0 (R + , H -1 (Ω) 9 ). Moreover u(0) = u 0 , S(0) = 0 .

Proof. Corollary 5.2 states that S ∈ C 0 (R + , ∆ θ-ω ). Therefore (see Section 4), the mapping of R + into 9 , is continuous; the Corollary statement follows right away.

H -1 (Ω) 9 defined by R + S → ∆ θ-ω i ֒→ ∆ -1 r → H -1 (Ω)
Hence, to the first existence and uniqueness theorem, we can add the following conclusion: S ∈ C 0 (R + , ∆ -ω ) ֒→ C 0 (R + , H -1 (Ω) 9 ) and S(0) = S 0 .

We now give a second existence and uniqueness Theorem in H γ+ω × D γ spaces.

Theorem 5.2 (Second Existence and Uniqueness Theorem). Let γ ≥ 0. Assume that u 0 ∈ H γ+ω , S 0 ∈ D γ . Then the boundary value problem given by the system of equations (2.1)-( 2.2) has a unique solution

(u, S) ∈ C 0 (R + , H γ+ω ) ∩ C 0 b (R * + , V ) ∩ L 1 loc (R + , V ) × C 0 (R + , D γ ) ∩ L ∞ (R + , D γ ) (5.23)
Moreover, there exists A > 0, independent of u, such that, for any t ≥ 0, u(t) H γ+ω + S(t) Dγ ≤ A u 0 H γ+ω + S 0 Dγ (5.24)

Lastly, u(0) = u 0 , S(0) = S 0 .

Proof. The solution uniqueness is a consequence of the following inclusions:

H γ+ω ֒→ H, D γ ֒→ L 2 (Ω) 9 , C 0 (R + , H γ+ω ) ∩ C 0 b (R * + , V ) ∩ L 1 loc (R + , V ) × [C 0 (R + , D γ ) ∩ L ∞ (R + , D γ )] ⊂ F ,
(R + , H γ+ω ) ∩ C 0 (R * + , V ) ∩ L 1 loc (R + , V ) × [C 0 (R + , D γ )
] follows from Theorem 5.1. In addition, the last estimate in Theorem 5.1 grants that S ∈ L ∞ (R + , D γ ). Next, that u(t) V -→ t→+∞ 0 was proved in the first existence and uniqueness Theorem presented above. Based on this fact, we infer that u ∈ C 0 b (R * + , V ), which ends the proof of solution existence. The estimate Eq.(5.24) is a consequence of Theorem 5.1.

6 The smoothness of solutions.

The following estimates will be used in proving the smoothness of solutions. They generalize those previously obtained in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF].

Proposition 6.1. ∃M > 0, s.t. ∀(τ, χ) ∈ [0, 1] 2 , ∀(x, t) ∈ R + , ∀µ ≥ λ 1 , one has: (a) lim A→+∞ +A -A T µ (x + iy)e (x+iy)t dy ≤ M µ χ/(2-δ) t χ (b) √ µ lim A→+∞ +A -A T µ w(x + iy)e (x+iy)t dy ≤ M µ [ω(1-τ )+τ ]/2 t τ (1-δ)
Proof. We only have to prove these estimates for χ = 0 and χ = 1, τ = 0 and τ = 1.

(a)

The case χ = 0 has already been addressed in Lemma 7.4 and Lemma 7.2 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]. We now prove the case χ = 1. Lemma 7.4 and inequality 70 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] give, for suitable κ > 0 and B > 0,

lim A→+∞ +A -A T µ (x + iy)e (x+iy)t dy ≤ K +∞ 0 T µ (ze iπ ) -T µ (ze -iπ ) e -zt dz + e -κtµ 1/(2-δ) ≤ K +∞ 0 u 1-δ + λ -(1-δ)/(2-δ) 1 (u 2-δ + B) 2
e -utµ 1/(2-δ) du + e -κtµ 1/(2-δ) (6.1)

It implies that:

lim A→+∞ +A -A T µ (x + iy)e (x+iy)t dy ≤ K +∞ 0 e -utµ 1/(2-δ) du + 1 tµ 1/(2-δ) ≤ A tµ 1/(2-δ) (6.2)
which gives the statement in (a) for χ = 1.

(b)

The case τ = 0 is addressed in Lemma 5.1. The case τ = 1: from (iii) in Lemma 7.5 in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] we get

√ µ lim A→+∞ +A -A T µ w(x + iy)e (x+iy)t dy ≤ K µt 1-δ + Ke -atµ 1/(2-δ) µ 1/2-δ ≤ M µt 1-δ (6.3)
as sup

tµ 1/(2-δ) ≥0 tµ 1/(2-δ) 1-δ e -atµ 1/(2-δ) ≤ M < +∞.
As a consequence we have the following extensions of estimates (iii) and (iv) of Theorem 5.1.

Corollary 6.1. Let u 0 ∈ H, S 0 ∈ L 2 (Ω) 9 . Then:

(a) ∃M > 0, ∀T ≥ 0, , s.t. ∀γ ∈ [0, 1 2 -δ ], ∀τ ∈ [0, 1], ∀t ∈ [0, T ] and ∀k ∈ N * , |α k (t)| 2 ≤ M |α 0 k | 2 λ 2γ k t 2γ(2-δ) + |b k | 2 λ τ +ω(1-τ ) k t 2τ (1-δ) (b) ∃M > 0, ∀T ≥ 0, s.t. ∀γ ∈ [0, 1], ∀τ ∈ [0, 1 2 -δ ], ∀t ∈ [0, T ] and ∀k ∈ N * , λ k |ρ * α k | 2 (t) ≤ M |α 0 k | 2 λ 2γ-1 k t 2[γ(2-δ)+δ-1] + |b k | 2 λ -(1-τ )(1-ω) k t -2(1-τ )(1-δ)
Proof. (a)

The statement in (a) follows from (5.1) and Proposition 6.1 (with γ = χ 2 -δ

).

(b)

The statement follows from (a) above by convolution.

We deduce from Corollary 6.1:

Proposition 6.2. Let γ ≥ 0, η > 0. Then: (a) Assume u 0 ∈ H 1+γ+ω and S 0 ∈ D 1+γ . Then u ∈ C 1 (R + , H γ ). (b) Assume u 0 ∈ H 3+γ-ω+η and S 0 ∈ D 3+γ . Then u ∈ C 2 (R * + , H γ ) ∩ W 2,p loc (R + , H γ ), p ∈ [1, 1/(1 -α)[.

Proof. (a)

Based on eq.(3.11a) and W 0 ∈ L ∞ (R + ), we infer that:

λ γ k |α ′ k (t)| 2 ≤ λ 1+γ k λ k |ρ * α k | 2 (t) + λ 1+γ k |b k | 2 W 0 2 ∞ (6.4)
Now, Lemma 5.2 with θ = γ + ω, leads to

λ 1+γ k λ k |ρ * α k | 2 (t) ≤ Aλ k λ γ+ω k |α 0 k | 2 + λ γ k |b k | 2 (6.5)
From Eqs.(6.4)-(6.5),

λ γ k |α ′ k (t)| 2 ≤ A λ 1+γ+ω k |α 0 k | 2 + λ 1+γ k |b k | 2 (6.6) Since u 0 ∈ H 1+γ+ω and S 0 ∈ D 1+γ , λ γ k |α ′ k (t)| 2 < +∞. Hence +∞ k=1 α ′ k ⊗ w k converges in C 0 (R + , H γ ).
Since by Lemma 5.2

+∞ k=1 α k ⊗ w k converges in C 0 (R + , H 1+γ ), it also converges in C 0 (R + , H γ ). Finally u ∈ C 1 (R + , H γ ). (b)
Observe that 1 + γ + ω ≤ 3 + γ -ω + η and 1 + γ ≤ 3 + γ. Consequently u 0 ∈ H 1+γ+ω , S 0 ∈ H 1+γ . Next, (a) above ensures that u ∈ C 1 (R + , H γ ). We now deduce several estimates for the second order derivatives. From eq.(3.11a) it follows that, for t > 0, α

′′ k (t) = -λ k (ρ * α ′ k ) (t) - √ λ k b k W ′ 0 (t) = λ 2 k (ρ * ρ * α k ) (t) + λ 3/2 k b k (ρ * W 0 ) (t) - √ λ k b k W ′ 0 (t) ∈ C 0 (R * + ) since α k ∈ C 0 (R + ), W 0 ∈ C 0 (R + ) ∩ C 1 (R * + ) and 0 ≤ ρ(t) ≤ kt -δ .
Let now ǫ > 0 be small enough. Part (a) in Corollary 6.1 with γ [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]), gives

= (1 -ǫ)/(2 -δ) ∈ 0, 1 2 -δ , τ = 1 and |W ′ 0 (t)| ≤ k t 1-α (see
λ γ/2 k |α ′′ k (t)| ≤ M T λ 2+γ/2 k t 1-2δ * |α 0 k | λ (1-ǫ)/(2-δ) k t 1-ǫ + |b k | λ 1/2 k t 1-δ + M T λ (1+γ)/2 k λ k t δ-1 + 1 t 1-α |b k | ≤ M T λ (3+γ)/2 k t δ-1 + λ (1+γ)/2 k t 1-α |b k | + M T λ 2-(1-ǫ)/(2-δ)+γ/2 k t 2δ-ǫ-1 |α 0 k | (6.7)
Observe that, for ǫ > 0 small enough, 2(3 + γ)/2 ≤ 3 + γ and 2(γ + 1)/2 ≤ 3 + γ. Also, for ǫ > 0 small enough, 2 [2 + γ/2 -(1 -ǫ)/(2 -δ)] ≤ 4 + γ -2/(2 -δ) + η.

Hence, by (6.7), and since u 0 ∈ H 3+γ-ω+η = H 4+γ-2/(2-δ)+η and S 0 ∈ D 3+γ , we get u ∈ C 2 (R * + , H γ ).

Proceeding as before (see Proposition 6.2) one obtains the following smoothness properties: Proof. (a)

We limit the proof to the case 0 < η < ω. Denote ω = ω -η; hence 0 < ω < ω < 1. Since u 0 ∈ H 2+γ-ω , S 0 ∈ D 2+γ-ω-ω , based on Theorem 5.1, we have that S ∈ C 0 (R + , D 2+γ-ω-ω ). Next, as 0 < ω < ω < 1 entails γ < 2 + γω -ω, one gets S ∈ C 0 (R + , D γ ).

Next we obtain an estimate for S ′ (t). Observe first that W 0 ∈ L ∞ (R + ) and that α ′ k (t) = -λ k (ρ * α k ) (t)-√ λ k b k W 0 (t). Therefore: Moreover, from [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF] we observe that W ′ 0 (t)S 0 Dγ ≤ K/t 1-α . It implies that W ′ 0 ⊗ S 0 ∈ L q loc (R + , D γ ) for any q ∈ [1, 1/(1 -α)[. Therefore

λ γ/2 k λ k |ρ * α ′ k | ≤ Aλ
λ k |ρ * α ′ k | (t) ≤ M T λ 1+γ/2 k λ k 1 t 2δ-1 * |α 0 k | λ 1/(2-δ) k t (2-δ)/(2-δ) + λ k 1 t 2δ-1 * |b k | √ λ k t 1-δ + |b k | t δ-1 ≤ M T λ 1+γ

  µ w)(iy)|dy. Combining these results achieves the proof.

  H γ+1 ). In addition, (a) whenever 0 ≤ γ ≤ θ ≤ γ + ω, u ∈ C 0 (R + , H θ ), and S ∈ C 0 (R * + , D γ ); moreover, u(0) = u 0 . (b) whenever 0 ≤ γ + ω ≤ θ, u ∈ C 0 (R + , H γ+ω ), and S ∈ C 0 (R + , D γ ); moreover, u(0) = u 0 and S(0) = S 0 .

  that S ∈ C 0 (R * + , D γ ). For any N ≤ M, by (c) of Lemma 4.1 and by (ii) of Lemma 5.2, M k=N

  ) follows from part (i) in Proposition 5.1. Next, we use part (iv) in Lemma 5.1 with µ = 1 -ω and τ = 1. It gives µ -1 = -ω and µ(2

  u ∈ C 0 (R + , H θ ) is a consequence of part (a)of Corollary 5.1. Next, Lemma 5.3 and part (e) of Lemma 4.1 imply that, for any M ≤ N, N k=M

  and of Corollary 3.1. The existence of a solution (u, S) ∈ C 0

Proposition 6 . 3 .

 63 Let γ ≥ 0, η > 0. (a) Assume u 0 ∈ H 2+γ-ω+η , S 0 ∈ D 2+γ . Then S ∈ C 1 (R * + , D γ ) ∩ W 1,p loc (R + , D γ ), for any p ∈ [1, 1/(1 -α)[. (b) Assume u 0 ∈ H 4+γ-ω+η , S 0 ∈ D 4+γ . Then S ∈ C 2 (R * + , D γ ).

(γ+1)/ 2 k

 2 λ k (ρ * ρ * |α k |) + λ k (ρ * |b k |) (t) Now, part (a) of Corollary 6.1, with δ = 2ω/(1 + ω) < 2ω/(1 + ω) = δ, γ = 1/(2 -δ), δ ∈]0, δ[, and τ = 1, gives |α k (t)| ≤ M T |α 0 k | λ 1/(2-δ) k t (2-δ)/(2-δ) + |b k | √ λ k t 1-δ However, |ρ * ρ| (t) ≤ Kt 1-2δ . Also, since 0 < δ < δ, then (2 -δ)/(2 -δ) ∈ [0, 1[. One infers that λ γ/2 k

t 2 - 2 (

 22 2δ-(2-δ)/(2-δ) |α 0 k | + t 1-δ |b k | (6.8) Let a = 2 -2δ -(2 -δ)/(2 -δ) = 1 -2δ + (δ -δ)/(2 -δ), and b = 1 -δ > 0. Then:λ γ/2 k λ k (ρ * α ′ k ) t) ≤ M T λ 2+γ-ω k |α 0 k | 2 t 2a + λ 2+γ k t 2b |b k | 2 (6.9)Recall that -as stated in (c) of Lemma 4.1 -that ∇w k + ∇ T w k k∈N * is an orthogonal sequence of functions that belongs to D γ , and∇w k + ∇ T w k 2 Dγ = (1+λ γ k )λ k .Consequently, using the estimate given above and that u 0 ∈ H 2+γ-ω , S 0 ∈ D 2+γ , leads to the fact that+∞ k=1 (ρ * α ′ k ) ⊗ ∇w k + ∇ T w k converges in C 0 (R * + , D γ ). Next, from Eq in [15], W ′ 0 ⊗ S 0 ∈ C 0 (R * + , D γ ). Therefore +∞ k=1 (ρ * α ′ k ) ⊗ ∇w k + ∇ T w k + W ′ 0 ⊗ S 0 ∈ C 0 (R * + , D γ ). Hence S ∈ C 1 (R * + , D γ ).Whenever a ≥ 0, by Eq.(6.9),+∞ k=1 (ρ * α ′ k ) ⊗ ∇w k + ∇ T w k + W ′ 0 ⊗ S 0 belongs to C 0 (R + , D γ ), thus belongs to L p loc (R + , D γ ) for any 1 ≤ p < +∞. Now, whenever a < 0, -a-(1-α) = 2δ-2-(δ-δ)/(2-δ)-1+α = [-3-(δ-δ)/(2-δ)+(2δ+α)] < 0.We conclude that +∞ k=1 (ρ * α ′ k ) ⊗ ∇w k + ∇ T w k converges in L q loc (R + , D γ ), for any q ∈ [1, 1/(1 -α)].

, for any q ∈ [1, 1/(1 -α)[ and irrespective of whether a is positive or negative. Eventually S ∈ W 1,q loc (R + , D γ ) for any q ∈ [1, 1/(1 -α)[.

(b)

The proof is omitted.

From Proposition 6.2 we can infer the existence of smooth solutions to eqs.(1.9). Assume that u 0 ∈ H 5+ω and S 0 ∈ D 5 ∩ C 1 (Ω) 9 . Then, the solution (u, S) the existence of which is granted by Theorem 3.1 of Section 3, complies with the statement (a) of Proposition 6.2, that is u

). All the precedent arguments eventually lead to the conclusion that (u, S)

Final comments

Fractional calculus has a long history that parallels the classical analysis [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF][START_REF] Podlubny | Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications[END_REF][START_REF] Samko | Fractional Integrals and Derivatives: Theory and Applications[END_REF]. It has long been used in modeling natural phenomena: for a quick glimpse see for example [START_REF] Adolfsson | Fractional derivative viscoelasticity at large deformations[END_REF][START_REF] Adolfsson | Nonlinear fractional order viscoelasticity at large strains[END_REF][START_REF] Adolfsson | On the fractional order model of viscoelasticity[END_REF][START_REF] Agarwal | A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions[END_REF][START_REF] Agrawall | Solution for a fractional diffusion-wave equation defined in a bounded domain[END_REF][START_REF] Baleanu | Fractional Hamiltonian analysis of higher order derivatives systems[END_REF][START_REF] Carpinteri | Fractals and fractional calculus in continuum mechanics[END_REF][START_REF] Douglas | Some applications of fractional calculus to polymer science[END_REF][START_REF] Douglas | Polymer science applications of path-integration, integral equations, and fractional calculus[END_REF][START_REF] Drozdov | Fractional differential models in finite viscoelasticity[END_REF][START_REF] Guyomar | Time fractional derivatives for voltage creep in ferroelectric materials: theory and experiment[END_REF][START_REF] Hanyga | Fractional-order relaxation laws in non-linear viscoelasticity[END_REF][START_REF] Kneller | Fractional Brownian dynamics in proteins[END_REF][START_REF] Koeller | A theory relating creep and relaxation for linear materials with memory[END_REF][START_REF] Lion | On the thermodynamics of fractional damping elements[END_REF][START_REF] Logvinova | A fractional equation for anomalous diffusion in a randomly heterogeneous porous medium[END_REF][START_REF] Makris | Three-dimensional constitutive viscoelastic laws with fractional order time derivatives[END_REF][START_REF] Özdemir | Fractional diffusion-wave problem in cylindrical coordinates[END_REF][START_REF] Podlubny | Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications[END_REF][START_REF] Robles-Vasquez | A composition rule to predict the linear viscoelastic properties of polybutadienes with varying microstructure[END_REF][START_REF] Seredyńska | Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum[END_REF][START_REF] Tan | A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates[END_REF][START_REF] Xu | Analysis of nonlinear fractional partial differential equations with the homotopy analysis method[END_REF][START_REF] Yang | Start-up flow of a viscoelastic fluid in a pipe with a fractional Maxwell's model[END_REF][START_REF] Yin | Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model[END_REF], and references cited therein. In particular, fractional derivative CEs have been found to accurately predict stress relaxation of viscoelastic fluids in the glass transition and glassy (high frequency) states.

The results presented here enrich and complement the linear stability analysis within the framework of variational/weak solutions initiated in [START_REF] Heibig | On the rest state stability of an objective fractional derivative viscoelastic fluid model[END_REF]. We have proved results regarding existence, uniqueness, smoothness and continuity at t = 0 of the solution to the initial boundary value problem stated in Section 1. Moreover, this work is related to that of Shaw, Whiteman and co-workers on the well posedness, existence and uniqueness of weak solutions for similar in nature hereditary -type integral models (see for example [START_REF] Karamanou | Computational modelling of thermoforming processes in the case of finite viscoelastic materials[END_REF], [START_REF] Shaw | A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems[END_REF], [START_REF] Shaw | Adaptive space-time finite element solution for Volterra equations arising in viscoelasticity problems[END_REF], [START_REF] Shaw | Applications and numerical analysis of partial differential Volterra equations: A brief survey[END_REF]), as well as to that reported in [START_REF] Palade | A new constitutive equation that models extensional flow strain hardening based on evolving natural configurations: stability analysis[END_REF], [START_REF] Palade | An integral constitutive law for viscoelastic fluids based on the concept of evolving natural configurations: stability analysis[END_REF], [START_REF] Voyiatzis | On Hadamard stability and dissipative stability of the molecular stress function model of non-linear viscoelasticity[END_REF].

The matter of the stability of the original nonlinear CE is an open question on which future work shall focus.

Aknowledgements

The authors are grateful to Emeritus Associate Professor Michel Charnay, Pôle de Mathématiques, INSA-Lyon, for kind support and encouragements.