Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping

Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.


Introduction
There are several mathematical models representing physical damping. The most often encountered type of damping in vibration studies are linear viscous damping and Kelvin-Voigt damping which are special cases of proportional damping. Viscous damping usually models external friction forces such as air resistance acting on the vibrating structures and is thus called "external damping", while Kelvin-Voigt damping originate from the internal friction of the material of the vibrating structures and thus called "internal damping" or "material damping".
The study of the stabilization problem for coupled systems has attracted a lot of attention in recent years e.g. [AN10, AV09, Duy07, Fat11, Has15b, LZ99, RZZ05, Teb12, ZZ07, ZZ06,ZZ15]. The systems discussed in those paper involve thermoelastic systems, fluid-structure interaction systems, and coupled wave-wave, plate-plate, or plate-wave equations. But in the case of platewave and for the multi-dimensional space (of interest in this paper), and as far as we know, the only models which has been treated in this subject are the model of coupled Euler-Bernoulli and wave equations with indirect damping mechanisms (see [Teb12]) and the model arising in the control of noise, coupling the damped wave equation with a damped Kirchhoff thin plate equation (see [AN10]). The system that we are going to discuss in the present paper is coupling the transversal vibration of the Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subdomain with the elastic wave equation.
A relevant question raised about the transmission problems and problems with locally distributed damping, is the asymptotic behavior of the solutions. Does the solution goes to zero uniformly? If this is the case, what is the rate of decay?
In [LL98] and recently in [Has15c] for the transmission problem case, longitudinal and transversal vibrations of a clamped elastic beam was studied as problems with locally distributed damping. It was shown, for the one-dimensional case, that when viscoelastic damping is distributed only on a subinterval in the interior of the domain, the exponential stability holds for the transversal but not for the longitudinal motion. Besides, an optimality result was shown for longitudinal case.
This vibrating system is assumed to coupling the transversal and longitudinal motions (with dissipation on the plate affects an arbitrarily small open subset of its interior) through the transmission conditions as given in the third forth and fifth line of (1.1): The first, is called the continuity condition, the second, is described by the fact that the slope of the beam is null and the third, says that the transverse force caused by the tension is equal to the transverse force due to shear. This problem was studied in [Has15a] for one-dimensional case. It was proved that the energy of the solution is decreasing with a polynomial rate for the two cases where the damping arising from the transversal motion and the damping arising from the longitudinal motion.
The energy of a solution (u 1 , u 2 ) of the system (1.1) at the time t ≥ 0 is given by By means of the classical energy method, we show that Therefore, the energy is a non-increasing function of the time variable t and our system (1.1) is dissipative. We define the Hilbert space H = X × H where H = L 2 (Ω 1 ) × L 2 (Ω 2 ) and where H 1 Γ (Ω 2 ) is the space of elements in H 1 (Ω 2 ) whose trace is zero on the boundary Γ. The space H is equipped with the norm whose domain is given by Our main result is the following Theorem 1.1 For any k ∈ N there exists C > 0 such that for any initial data (u 0 We should mention here that the subject of stabilization of transmission problems with localized Kelvin-Voigt dissipation is perhaps not intensively studied but is not new in fact, in [ARSV11] the authors consider the transmission problem of a material composed by three components, one of them is a Kelvin-Voigt viscoelastic material, the second is an elastic material (no dissipation) and the third is an elastic material inserted with a frictional damping mechanism where they show different types of decay rate of energy depends on which component is in the middle, and in [RBA11] the authors consider a transmission problem for the longitudinal displacement of a Euler-Bernoulli beam, where one small part of the beam is made of a viscoelastic material with Kelvin-Voigt constitutive relation in which they show that the semigroup associated to the system is exponentially stable.

The method of analysis
Besides the fact that the Kelvin-Voigt damping is unbounded in the energy space and the fact that the resolvent of the system operator is not compact, the main difficulty of our problem is none other than the different speeds of propagation due to the coupling between the wave equation and the plate equation. The method that we consider here consist to the use of the Burq's result [Bur98] (see also [BD08]) which links, for dissipative operators, logarithmic decay to resolvent estimates with exponential loss. The main idea, as introduced by Lebeau [Leb96] is to use the what's called, Carleman estimates (see also [Duy07] for the case of non linear damping and [ET12] for the case of hyperbolic systems). Unlike to the works of [Bel03,Duy07,Fat11], here Carleman estimate does not seem to be enough, that is why we have combined it with some contradiction arguments to establish the kind of resolvent estimate cited above. Moreover, to deal with the high order of the plate equation, Carleman estimate (Theorem 3.2) is established for system of second order (3.4) which is derived from the resolvent problem (4.8) by decomposing the plate equation into two second order operators (4.11). The outline of this paper is as follows: In section 2 we show that the corresponding model are well posed, in section 3 we give the Carleman estimates and we construct a suitable weight functions that satisfy the Hörmander's assumption. In section 4 we prove the resolvent estimate which provides the logarithmic decay result given by Theorem 1.1.

Existence and uniqueness
In this section and through the semigroup theory we will show that the problem (1.1) is Wellposed. The system (1.1) can be written in the abstract form as a Cauchy problem as follows where we recall that the operator A is defined by with domain D(A) = (u 1 , u 2 , v 1 , v 2 ) ∈ H : (v 1 , v 2 , ∆(∆u 1 + a∆v 1 ), ∆u 2 ) ∈ H, ∂ ν ∆u 1 |S + ∂ ν u 2 |S = 0 .
In the space Y = L 2 (Ω 1 ) × H 1 Γ (Ω 2 ) we define the operator G as follows We define a norm in the space X by The graph norm of G is given by Lemma 2.1 (X, . X ) is a Hilbert space with a norm equivalent to the graph norm of G.

Proof :
Let's note first, by setting u = 1 Ω 1 u 1 + 1 Ω 2 u 2 , that the continuity transmission condition u 1 |S = u 2 |S allows to look at u as an element of H 1 (Ω). Hence by Green's formula and Poincaré inequality there exists C > 0 such that for every (u 1 , u 2 ) ∈ X we have In particular G is a strictly positive operator on Y . Besides, since for every (u 1 , u 2 ) ∈ X we have then the equivalence of the two norms holds.
To finish the proof we have only to prove that G is a closed operator on Y . Let (u 1,n , u 2,n ) ∈ Y , (u 1 , u 2 ) and (f 1 , f 2 ) such that then we obtain also −∆u 1 = f 1 ∈ L 2 (Ω 1 ). In the other hand, while then u 1,n is a Cauchy sequence in H 1 (Ω 1 ), that converge to u 1 in H 1 (Ω 1 ) where we argue this fact as follows, For the transmission conditions we have where we have used here [TW09, Theorem 13.7.6]. This show now that G is a closed operator and this conclude the proof.
Theorem 2.1 The operator A is m-dissipative and especially it generates a strongly semigroup of contraction in H.

.4]) we have only to prove that
This amounts to finding (u 1 , u 2 , v 1 , v 2 ) ∈ D(A) that satisfies the following system From Lemma 2.1 and the Riesz representation theorem, we can find a unique (u 1 , u 2 ) ∈ X such that for all (ϕ 1 , ϕ 2 ) ∈ X we have Then by Green's formula we obtain Then from (2.1) and (2.2) we obtain and this show the following equality And this give end to our proof.
One consequence of this last result is that if we assume that (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ D(A), there exists a unique solution of (1.1) which can be expressed by means of a semigroup on H as follows where e tA is the semigroup of the operator A. And we have the following regularity of the solution  And if (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ H, the function (u 1 (t), u 2 (t)) given by (2.3) is the mild solution of (1.1).

Carleman estimate near the surface
This section is devoted to establish the Carleman estimate. We set the operator with h is a small semi-classical parameter and where is a second order polynomial in ξ ′ with coefficients in R with principal symbol We consider the following transmission problem We let ϕ a weight function and we define in both side of S the conjugate operator We suppose that the weight function ϕ is in 3. The sub-ellipticity condition: Follows to Le Rousseau and Robbiano result [RR10] we can prove by using the same argument and the exactly the same steps we can prove the following result Theorem 3.1 Let K be a compact subset of V and ϕ a weight function satisfying the above assumption, then there exist C > 0 and h 0 > 0 such that for all 0 < h ≤ h 0 , w and f satisfying (3.1) where w |R n − ∈ C ∞ c (K g ) and w |R n The proof of Carleman estimate is the same for both, in this paper and in the Le Rousseau and Robbiano paper [RR10] even that the transmission conditions are different. In fact, while in [RR10] they are depending on some diffusion coefficients where an additional assumption ( [RR10, (2.2)]), on the jump at the interface of the weight functions, is assumed in addition to that given above, here the transmission conditions depend on the pseudodifferential parameter h where, for h small enough this scaling coefficient allows us to ensure the assumption of Le Rousseau and Robbiano [RR10,(2.2)] which became in our case Thus we may notice how the scaling coefficient h is playing the same role as the diffusion coefficients in the Rousseau and Robbiano paper [RR10]. Noting also another version of this analysis appeared more recently in [RLR13].
The purpose of the rest of this section is to provide a global Carleman estimate for a transmission problem with three entries governed by a three elliptic operators. Besides, we will try to construct a suitable weight functions that will be needed in the next section.
6. The sub-ellipticity condition: Under these assumption the global Carleman estimate is given by the following Theorem 3.2 Let ϕ 1 and ϕ 2 the two weight functions as described above, then there exist C > 0 and h 0 > 0 such that f ′′ 1 and f 2 satisfying to the system (3.4).

Proof :
The proof follows easily from Theorem 3.1 in fact, system (3.4) can be shown as a combination of two transmission problems, the first is by consider only the equation with entries y ′ 1 and y 2 only, where in the first transmission equation the term θ ′ = hy ′′ 1 should be seen as an error in the continuity of the trace of y ′ 1 and y 2 , namely we have and the second problem is with the entries y ′′ 1 and y 2 only as follow We apply Theorem 3.1 for each of these systems by taking into account [LR97, Proposition 1] and [LR95, Proposition 2] then we get and h e ϕ 1 /h y ′′ 1 2 The result follows easily now by combing the two last estimates where the terms θ ′ = hy ′′ 1 are absorbed to the left hand side for h > 0 small enough.
We return now to our geometric baseline as described in the introduction of this paper and we denote by Ω 1 = Ω 1 \B r where B r is an open ball of Ω 1 with radius r > 0 such that B r ⊂ Ω 1 . We try to find four phases ϕ 1,1 , ϕ 1,2 , ϕ 2,1 and ϕ 2,2 satisfying the Hörmander's condition except in a finite number of balls where ϕ 1,1 or ϕ 1,2 (resp. ϕ 2,1 or ϕ 2,2 ) do not satisfy this condition the other does and is strictly greater.
And that concludes our construction of the weight functions.

Resolvent Estimate
This section is devoted to prove a resolvent estimate, precisely we will show that for some constant C > 0 we have for every µ ∈ R large enough in absolute value, Which by Burq's result follow the kind of energy decay rate given in Theorem 1.1. We suppose that the resolvent estimate (4.1) is false. Then there exist two sequences K m > 0 and µ m ∈ R and two families (u 1,m , u 2,m , v 1,m , v 2,m ) ∈ D(A) and (f 1,m , f 2,m , g 1,m , g 2,m ) ∈ H, m = 1, 2, . . . such that This implies that From (4.2) and (4.3), we get Then from the first equation of (4.4) and (4.5), we obtain (4.6) |µ m | 2 e Km|µm| which mean that v 1,m H 2 (ω) is bounded. We multiply the third equation of (4.4) by µ −1 m ψ.v 1,m where ψ ∈ C ∞ (Ω 1 ) and supp(ψ) ⊂ ω then from (4.5) and (4.6) we obtain |u 1,m | 2 dx −→ 0.
By setting (4.9) then (4.8) can be recast as follows (4.10) We denote by (4.11) it follows from (4.10) that z ′ 1 , z ′′ 1 and z 2 are solution of the following transmission problem (4.12) where θ = −z ′ 1 + 2z ′′ 1 . We set B 5r a ball of radius 5r > 0 such that a(x) > 0 in B 5r ⊂ ω. We set as the previous section Ω 1 = Ω 1 \B r . The most important ingredient of the proof of the resolvent estimate (4.1) is the following lemma which is essentially the result of the Carleman estimate.
Lemma 4.1 There exist a constant C > 0 such that for any (u 1 , u 2 , v 1 , v 2 ) ∈ D(A) solution of (4.8) the following estimate holds for all |µ| > 0 large enough.
Since v 1 = 0 in supp(a) and µ = 0 the top line of (4.28) yields u 1 = 0 in supp(a). The third line of (4.28) combined with the first one could be written as ∆z + µz = 0 in Ω 1 and z = 0 in supp(a), where we denoted by z = ∆u 1 − µu 1 . Since ω ⊂ supp(a) then by Calderón's theorem [RL09,Theorem 4.2] for elliptic operators we find that z = 0, this mean that ∆u 1 − µu 1 = 0 which imply for the same argument as previously that u 1 = 0 in Ω 1 . Reporting that in the first line of (4.28), we derive v 1 = 0 in Ω 1 . The second and fourth line of (4.28) lead to ∆u 2 + µ 2 u 2 = 0 in Ω 2 with the boundary conditions u 2 = 0 on ∂Ω 2 and ∂ ν u 2 = 0 on S.
By standard theory in linear elliptic equations u 2 = 0 in Ω 2 . Using the second line of (4.28), we get v 2 = 0 in Ω 2 ; hence U = 0. Therefore, A has no purely eigenvalue.