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Logarithmic stabilization of an acoustic system with a damping term of Brinkman type

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Without imposing any hypotheses on the structural properties of the damping term, we identify logarithmic decay of solutions with growing time. Logarithmic decay rate is shown by using a frequency domain method and combines a contradiction argument with the multiplier technique and a new Carleman estimate to carry out a special analysis for the resolvent.


Introduction
We consider the following system of equations: where Ω is a bounded domain in R d , d ≥ 2, with a smooth boundary Γ, div = ∇· is the divergence operator and b ∈ L ∞ (Ω), with b ≥ 0 on Ω and such that Here ω = ∅ stands for the open subset of Ω on which the feedback is active. As usual n denotes the unit outward normal vector along Γ.
The system of equations (1.1) is a linearization of the acoustic equation governing the propagation of acoustic waves in a compressible medium, see Lighthill [21,22,23], where b u represents a damping term of Brinkman type. This kind of damping arises also in the process of homogenization (see Allaire [1]), and is frequently used as a suitable penalization in fluid mechanics models, see Angot, Bruneau, and Fabrie [4]. Our main goal is to prove the logarithmic decay of solutions of (1.1) with growing time.
Accordingly, the problem (1.1) can be recasted in an abstract form: where Z = (u, r), or, equivalently, It can be shown (see [2]) that for any initial data (u 0 , r 0 ) ∈ D(A) the problem (1.1) admits a unique solution (u, r) ∈ C([0, ∞); D(A)) ∩ C 1 ([0, ∞); H). Moreover, the solution (u, r) satisfies, the energy identity Using (1.5) and a standard density argument, we can extend the solution operator for data (u 0 , r 0 ) ∈ H. Consequently, we associate with the problem (1.1) (or to the abstract Cauchy problems (1.3) or (1.4)) a semi-group that is globally bounded in H.
As the energy E is nonincreasing along trajectories, we want to determine the set of initial data (u 0 , r 0 ) for which (1.7) E(t) → 0 as t → ∞.
Such a question is of course intimately related to the structural properties of the function b, notably to the geometry of the set ω on which the damping is effective. In fact, when the damping term is globally distributed Ammari, Feireisl and Nicaise [2] showed an exponential decay rate of the energy by the means of an observability inequality associated with the conservative problem of (1.1). Besides, it is also shown that if the damping coefficient is not uniformly positive definite (i.e inf x∈Ω b(x) = 0) then the system (1.1) is not exponentially stable. In this paper we consider a damping which is locally distributed over the domain Ω without any geometrical control condition in particular this including the case when the damping coefficient is not uniformly positive defined. So we expect to prove a weaker decay rate then given in [2]. More precisely, we prove a logarithm decay rate of the energy. Our approach is based in the frequency domain method which consist to prove an exponential loss on the resolvent estimate [7,6,10] where the main tool for establishing a such decay is the Carleman estimate.
The theory of Carleman estimates for scalar equations is rather well developed by now. We refer to Hörmander [14] and Lebeau and Robbiano [17,18,19] for the second-order elliptic and hyperbolic PDE's and to Isakov [15] second-order parabolic and Schrödinger operators. However, it turned out that Carleman estimates for systems in more than two variables are difficult to obtain and still somehow very limited: The first results to systems go back to then Carleman's original work [9] which is written for a system in two independent variables, and we refer to Calderón [8] and Kreiss [16] for more relevant systems. Recently, Eller and Toundykov [11] have established a Carleman estimate for some first-order elliptic systems. This estimate is extended to elliptic boundary value problems provided the boundary condition satisfies a Lopatinskii-type requirement. In this paper we provide a Carleman estimate for a system of first-order which does not fit into the same framework as that of Eller and Toundykov [11]. Unlike their approach, our method is based into the Hörmander approach which is essentially based on the sub-ellipticity condition and the Gårding inequality in order to control the non-elliptical regions.
The paper is organized as follows. Section 2 summarizes some well known facts concerning the acoustic system (1.1). In section 3, we establish a new Carleman estimate needed for the stabilization problem of the system (1.1). In Section 4, we prove the logarithmic stability for the system (1.1).

Preliminaries
We start with a simple observation that the problem (1.1) can be viewed as a bounded (in H) perturbation of the conservative system which can be recast as the standard wave equation Consequently, the basic existence theory for (1.1) derives from that of (2.8). Hence A d generates a C 0 -semigroup (S(t)) t≥0 in H that is even of contraction because A d is dissipative (see (1.5)).
The first main difficulty is that the operator A d possesses a non-trivial (and large) kernel that is left invariant by the evolution. Indeed if (u, r) belongs to ker A d , then it is solution of the "stationary" problem (2.9) ∇r + bu = 0, div u = 0, in Ω.
By an integration by parts, using the fact that u is solenoidal and the boundary condition u · n = 0 on Γ, we get Ω ∇r · u dx = 0, and therefore we obtain In other words, we have u = 0 on supp b, and coming back to (2.9), we find ∇r = 0.
Accordingly, we have shown that For shortness set E = ker A d and introduce also its orthogonal complement H 0 in H.
It is easy to check that in particular, the semi-group associated with (1.1) leaves both E and H 0 invariant. Consequently, the decay property (1.7) may only hold for initial data emenating from the set H 0 .
The following observation can be shown by a simple density argument: Therefore the energy is non-increasing and (1.5) holds for all initial datum in H.
As already shown in the above, the strong stability result (1.7) may hold only if we take the initial data There are several ways how to show (1.7), here we make use of the following result due to Arendt and Batty [5]: Theorem 2.1. Let (T (t)) t≥0 be a bounded C 0 -semigroup on a reflexive Banach space X. Denote by A the generator of (T (t)) and by σ(A) the spectrum of A. If σ(A) ∩ iR is countable and no eigenvalue of A lies on the imaginary axis, then lim t→+∞ T (t)x = 0 for all x ∈ X.
In view of this theorem we need to identify the spectrum of A d lying on the imaginary axis, and we have according to [2]: • Suppose that |ω| > 0. If λ is a non-zero real number, then iλ is not an eigenvalue of A d .
• Suppose that |ω| > 0. If λ is a non-zero real number, then iλ belongs to the resolvent Now, Theorem 2.1 leads to ). Let (u, r) be the unique semi-group solution of the problem (1.1) emanating from the initial data (u 0 , r 0 ) ∈ H. Let P E be the orthogonal projection onto the space E = ker[A d ] in H, and let (w, s) = P E (u 0 , r 0 ).
We now state the main result of this article. We begin by a proposition on an estimate of the resolvent. Proposition 2.1. There exist C > 0 such that for every |µ| ≥ 1, and (f, or equivalently We recall the following result. Theorem 2.2. Let B a generator of a C 0 -semigroup (T (t)) t≥0 on H, a Hilbert space. We assume Then there exist C 1 > 0 such that for all u ∈ D(B) we have , ∀ t ≥ 0.
One has also, for every k ≥ 1 there exists A weak version of this theorem was first proven by Lebeau [17], next Burq [6] gives the precise statement. We also refer to Batty and Duyckaerts [7] for some generalizations.
On H 0 = ker[A d ] ⊥ , as seen above A d + iµ is invertible on H 0 , in fact A d is invertible on H 0 and A d + iµ is invertible on H for µ = 0. The semigroup is bounded as the norm on H is non-increasing by (1.5). With Proposition 2.1, we can apply Theorem 2.2. We then obtain. Theorem 2.3. Let (u, r) be the unique semi-group solution of the problem (1.1) emanating from the initial data (u 0 , r 0 ) ∈ D(A). Let P E be the orthogonal projection onto the space E = ker[A d ] in H, and let (w, s) = P E (u 0 , r 0 ).
Proposition 2.1 is obtained from Carleman estimates. We need two kinds of such estimates, first an estimate far away the boundary, second an estimate up to the boundary. Both estimates are proven in the next section.

Carleman estimates
Let Ω be an open bounded subset of R d . Let (u, r) be a solution of the resolvent problem (A 0 + iµ)(u, r) = (f, g) ∈ (L 2 (Ω)) d × L 2 m (Ω), that is Here we moreover assume that (u, r) are supported in K ⊂ Ω where K is a compact set. Taking the divergence of the first line and using that div u = iµr − g, we obtain We have to give a Carleman estimate for the solution of this type of equation. This is done in Section 3.2. to do that we need some tools on pseudo-differential operators we recall below.
3.1. Pseudo-differential operators. We start this section with some useful notations. If α = (α 1 , . . . , α n ) ∈ N n is a multi-index, we introduce the following notation: is the set of functions of C ∞ class with rapid decay rate. Its dual, S ′ (R n ) is the set of temperate distributions. If u ∈ S (R n ) its Fourier transform denoted byû is defined byû(ξ) = Definition 3.1. Let a( . , . , τ ) ∈ C ∞ (R n × R n ) where τ ≥ 1 is a large parameter, such that for every muti-index α, β ∈ N n we have where we denoted by ξ, τ = (|ξ| 2 + τ 2 ) 1 2 . In this case we say that a is a symbol of order m and we write a ∈ S m τ . We call principal symbol of a ∈ S m τ the equivalence class of a in S m τ /S m−1 τ . We also define S −∞ τ = r∈R S r τ and S +∞ τ = r∈R S r τ .
Definition 3.2. We define the pseudo-differential operator of order m by where a ∈ S m τ . The set of the pseudo-differential operator of order m is denoted by Ψ m τ . If A ∈ Ψ m τ , we denote by σ p (A) his principal symbol. Remarks 3.1. Let s ∈ R for u ∈ S ′ (R n ) we set the following norm u τ,s = Λ s τ u 0 with Λ s τ := Op( ξ, τ s ). Hence we can define the corresponding space We then write a ∼ j a j . The symbol a is unique up to S −∞ τ in the sens that the difference of two symbols is in S −M τ for all M ∈ N. Hence, we identify a 0 with the principal symbol of a.
which admits the following asymptotic expansion which admits the following asymptotic expansion In particular we have σ p (Op(a) * ) =ā.
Theorem 3.5 (Gårding inequality). Let K be a compact subset of R n and a(x, ξ, τ ) ∈ S m τ , of principal symbol a m . We suppose that there exist C > 0 and R > 0 such that Then for any 0 < C ′ < C there exists τ * > 0 we have 3.2. Local Carleman estimate away from the boundary. We set the operator a real values function ϕ and then we define the conjugate operator by where µ is a parameter that depends on τ , precisely we suppose that and with principal symbol p ϕ given by We define the following self-adjoint operators with principal symbols respectively q 2 (x, ξ, τ ) = |ξ| 2 − τ 2 |∇ϕ| 2 − µ 2 and q 1 (x, ξ, τ ) = 2τ ∇ϕ.ξ.
We assume that the weight function ϕ ∈ C ∞ (R n , R) satisfies the following sub-ellipticity condition in K a compact set of R d , if Note that the constant C does not depend on µ assuming (3.3).
Then for λ large enough ϕ = e λψ satisfies the sub-ellipticity assumption in K.
Proof. We can assume that ψ ≥ 0, as we can add a constant to ψ and ϕ is multiplied by a constant β. Changing τ in τ /β we can see that sub-ellipticity condition is also satisfied for a different constant C. A straightforward calculation shows that Using the fact that ϕ = e λψ then we have Besides, we have Then it follows from these estimates that Since |∇ψ| > 0 in the compact set K then for λ large enough we have {q 2 , q 1 } ≥ C λ τ 3 > 0. As |ξ| is comparable to τ on p ϕ = 0, we obtain the result.
Lemma 3.3. Let f and g be two real continuous functions defined in K such that f is positive on a compact subset K of R d and verifies that We set h κ = κf + g, then for κ sufficiently large then h κ ≥ C for some constant C > 0.
Proof. Let y 0 ∈ K to prove the result we distinguish two cases. Case 1: We assume f (y 0 ) = 0. Then according to the assumption made in this lemma we have h κ (y 0 ) = g(y 0 ) ≥ L. Then there exists a neighborhood of y 0 , V y0 such that for y ∈ V y0 and every κ > 0, h κ (y) ≥ g(y) ≥ L/2. Let κ y0 = 1. Case 2: We assume f (y 0 ) > 0. Since f and g are continuous, there exist V y0 a neighborhood of y 0 and C 1 , C 2 > 0 such that f (y) ≥ C 1 and |g(y)| ≤ C 2 for all y ∈ V y0 . Then for all We cover K, by compactness argument, by a finite number of such neighborhoods V y1 , . . . , V yp with associated κ j = κ yj . Taking κ = max 1≤j≤p {κ j }, we have h κ (y) ≥ L/2 on each V yj , then on K.
This completes the proof.
Proof. First we assume |ξ| large with respect τ , that is |ξ| ≥ βτ for β sufficiently large, to be fixed below. We have If β is sufficiently large we obtain Second we assume |ξ| ≤ βτ . As ρ κ is homogeneous of degree 4 in (ξ, τ, µ), we can prove the estimate on As K ′ is a compact set, we can apply Lemma 3.3 by taking f = q 2 2 + q 2 1 and g = τ {q 2 , q 1 }. This completes the proof. Theorem 3.6. Let Ω be an open bounded set of R n and ϕ be a function that satisfies the sub-ellipticity assumption in K. Then there exist τ * > 0 and C > 0 such that This theorem is classical, and we can find in Hörmander [14]. Here we give a proof to be complete.
Proof. Let's take w = e τ ϕ r then P r = f can be written as follow P ϕ w = g = f e τ ϕ . Since g = Q 2 w + iQ 1 w and Q 1 and Q 2 are symmetric then we have We fix κ large enough such that the statement of Lemma 3.4 is fulfilled, then for τ sufficiently large satisfying κτ −1 ≤ 1 then from (3.7) we obtain Since we have Thus, estimate (3.6) follows by replacing (3.11) and (3.12) into (3.10). This concludes the proof.
Let Ω be an open bounded set of R d , let K ⋐ Ω and ϕ be a function that satisfies the sub-ellipticity assumption in K. Then there exist τ * > 0 and C > 0 such that From this theorem we can deduce this corollary.
Let Ω be an open bounded set of R d , let K ⋐ Ω and ϕ be a function that satisfies the sub-ellipticity assumption in K. Then there exist τ * > 0 and C > 0 such that Proof. As r satisfies (3.2), with Theorem 3.7, we only have to estimate τ 3 e τ ϕ u 2 0 . From the first equation of (3.1) we have τ 3 e τ ϕ u 2 0 τ 3 e τ ϕ µ −1 (f + ∇r) 2 0 , which gives the result using (3.3).
As w τ,1 is equivalent to τ e τ ϕ r 0 + e τ ϕ ∇r 0 arguing as in the proof of (3.11) we obtain the result of the theorem.

Local Carleman estimate at the boundary.
In this section because the boundary, we use a tangential pseudo-differential calculus. This calculus is completely analogous to the one presented in Section 3.1 except that a function a(x ′ , x d , ξ ′ ) is a symbol in (x ′ , ξ ′ ) in the sense of Definition 3.1 where x d is a parameter and the estimates given in Definition 3.1 are uniform with respect x d . To avoid confusion we denote by S m T,τ the class of tangential symbol of order m, Op T (a) the operator associated with the symbol a ∈ S m T,τ . The class of operators associated with symbols in S m T,τ is denoted by Ψ m T,τ . We refer to [20] for details on these symbols and operators. We consider functions in a half space R d−1 × (0, +∞) = R d + , and we denote by . + = . L 2 (R d + ) the L 2 -norm and (., .) + = (., .) L 2 (R d + ) the associated inner product. At the boundary x d = 0 we denote the L 2 -norm by |g| 2 = R d−1 |g(x ′ )| 2 dx ′ and the inner product associated by (., and Γ an open set in R d−1 × R + such that for all (ξ ′ , τ ) ∈ Γ and λ > 0 then (λξ ′ , λτ ) ∈ Γ. For s ∈ R we denote by Λ s T,τ the tangential operator defined by Λ s T,τ = Op T ( ξ ′ , τ s ). We recall the following microlocal Gårding inequality obtained, for instance, by applying sharp Gårding inequality.
Let a(x, ξ ′ , τ ) ∈ S m T,τ , with principal part a m homogeneous of order m. If there exist C 0 > 0 and R > 0 such that then for any 0 < C 1 < C 0 , N ∈ N, there exist C N and τ * ≥ 1 such that for u ∈ S (R d ) and τ ≥ τ * .
As we want to change the variables in order to have a flat boundary which is convenient to do the computations, we use the language and usual tools of Riemannian geometry. In this framework the gradient and divergence operators keep forms we can follow after a change of variables. Our purpose is to use these tools locally and we do not use manifold tools as charts, atlas and etc. To fix the notation, let V be an open set in R d . Let g(x) = g ij (x) 1≤i,j≤d be a positive symmetric matrix called the metric, we denote by g −1 (x) = g ij (x) 1≤i,j≤d the inverse of g(x). For a smooth function r, we denote by (∇ g r(x)) i = 1≤j≤d g ij (x)∂ xj r(x) the gradient of r. We have ∇ g r(x) ∈ T x V , this means that ∇ g r is a tangent vector field.
For u(x) = u 1 (x), . . . , u d (x) a smooth tangent vector field, we define the divergence operator by For a smooth function r and a smooth tangent vector field u, we have (3.14) div g (ru) = r div g u + g(∇ g r, u), where g(∇ g r, u) = 1≤i,j≤d For two smooth functions r 1 and r 2 we have (3.15) ∇ g (r 1 r 2 ) = r 1 ∇ g r 2 + r 2 ∇ g r 1 .
It is well-known that there exist coordinates (called normal geodesic coordinates) such that the boundary is defined locally by x d = 0, the open set Ω ∩ V is defined by x d > 0, the metric g is such that g id = g di = 0 for i = 1, . . . , d − 1 and, g dd = 1. We denote byg = (g ij ) 1≤i,j≤d−1 the metric g on x d fixed.
We can define on each manifold x d = const the gradient and divergence operators associated withg and for r a smooth function andũ = (u 1 , . . . , u d−1 ) a smooth vector field on x d = const, we have In such coordinates, we have det g = detg. The gradient and divergence operators take the following form.
Then System (3.17) takes the form where F = e τ ϕ f and G = e τ ϕ g.

from (3.3).
We begin by reducing the system in a 2 × 2 system. We denote by ζ ′ ∈ S 1 T,τ the tangential symbol of the operator −i∇g + iτ ∇gϕ. We have Applying in both side of this equation the operator Op T (δ), we obtain

From this equation and second and third of (3.19) we obtain two equations on w and
Let U = (w, v d ), the system (3.21) has the form and B is a tangential matrix operators with principal symbol The definition of α is ambiguous when q − µ 2 ≤ 0 but in this case if q − µ 2 < 0 the root are simple and the analysis below is independent of the choice of root. In particular the roots are smooth, or if q − µ 2 = 0 the root is double and below, we give a specific analysis in this case. The root of P (λ) are iτ ∂ x d ϕ ± iα and the analysis in what follows depends on the location of roots in complex plane. We have the following result, denoting s = t 2 where t, s ∈ C we have for r 0 > 0, Indeed, let t = x+iy, we have Re s = x 2 −y 2 and Im s = 2xy, we obtain 4r 2 0 Re s−4r 4 0 +(Im s) 2 = 4(r 2 0 + y 2 )(x 2 − r 2 0 ) which gives the result.
From (3.22), we obtain that | Re α| τ |∂ x d ϕ| is equivalent to where, from the definition of q, we have We prove a microlocal Carleman estimate.
Letα = χ 0 α, as α is a smooth homogenous function of order 1 in W ,α ∈ S 1 T,τ . Recall the notation Λ s T,τ = Op T ( ξ ′ , τ s ), according with the above algebraic computations and with the left vector found, we define to the second equation and summing up, we obtain If j = 2, we have τ ∂ x d ϕ + Re α τ + |ξ ′ | in W . Let χ 2 ∈ S 0 T,τ supported in χ 0 = 1 and χ 2 = 1 on the support of χ 1 . From symbolic calculus we have Then the tangential Gårding inequality of Theorem 3.8 applies and we have , for C 1 > 0, for every N > 0 and C N > 0, uniformly with respect to τ chosen sufficiently large. This implies Proof. We have to distinguish three cases, that is | Re α| τ |∂ x d ϕ| at (x 0 , ξ ′ 0 , τ 0 ).
As −τ ∂ x d ϕ + Re α τ + |ξ ′ | in W , from (3.26), we can introduce a cutoff as in (3.28) to apply the tangential Gårding inequality of Theorem 3.8, we deduce which implies (3.31). (3.22) and as Re α and ∂ x d ϕ are positive, this is equivalent to We use Carleman technics to obtain an estimate. Before doing that we must translate subellipticity assumption (3.4) on p ϕ on analogous condition on α. First observe that .
Second, for a smooth function q = q r + iq i where q r , q i are real valued, we have {q,q} = 2i{q i , q r }. Thus on p ϕ = 0 we have Thus sub-ellipticity condition reads in W , there exists C > 0 such that At (x 0 , ξ ′ 0 , τ 0 ), observe that we can choose ξ d such that ξ d + Im α = 0 and as τ 0 ∂ x d ϕ − Re α = 0, the condition (3.33) means, by continuity and homogeneity, there exists C > 0 such that As the principal symbol of B is τ ∂ x d ϕ − Reα, we obtain Then from this estimate and (3.36), we obtain then from (3.35), (3.37) and (3.38) we obtain as we can absorb the remainder term z 2 + by the left hand side. Recalling the definition of z 1 given by formula (3.24), the symbolic calculus yields Symbolic calculus also gives which is (3.32). This achieves the proof of Lemma 3.6 as we have treated the three cases.
We can prove Lemma 3.5 in the case α = 0.

From the first equation of (3.19) and from (3.3) we have
, arguing as from above and using the Gårding estimate of Theorem 3.5, we have From (3.44)-(3.48) we obtain Lemma 3.5.

Logarithmic stability
The exponential estimate of Proposition 2.1 is the consequence of the two following results. First a global Carleman estimate with an observability term and second an estimate of the observability term coming from the dissipation.
We can estimate u∇χ and χu by √ bu and by the first estimate of Proposition 4.1 we obtain the second estimate.
This gives the sought result as we can absorb the term τ e τ ϕ u 0 + τ e τ ϕ r 0 with the left hand side.