STABILITY RESULTS OF SOME FIRST ORDER VISCOUS HYPERBOLIC SYSTEMS

In this paper, we first introduce an abstract viscous hyperbolic problem for which we prove exponential decay under appropriated assumptions. We then give some illustrative examples, like the linearized viscous Saint-Venant system. In order to achieve the optimal decay rate, we also perform a detailed spectral analysis of our abstract problem under a natural assumption satisfied by various examples. We finally consider the boundary stabilizability of the linearized viscous Saint-Venant system on trees. Mathematics Subject Classification. 35L50, 93D15, 35B37 Received November 21, 2016. Accepted March 10, 2018.


Introduction
Stability of hyperbolic systems becomes a very important area of research due to various applications in fluid dynamics, electromagnetism, wave propagation, traffic flow, etc., see [3,6,14,16,29]. Obviously the stability property of a given system is closely related to the chosen dissipation law. In this paper, we concentrate on viscous dissipation laws that appear for instance for the linearized compressible Navier-Stokes system [5,13,41] or the viscous two-phase model [23]. In order to avoid repetitive proofs for each model, we perform an unified analysis by designing an abstract setting that includes these models and even allows to treat new ones like the linearized viscous Saint-Venant system on trees and the Maxwell system with a viscous damping.
More precisely, we first study an abstract evolution equation in the form in an appropriate Hilbert setting with natural assumptions on A and B (see Sect. 2 for the details). First using semi-group theory we show that such a system is well-posed. Second using a frequency domain approach [27,40], we show that this system is exponentially stable (in an appropriated subspace of the starting one). Then we illustrate our theoretical results by four examples. In each case, we give the exact Hilbert setting and check the different assumptions. The first example is the linearized compressible Navier-Stokes systems on a bounded domain of R d for which an exponential stability was proved in [41] by showing that the associated semi-group is analytic; here we give an alternative proof that is much simpler. The second example concerns the linearized viscous Saint-Venant (or compressible Navier-Stokes) system on trees, such a problem is an extension of the same problem set in an interval [5,13], where the authors showed the exponential decay by performing a precise spectral analysis. There is a quite large literature on the Saint-Venant system on networks, see for instance [7,19,31,37], but, as far as we know, the authors concentrate on nonlinear problems with boundary dissipation or on linear models with zero order dissipative terms [6,7], but without viscosity. The third application is the study of a quite general one-dimensional hyperbolic system with a viscous damping that covers the case of the linearized viscous Saint-Venant system on an interval as well as the linearized viscous two-phase model [23]; to our best knowledge, no decay results are available for this last problem. The final example concerns the Maxwell system in a bounded domain of R 3 with a viscosity term. Such a dissipation law is not considered in the literature, since usually either Ohm's law (an internal damping of the form σE) or a dissipative boundary condition (Silver-Müller boundary) are used, see [28,38,39].
The drawback of the frequency domain approach is that it does not furnish the exact decay of the energy. One possibility to overcome this difficulty and get the exact decay is to perform a detailed spectral analysis, namely by obtaining the eigenvalues and the (generalized) eigenvectors of the associated operator and by showing that these (generalized) eigenvectors form a Riesz basis of the energy space, see [5,15]. In such a situation, the decay rate is equal to the spectral abscissa. For our system (1.1), with the assumption that B * B is equal to A up to a positive multiplicative factor, we perform the full spectral analysis of the associated operator and therefore conclude that the decay rate of system (1.1) is equal to the spectral abscissa. The last three above examples enter in this framework and therefore a precise decay rate can be obtained for them.
Finally we come back to the linearized viscous Saint-Venant system on a tree, and inspired from [5], we are interested in the advection dominating case, for which the decay rate of the energy norm of the solution is slow due to the advection modes. Similarly to [5], our goal is to restore the optimal decay by building a Dirichlet control at all except one extremities of the tree. This method is based on an extension method and allows to obtain an exponential decay rate with, contrary to [5], an arbitrary initial datum in the energy space.
Our main contributions can be summarized as follows: In Theorem 2.7, we show the exponential decay of the abstract problem (1.1). We illustrate this result by various examples, in particular new ones, like the linearized viscous Saint-Venant system on trees, the linearized viscous two-phase model and the Maxwell system with a viscous damping, are considered. Under a realistic assumption between A and B, we perform a detailed spectral analysis that allows to show that the (generalized) eigenvectors of the (non-selfadjoint) operator associated with problem (1.1) form a Riesz basis and consequently that the decay rate coincides with the spectral abscissa (see Thm. 4.4). We extend the result from [5] to the linearized viscous Saint-Venant system on a tree in the advection dominating case.
The paper is organized as follows: In Section 2, we introduce the abstract setting, some notations and the general problem studied later on. Its well-posedness is proved and its exponential decay is obtained. Section 3 is devoted to the analysis of some illustrative examples. The spectral analysis of our abstract problem is performed in Section 4. We end up the paper with the boundary stabilizability of the linearized viscous Saint-Venant system on trees in Section 5.
Let us finish this section with some notations used in the remainder of the paper. For a bounded domain D, the usual norm and semi-norm of H s (D) (s ≥ 0) are denoted by · s,D and | · | s,D , respectively. For s = 0, we will drop the index s. Furthermore, the notation A B (resp. A B) means the existence of a positive constant C 1 (resp. C 2 ), which is independent of A and B such that A ≤ C 1 B (resp. A ≥ C 2 B). The notation A ∼ B means that A B and A B hold simultaneously.

An abstract framework
Let H 1 (resp. H 2 , V ) be a complex Hilbert space with norm and inner product denoted respectively by · 1 and (·, ·) 1 (resp. · 2 and (·, ·) 2 , · V and (·, ·) V ) such that V is a dense subspace of H 1 and is compactly embedded into H 1 . Denote by V the dual space of V with respect to the pivot space H 1 .
Let us now fix a sesquilinear (linear in the first variable and conjugate linear in the second one) and symmetric form a from V × V to C that is supposed to be strictly coercive, in the sense that Then we denote by A the operator from V into V by Let us further fix a sesquilinear and continuous form b from V × H 2 into C. Denote by B the linear and continuous operator from V into H 2 defined by and let B * be its adjoint operator (that is continuous from H 2 into V ). We finally assume that the next inf-sup condition is valid In this setting, we consider the following evolution system: find u and ρ solutions of where u t represents the time derivative, while u 0 and ρ 0 are the initial data.
The existence of a solution to (2.3) in an appropriated Hilbert setting is obtained using semigroup theory. Indeed let us introduce the Hilbert space H = H 1 × H 2 with its natural inner product and introduce the (unbounded) operator A from H into itself by Theorem 2.1. Under the above assumptions, the operator A generates a C 0 -semigroup of contractions (T (t)) t≥0 on H. Therefore for all U 0 ∈ H, the problem Proof. It suffices to prove that A is a maximal dissipative operator, hence by Lumer-Phillips' theorem it generates a C 0 -semigroup of contractions on H.
Let us start with the dissipativity. For U = (u, ρ) ∈ D(A), we have Hence Let us go on with the maximality. Let λ > 0 be fixed. Given F = (f, g) ∈ H, we look for U = (u, ρ) ∈ D(A) such that (λ − A)U = F , or equivalently (2.5) Assume for the moment that such a U exists. Then the second identity is equivalent to This expression in the first identity implies that Taking the duality pairing with v ∈ V , we find that u ∈ V satisfies Now this problem has a unique solution u ∈ V , by Lax-Milgram Lemma because the left-hand side is a continuous and coercive sesquilinear form on V , since and because the right-hand side is a continuous form on V . But by the definition of A, we deduce that u is a solution of (2.7). Now defining ρ by (2.6), then the identity (2.7) means that that clearly belongs to H 1 .
As usual the energy associated with (2.3) is defined by that is equal to 1/2 of the norm of (u, ρ) in H.
Proposition 2.2. The solution (u, ρ) of (2.3) with initial datum in D(A) satisfies therefore the energy is nonincreasing.
Note that system (2.3) is not strongly stable in the whole space since the kernel of A is not necessarily reduced to {0} as the next lemma shows.

Accordingly we denote by
and Note that A 0 is well-defined from D(A 0 ) into H 0 since the inf-sup condition (2.2) implies that R(B) = (ker B * ) ⊥ = H 3 (see Lem. I.4.1 of [25]). At this stage, we want to prove the uniform stability of system (2.3) in H 0 . Our proof is based on a frequency domain approach, namely the exponential decay of the energy is deduced from the following result (see [40] or [27]): Lemma 2.4. A C 0 semigroup (e tL ) t≥0 of contractions on a Hilbert space H is exponentially stable, i.e., satisfies where ρ(L) denotes the resolvent set of the operator L.
Let us check that A 0 satisfies the first assumption of Lemma 2.4. Proof. For any z ∈ R and an arbitrary F = (f, g) ∈ H 0 , we look for U = (u, ρ) ∈ D(A) solution of izU − AU = F, (2.13) or equivalently (2.14) Now we distinguish the case z = 0 to the other cases. First if z = 0, then in (2.14) we can eliminate η by 1 iz (Bu + g) and the first equation becomes (compare with the proof of Lem. 2.1) Taking the duality pairing with v ∈ V , we find that u ∈ V satisfies Now this problem has a unique solution u ∈ V , by Lax-Milgram Lemma because the left-hand side is a continuous and coercive sesquilinear form on V , since and because the right-hand side is a continuous form on V . As before we deduce that u is a solution of (2.15) and defining ρ = 1 iz (Bu + g), we deduce that (u, ρ) belongs to D(A) and satisfies (2.13). Finally ρ is indeed in H 3 because Bu and g both belong to H 3 .
In the case z = 0, problem (2.14) reduces to that is a standard saddle point problem (see [25], §I.4). By the coercivity assumption on a and the inf-sup condition (2.2), we deduce that this problem has a unique solution (u, ρ) ∈ V × H 3 (see [25], Thm I.4, p. 59). This pair is clearly in Now we need to analyze the behaviour of the resolvent of A 0 on the imaginary axis.
Lemma 2.6. The resolvent of the operator of A 0 satisfies condition (2.12).
Proof. We use a contradiction argument, i.e., we suppose that (2.12) is false. Then there exist a sequence of real numbers β n → +∞ and a sequence of vectors Z n = (u n , ρ n ) in D(A 0 ) with Z n H = 1 such that By the definition of A, this directly implies that iβ n u n + Au n + B * ρ n 1 → 0, (2.18) iβ n ρ n − Bu n 2 → 0. (2.19) We first notice that by the dissipativeness of A, we have a(u n , u n ) = ((iβ n − A)Z n , Z n ) H ≤ (iβ n − A)Z n H → 0, that leads to As B is continuous from V into H 2 , the property (2.20) allows to conclude that This proves that (u n , ρ n ) → 0 in H and yields a contradiction.
These two lemmas directly imply the following energy decay.
Theorem 2.7. There exists two positive constants C and ω such that for all U 0 ∈ H, where P (u 0 , ρ 0 ) = (0, Qρ 0 ), Q being the orthogonal projection in H 2 on ker B * .
Proof. Lemmas 2.5 and 2.6 show that A 0 satisfies the necessary and sufficient conditions from Lemma 2.4, therefore the semigroup generated by A 0 is exponentially decaying. But for U 0 ∈ H, U 0 − P U 0 belongs to H 0 and therefore The conclusion follows from the fact that and the trivial estimate U 0 − P U 0 H ≤ U 0 H .

Some examples
In this section, we give some concrete examples that enter in our abstract framework.

The linearized compressible Navier-Stokes system on a bounded domain of R d
Let Ω be a bounded domain of R d , d ∈ N * , with a Lipschitz boundary. On this domain, we examine the problem in Ω, where u(x, t) (resp. ρ(x, t)) represents the velocity (resp. density) of the fluid at the point x and time t. As usual µ and λ are viscosity coefficients satisfying while a and b are positive constants. This problem corresponds to the linearization of the compressible Navier-Stokes equation around a constant steady state (0, ρ 0 ) with ρ 0 > 0, see [13,26,34,41]. This system enters into our abstract framework in the following way. We take H 1 = L 2 (Ω) d , H 2 = L 2 (Ω) and V = H 1 0 (Ω) d and choose the sesquilinear forms 1 The coerciveness of a is direct as On the other hand, it is easy to see that and therefore : Then the inf-sup condition (2.2) is the standard inf-sup condition for the Stokes system (direct consequence of Cor. I.2.4 of [25], see [25], p. 81). Finally, if H 1 is equipped with its natural inner product and H 2 with the inner product then it is easy to check that and that According to Theorem 2.1, problem (3.1) is well-defined in a weak sense for initial data in L 2 (Ω) d × L 2 (Ω) and in a strong sense for initial data in D(A). Additionally, by Theorem 2.7, the solution (u, ρ) of problem (3.1) tends exponentially to the stationary solution (0, c 0 ), where c 0 = 1 |Ω| Ω ρ 0 dx, in other words, there exist two positive constants C and ω such that the solution (u, ρ) of (3.1) satisfies This decay is not new and is proved in Theorem 1.1 of [41] (see also [5], Cor. 1 or [13], Sect. 2.2 in the case d = 1) but our proof is much simpler. In dimension 1, the decay rate ω is explicit in Corollary 1 of [5], using the results from Section 4, we indeed recover this decay rate, this will be done below.

The linearized viscous Saint-Venant system on trees
We first recall the notion of C 2 -networks, which is simply those of [9], we refer to [1,2,8,10,11,32,35] for more details.
All graphs considered here are non empty, finite and simple. Let G be a connected topological graph imbedded in R m , m ∈ N * , with n vertices V = {v i : 1 ≤ i ≤ n} and N edges E = {e j : 1 ≤ j ≤ N }. Each edge e j is a Jordan curve in R m and is assumed to be parametrized by its arc length parameter x j , such that the parametrization We now define the C 2 -network Γ associated with G as the union Γ = E ∪ V.
The valency of each vertex v is denoted by γ(v). For shortness, we later on denote by V ext = {v ∈ V : γ(v) = 1} the set of boundary (or exterior) vertices and V int = V \ V ext , corresponding to the set of interior vertices. For each vertex v, we also denote by For a function u : Γ → C, we set u j = u • π j : [0, l j ] → C, its "restriction" to the edge e j and use the abbreviations: for a vertex v ∈ e j . Finally, differentiations are carried out on each edge e j with respect to the arc length parameter x j . For any p ∈ [1, ∞), we denote by L p (Γ) the set of measurable functions on Γ such that u j ∈ L p (0, l j ), for all j = 1, . . . , J. For shortness we write Now we denote by P C(Γ) the set of piecewise continuous functions on Γ, which means that u : Γ → C belongs to P C(Γ) if and only if u j ∈ C([0, l j ]), for all j = 1, . . . , N . Further C(Γ) is the set of continuous functions on Γ, which means that u ∈ C(Γ) if and only if u ∈ P C(Γ) and Similarly we denote by P H 1 (Γ) the set of piecewise H 1 functions on Γ, in other words u ∈ P H 1 (Γ) if and only if u j ∈ H 1 (0, l j ), for all j = 1, . . . , N . Further let us set H 1 (Γ) = P H 1 (Γ) ∩ C(Γ). These two spaces are clearly Hilbert spaces with their natural inner product.
Let us now fix a C 2 -network Γ that is a tree (graph without cycle, so that V ext is non empty) and two positive constants a and ν corresponding to the advection and viscosity coefficients respectively. With these assumptions, we consider the linearized viscous Saint-Venant system on Γ (see [5] for one interval) where Q j := (0, l j ) × (0, ∞). As before u and ρ represent the velocity and the water height of the fluid respectively.
In the above system, the first transmission condition at interior nodes physically means the conservation of mass through the vertices, while the second one corresponds to the stress balance equation.
Remark 3.1. This system also corresponds to the linearized compressible Navier-Stokes system on trees, where in that case, ρ represent the density of the fluid, see [13]. Note that in [6,7], the damping appears in the model as a zero order term; but such a case is outside the scope of our analysis. We may further mention that the derivation of water wave model from Navier-Stokes equations leads to the introduction of a non-local term [21,30], this term is here neglected, since it requires additional investigations.
As before, this system is covered by our abstract setting with the next choices. Take the Hilbert spaces H 1 = H 2 = L 2 (Γ) and Choose the sesquilinear forms The coerciveness of a is a direct consequence of the compact embedding of V into L 2 (Γ) that yields since Γ is a tree (Poincaré type inequality on Γ [2]). From the definition of b, we see that B is given by and is clearly a linear continuous operator from V into L 2 (Γ). Moreover again due to Poincaré's inequality, B is injective and its range is closed. Consequently according to Theorem 2.19 of [12], we have R(B) = (ker B * ) ⊥ . This means that B is actually an isomorphism from V into H 3 and by Lemma I.4.1 of [25], the inf-sup condition (2.2) holds. In order to have an explicit form of the space H 3 , let us characterize the kernel of B * .
Proof. Let ρ be in ker B * , then it satisfies In a first step for any j = 1, . . . , N , we take u ∈ V such that u k = 0 for all k = j and u j =φ ∈ D(0, l j ). Then This implies in particular that ρ belongs to P H 1 (Γ). In a second step, we come back to (3.3) and apply Green's formula on each edge to find Since for any v ∈ V int and any X ∈ C Nv orthogonal to (1, . . . , 1) , there exists a function u ∈ V such that by taking such a test function in the previous identity, we find that (ρ j ) j∈Jv is orthogonal to X. Since this holds for all X orthogonal to (1, . . . , 1) , we deduce that (ρ j ) j∈Jv is a multiple of (1, . . . , 1) or equivalently As this holds for all interior vertices and Γ is connected, this proves the inclusion The other inclusion being a direct consequence of Green's formula, the proof is complete.
A direct consequence of the previous lemma is that exactly as in the previous subsection. Let us further characterize the domain of the operator A.
and the equivalence Proof. By its definition, (u, ρ) belongs to D(A) if and only if (u, ρ) ∈ V × L 2 (Γ) and satisfies In other words, there exists h ∈ L 2 (Γ) such that From the definition of A and B, this equivalently means that As in Lemma 3.2, by taking first test functions equal to zero except at one edge where it coincides with a smooth function with a compact support, we deduce that and consequently The second step of the proof of Lemma 3.2 directly shows that νu − aρ belongs to H 1 (Γ). The equivalence (3.4) directly follows from the definition of A(u, ρ) and from (3.6).
In conclusion according to Theorem 2.1, problem (3.2) is well-defined in a weak sense for initial data in L 2 (Γ) × L 2 (Γ) and in a strong sense for initial data in D(A). Further by Theorem 2.7 and the above characterization of ker B * , the solution (u, ρ) tends exponentially to the stationary solution (0, Qρ 0 ), where Qρ 0 = 1 |Γ| Γ ρ 0 , |Γ| = Γ dx being the length of Γ. In other words, there exist two positive constants C and ω such that the solution (u, ρ) of (3.2) satisfies To our best knowledge this result is new, the only drawback is that the decay rate is not explicit. This drawback will be set up by a precise spectral analysis, and is based on the property This property follows from the definition of B. Indeed for any u, v ∈ V , one has which shows that (3.7) holds.

An one-dimensional hyperbolic system with a viscous damping
On a real interval (0, 1), we consider the system where u(x, t) (resp. ρ(x, t)) represents the vectorial unknown functions at the point x and time t with values in C n (resp. C m ), M is a n × n symmetric and positive definite matrix and C is a m × n matrix. This system is a first order linear hyperbolic system if M = 0 that is conservative [14], therefore the term −M u xx corresponds to a viscous damping that will be responsable of the exponential decay of the system. If n = m = 1, this system is nothing else than the linearized viscous Saint-Venant system in (0, 1) (with a = 1) [5,13]. If n = 1 and m = 2, it corresponds to a linearized viscous two-phase model where ρ 1 and ρ 2 are the density of phase 1 and 2 (with ρ = (ρ 1 , ρ 2 )), u is the common velocity, with the pressure law p(ρ 1 , ρ 2 ) = a 2 1 ρ 1 + a 2 2 ρ 2 , a 1 , a 2 being two given real numbers, hence the matrix C * is given by (a 2 1 , a 2 2 ) and M = µ > 0, see system (1.11) of [23]. Note that the linearization of this system is made around the point (m 0 , n 0 , 0) with m 0 , n 0 > 0.
Again, system (3.8) enters in our abstract setting with the next choices: Take the Hilbert spaces H 1 = L 2 (0, 1) n , H 2 = L 2 (0, 1) m and V = H 1 0 (0, 1) n equipped with their natural inner product and choose the sesquilinear forms: The coerciveness of a is a direct consequence of Poincaré's inequality in H 1 0 (0, 1) as a(u, u) =  u ∈ ker C in (0, 1), and therefore there exist α i ∈ H 1 0 (0, 1), for all i = 1, . . . , I such that This lemma allows us to characterize the space W = (ker B) ⊥ , the orthogonal of ker B in V once it is equipped with the inner product where (ker C) ⊥ means the orthogonal of ker C in C n .
Proof. By definition, u ∈ W if and only if Hence according to Lemma 3.4, this is equivalent to For a fixed j ∈ {1, . . . , I}, as test function in the above identity, we take α j = ϕ ∈ D(0, 1) and α i = 0 for any i = j, and obtain 1 0φ x u x ·ē j dx = 0, ∀ϕ ∈ D(0, 1).
We are ready to show a sort of Poincaré's inequality. For any u ∈ W , as (3.10) implies that u x belongs to (ker C) ⊥ almost everywhere in (0, 1), this estimate implies that a. x ∈ (0, 1).
Integrating this estimate in (0, 1), we conclude by using the standard Poincaré inequality.
At this stage, we consider the restriction of B to W , namely let be a basis of ker C * and {e * i } m i=I * +1 be a basis of (ker C * ) ⊥ (in C m ). Then it holds α i e * i with α i ∈ L 2 (0, 1), ∀i ≤ I * and α i ∈ C, ∀i > I * }.
From the previous lemma, we see that ker B * is finite-dimensional if ker C * is reduced to {0}, its dimension being equal to m, otherwise it is an infinite dimensional space. In any case, its projection Q can be easily computed as follows: Without loss of generality, we can assume that the basis {e * i } m i=1 of C m is orthonormal, then one has Indeed for an arbitrary ρ ∈ L 2 (0, 1) m , by the previous lemma with α i ∈ L 2 (0, 1), for all i ≤ I * and α i ∈ C for all i > I * fixed such that for all β i ∈ L 2 (0, 1), if i ≤ I * and all β i ∈ C if i > I * . By the orthogonality property of the e * i , this is equivalent to for all β i ∈ L 2 (0, 1), if i ≤ I * and all β i ∈ C if i > I * . In this property, first for any j ≤ I * , we fix β j = β arbitrary in L 2 (0, 1) and β i = 0 else, and find 1 0 α jβ dx = 1 0 ρ ·ē * jβ dx, ∀β ∈ L 2 (0, 1), which shows that Second for any j > I * , we fix β j = β arbitrary in C and β i = 0 else, and find again (3.15) that here reduces to Inserting (3.15) and (3.16) into the expression (3.14), we find (3.13).
In conclusion according to Theorem 2.1, problem (3.8) is well-defined in a weak sense for initial data in L 2 (0, 1) n × L 2 (0, 1) m and in a strong sense for initial data in D(A). By Theorem 2.7 and the above characterization of ker B * , the solution (u, ρ) tends exponentially to the stationary solution (0, Qρ 0 ), where Recall that Theorem 2.7 does not furnish an explicit decay rate but under the assumption that for some positive real number β, then the results of the next section allow to set up this drawback since we then have Indeed from the definition of B, for any u, v ∈ V , one has

The viscous Maxwell system in a bounded domain of R 3
Let Ω be a bounded domain of R 3 with a Lipschitz and simply connected boundary. On this domain, we consider the Maxwell system in Ω, (3.19) where E(x, t) (resp. H(x, t)) represents the electric (resp. magnetic) field at the point x and time t and the boundary conditions are standard electric boundary conditions. Here ν is a viscosity coefficient that is supposed to be positive. The case ν = 0 corresponds to the standard Maxwell system that is conservative [17,22,33], and consequently the term ν curl curl E is a viscous damping that will be responsable of the exponential decay of the system. This system enters into our abstract framework in the following way (see [22,36] in the case ν = 0). We first recall that both being Hilbert spaces with the inner product of L 2 (Ω) 3 . We further set V = H(div = 0, Ω) ∩ H 0 (curl, Ω), that is a Hilbert space with the inner product (due to Friedrich's inequality, see [33], Cor. 4.8) We now choose the sesquilinear forms The coerciveness of a is direct, since Further as for E ∈ V , curl E belongs to H 2 , we directly deduce that As before, let us characterize the kernel of B * and the domain of the operator A. Proof. Let H be in ker B * , then it satisfies (3.20) As D(Ω) 3 is not dense in V , while it is dense in X 0 (Ω) [4], p. 827, therefore in this last identity, we want to change the set of test functions into X 0 (Ω). For that purpose, for an arbitraryẼ in X 0 (Ω), we consider the unique solution ϕ ∈ H 1 0 (Ω) of Ω ∇ϕ · ∇ψ dx = ΩẼ · ∇ψ dx, ∀ψ ∈ H 1 0 (Ω).
This identity implying thatẼ − ∇ϕ is divergence free, we deduce that E =Ẽ − ∇ϕ belongs to V . Consequently The converse inclusion is a consequence of the next Green's formula  [20] in the smooth case and [4], Prop. 3.14 for a different expression of the basis), we deduce that ker B * is finite-dimensional, its dimension being equal to J, the number of cuts Σ j , j = 1, . . . , J, such that Ω • = Ω \ ∪ J j=1 Σ j becomes simply connected. More precisely there exist J functions q j ∈ H 1 (Ω • ), j = 1, . . . , J, such that where∇q j means that we take the gradient of q j in Ω • , the properties of q j (stated in Lem. 1.2 of [24]) implying that∇q j belongs to K T (Ω Proof. We first check that where ·, · Σj means the duality pairing between H where [q j ] k means the jump of q j through Σ k . As [q j ] k = δ jk (see Lem. 1.2 of [24]), we conclude that H · n, 1 Σj = 0, ∀j = 1, . . . , J.
Given H ∈ H 3 , owing to (3.24), we can apply Theorem 3.17 of [4] that yields a unique ψ ∈ V such that curl ψ = H, that trivially implies and proves (3.23).
Let us further characterize the domain of the operator A. In other words, there exists h ∈ H 1 such that From the definition of A and B, this equivalently means that Again as in Lemma 3.8, we can change the test functions to any element of X 0 (Ω) since for any ϕ ∈ H 1 0 (Ω), one has Ω h · ∇φ dx = − Ω div hφ dx = 0, due to Green's formula and recalling that h is divergence free.
In other words (3.26) is equivalent to Consequently, we find equivalently that curl(ν curl E − H) = h in the distributional sense and since h belongs to L 2 (Ω) 3 , ν curl E − H belongs to H(curl, Ω). The identity (3.25) directly follows from the previous considerations.
In conclusion, Theorem 2.1 guarantees that problem (3.19) is well-defined in a weak sense for initial data in As already mentioned before, Theorem 2.7 does not furnish an explicit decay rate but for this model as we immediately check that the results of the next section allow to give an explicit decay rate.

A spectral analysis
Here we come back to the abstract setting from Section 2 and show that a spectral analysis of the operator A is possible under the assumption B * B = βA for some positive real number β.
are eigenvalues of A. If λ 2 k − 4β = 0, then λ k+ and λ k− are simple eigenvalue of A and the corresponding eigenvectors (up to a multiplicative factor) are given by On the contrary, if λ 2 k − 4β = 0, for some k ∈ N * , then λ k+ = λ k− is an eigenvalue of A of geometric multiplicity one and algebraic multiplicity 2, namely is the corresponding eigenvector, while is a generalized eigenvector, namely it satisfies Proof. Since we have already characterized the kernel of A (see Lem. From the second equation, we can eliminate ρ given by and inserting it in the first identity, we get But recalling our assumption (4.1), we find that u is solution of Now we notice that the definition of the domain of A requires that Au + B * ρ has to be in H 1 , but due to (4.6) and (4.1), we get We remark that 1 + β λ cannot be equal to zero, otherwise by (4.7), u is zero and hence ρ = 0 by (4.6), which is not allowed. Consequently u belongs to D(A) and satisfies This means that u = 0 is an eigenvector of A with eigenvalue − λ 2 λ+β , and therefore for some k ∈ N * and u = ϕ k (up to a multiplicative factor). As this identity is equivalent to the second order equation we find that λ ∈ {λ k+ , λ k− }. By the identity u = ϕ k and (4.6), we find the expression (4.3) for the associated eigenvector, except if λ 2 k − 4β = 0. In that last case, we see that the expression of U k+ and U k− in (4.6) are the same and consequently, λ k+ has only one eigenvector. But simple calculations show that U k− given by (4.5) is a generalized eigenvector.
The next step is to show that the eigenvectors and generalized ones that we just found form a Riesz basis of H 0 . First we show that they generate the whole H 0 . For further uses, we introduce the exceptional set N 0 = {k ∈ N * : λ 2 k = 4β}, and the set Note that for k ∈ N 0 ∪ N 1 , the eigenvalues λ k+ and λ k− are real.
In summary we have shown that (4.11) holds for all k ∈ N * , or equivalently (u, ϕ k ) H1 = 0, ∀k ∈ N * , as well as The first property implies that u = 0, since {ϕ k } k∈N * is an orthonormal basis of H 1 . The second property implies that since {ϕ k } k∈N * is also an orthonormal basis of V . But as by assumption ρ ∈ (ker B * ) ⊥ , we conclude that ρ = 0.
We are ready to prove the Riesz basis property. H U k± } k∈N * is a Riesz basis of H 0 . Proof. For shortness, we denote by V k± = κ k± U k± , with κ k± = U k± −1 H , so that V k± H = 1. We first show that from the set {V k± } k∈N * we can build an orthonormal basis of H 0 . For that purpose, we calculate their inner product by distinguish the following cases: (a) For k, k ∈ N * \ N 0 , we have by (4.1) Hence if k = k , we deduce that On the contrary if k = k , we only deduce that that, as easily checked, is different from 0. But since V k+ and V k− are linearly independent, we can orthonormalize them, by settingṼ where 14) Note that for k ∈ N 1 from (4.9), we see that λ k+ λ k− = βλ 2 k , and therefore For k ∈ N * \ N 0 and k ∈ N 0 , by the previous considerations, we clearly have (c) For k, k ∈ N 0 , with k = k , by the previous considerations, we directly have Obviously V k+ and V k− are not orthogonal but as before, we can orthonormalize them by introducingṼ k− as in (4.13) with δ k and η k appropriately defined. These three points show that the family {V k+ ,Ṽ k− } k∈N * is an orthonormal basis of H 0 . Consequently any (u, ρ) ∈ H 0 can be written as Coming back to the family {V k± } k∈N * , we have that Hence {V k± } k∈N * will be a Riesz basis, if we can show that . (4.20) According to (4.17), this equivalence holds if and only if (4.21) with the relations (4.19). To prove this last equivalence we split up the sum into two terms, the first one for k small and the second one for k large. Namely for K fixed later on, we show that there exist two positive constants C 1 and C 2 depending only on K such that as well as If these two equivalences hold for one fixed K, then by summing up (4.23) on k > K and summing up with (4.22), we find that (4.21) is valid and the proof will be finished.
Hence it remains to prove the two equivalences above. The first equivalence is a simple consequence of the equivalence of norm in a finite dimensional space. Indeed it is clear that if and only ifα k =β k = 0, and consequently the mid term in (4.22) is a norm in C 2K . For the equivalence (4.23), we look at the behavior of δ k and η k for k large. Hence without loss of generality we can assume that k is in N 1 . For such a k, from (4.12), we have Hence we need the asymptotic behavior of λ k± . But from the expression (4.2), we have , and therefore, one has λ k+ ∼ −β for k large. (4.24) This behavior shows that Similarly and therefore These two equivalences in (4.16) yields By (4.15), we deduce that Using (4.19), we can write where (·, ·) C 2 means the standard inner product in C 2 and the matrix M k is given by As the equivalences on δ k and η k imply that M k is similar to the identity matrix for k large, we deduce that (4.23) holds for k > K with K large enough.
The Riesz basis property allows us to give an explicit decay rate of the semi-group generated by A 0 defined by (2.9)-(2.10). Proof. Owing to Theorem 4.3, any U ∈ H 0 can be written as with c k+ , c k− ∈ C (recalling that V k± = κ k± U k± ) such that Therefore e tA0 U is given by

Theorem 4.3 then yields
The conclusion will follow if we can show that and λ k+ < −τ, ∀k ∈ N 0 . (4.27) But the second situation occurs if and only if there exists k ∈ N * such that λ 2 k = 4β and in that case In the first situation, we distinguish two cases: 1. If k ∈ N * \ (N 0 ∪ N 1 ), then this is equivalent to the condition that λ 2 k < 4β and in such a case, we have 2. If k ∈ N 1 (equivalent to λ 2 k > 4β), then clearly On the other hand, it is easy to check that λ k+ is a non decreasing function of k, and therefore, by (4.24), we have as requested.

Boundary stabilizability of the linearized viscous Saint-Venant system on trees
If we come back to the linearized viscous Saint-Venant system (3.2) on a tree Γ, according to Remark (3.7), the assumption (4.1) holds with β = a ν . Consequently the decay of the semigroup generated by A 0 is given by where λ 2 1 is the first eigenvalue of the Laplace operator (up to the factor ν) on Γ with boundary and transmission conditions associated with V (compare with Cor. 1 of [5] in the case of an interval of length π).
As in [5], we are here interested in the advection dominating case, that here corresponds to the case when In that case, the decay rate of the energy norm of the solution of (3.2) is − λ 2 1 2 and is then slow due to the advection modes (corresponding to the complex eigenvalues of A with a real part smaller than a ν ). Inspired from [5], our goal is to restore the optimal decay − a ν for any tree by building a Dirichlet control at all except one extremities of the tree. This method allows to obtain an exponential decay rate −σ with σ as close as we want from a ν and with an arbitrary initial datum in the energy space. From now on we then fix a C 2 -network Γ that is a tree and fix V Diss ext a subset of V ext made of all vertices of V ext , except one, say r (the root of the tree), namely V Diss ext = V ext \ {r}. As in Section 3.2, we suppose given two positive constants a and ν corresponding to the advection and viscosity coefficients respectively. Then we consider the non-homogeneous problem ext , t > 0, u(·, 0) = u 0 , ρ(·, 0) = ρ 0 in Γ, with that is a Hilbert space with the norm (3.5), and for all v ∈ V Diss ext , q v ∈ C([0, ∞)) satisfies the compatibility condition The existence and decay of the solution to this problem will be given below by using an extension method (as in the proof of Thm. 4.1 of [5]). But before, let us introduce some notations and useful properties.
First, for all v ∈ V Diss ext , we extend the edge e jv of Γ having v as extremity into a longer edgeẽ jv (of length l iv strictly larger than l jv ). Denote byΓ the new tree obtained from Γ by simply replacing the edges e jv , with v ∈ V Diss ext , byẽ jv , see Fig. 1, where Γ is in black and the extension in red. Denote byṼ ext the set of exterior vertices ofΓ (note thatΓ has the same set V int of interior vertices as Γ). On this new treeΓ, we now denote bỹ λ 2 k , k ∈ N * , the eigenvalue of the operatorÃ (that corresponds to the operator A but defined inΓ) of associated eigenfunctionsφ k ,λ k± the associated eigenvalues ofÃ (that again corresponds to the operator A but defined inΓ).
We first build an extension operator from Γ toΓ.
Lemma 5.1. There exists a continuous operator E 1 from H toH such that and that is also continuous from D to D(Ã), i.e., satisfying Proof. With loss of generality, for all v ∈ V Diss ext , we can assume that the parametrization of the edge e jv is such that v corresponds to l jv and that the edgeẽ jv is parametrized by (0,l jv ). Fix a function η ∈ H 1 (Γ) (that plays the rule of a cut-off function) such that η j ∈ C ∞ ([0, l j ]), for all j = 1, . . . , N and satisfying, for all v ∈ V Diss ext , with ε > 0 fixed small enough so that 0 < l jv − 4ε < l jv + 4ε <l jv .
Furthermore for all v ∈ V Diss ext , we fix a cut-off function χ jv ∈ D(l jv , l jv + 4ε) such that For Y = (u, ρ) ∈ H, we take E 1 Y = Y on Γ and for all v ∈ V Diss ext , we set , ∀x ∈ (l jv , l jv + 4ε], (E 1 Y ) jv (x) = (0, 0), ∀x ∈ (l jv + 4ε,l jv ]. The continuity property of E 1 from H toH follows by simple changes of variable, while its continuity property from D to D(Ã) is a simple consequence of Leibniz' rule and again simple changes of variable.
We go on with a multiplicative result.
For any σ > 0, we finally introduce the finite set We finally need the next technical lemmas.
Proof. Assume that there exist coefficients c k such that k∈Ñσ c kφk = 0 onΓ \Γ. (5.4) Then by grouping the sum by packet corresponding to the same eigenvalue, we can write where K is a subset of N σ such that λ k = λ k for all k, k ∈ K such that k = k and Now for all v ∈ V Diss ext , we notice that ψ k can be written as with α k , β k ∈ C. With these notations, (5.4) is equivalent to ψ = 0 onΓ \Γ.
Going back to (5.6), this means that for any k ∈ K, one has ψ k = 0 on e jv , ∀v ∈ V Diss ext .
We now show from generation to generation that this implies that ψ k is zero onΓ, which automatically implies that c k = 0, for all k. Indeed we first notice that ψ k − λ 2 k ν ψ k = 0 on e j , ∀j ∈ {1, . . . , N }.
Then restricting this identity to an edge j of the last but one generation, as for the interior vertex v in common with the last generation, (5.7) and the transmission conditions j∈Jv ψ j (v)ν j (v) = 0, ∀v ∈ V int , ψ j (v) = ψ k (v), ∀v ∈ V int , j, k ∈ J v , imply that ψ k (v) = ψ k (v) = 0, by Holmgren uniqueness theorem, we deduce that ψ k = 0 on e j .
We are ready to state our stability result (compare with Thm. 4.1 of [5] in the case of an interval).
With the help of Lemma 5.1, we consider with E 2 Y 0 ∈ D(Ã) defined in such a way that (EY 0 , (0, 1))H = 0, (5.13) (EY 0 ,Ũ k± )H = 0, ∀k ∈Ñ σ . (5.14) Hence we look for E 2 Y 0 in the form 2 where ϕ is the function fixed in Lemma 5.2 (that allows to conclude that E 2 Y 0 belongs to D(Ã)) and the coefficients c k± and d are fixed in order to satisfy (5.13) and (5.14). Indeed these two conditions are equivalent to (E 2 Y 0 , (0, 1))H = −(E 1 Y 0 , (0, 1))H, From the expression (5.15) of E 2 Y 0 , this system is equivalent to a square linear system M X = C, 2 Let us notice that the assumption λ 2 1 2 < a ν implies thatÑσ is non empty if σ is sufficiently close to a ν . Indeed asΓ is larger than Γ, the space V defined on Γ can be viewed as a closed subspace of the space V defined onΓ and by Rayleigh quotient techniques, one hasλ 1 ≤ λ 1 .
where the unknown vector X is equal to (c k+ ) k∈Ñσ , (c k− ) k∈Ñσ , d and the coefficients of the matrix M are simply the inner product in L 2 (Γ \Γ) 2 of the functions √ ϕŨ k± , k ∈Ñ σ and √ ϕ(0, 1). Since, by Lemma 5.4, these functions are linearly independent onΓ \Γ, the matrix M is invertible, which guarantees the existence and uniqueness of the coefficients c k± and d.