Let Ω⊂lRn (n∈lN∗) be a bounded domain with C2 boundary Γ =∂Ω. Let Ω1 be a sub-domain
of Ω and set Ω2 = Ω\Ω1. We denote by γ the interface, by Γj =∂Ωj \γ (j = 1,2) the exterior
boundaries, and by νj the unit outward normal vector of Ωj (j= 1,2). We assumeγ 6=∅and γ is
of classC1.
Consider the following hyperbolic-parabolic coupled system:
(7.1) yt−∆y= 0 in (0,∞)×Ω1, ztt−∆z= 0 in (0,∞)×Ω2, y= 0 on (0,∞)×Γ1, z= 0 on (0,∞)×Γ2, y=zt, ∂ν∂y1 =−∂ν∂z2 on (0,∞)×γ, y(0) =y0 in Ω1, z(0) =z0, zt(0) =z1 in Ω2.
As we said above, this is a simplified and linearized model for fluid-structure interaction. In system (7.1), y may be viewed as the velocity of the fluid; while z and zt represent respectively the
displacement and velocity of the structure. More realistic models should involve the Stokes (resp.
the elasticity) equations instead of the heat (resp. the wave) ones.
In [170] and [197], the same system was considered but for the transmission condition y = z
on the interface instead of y =zt. But, from the point of view of fluid-structure interaction, the
transmission condition y = zt in (7.1) is more natural. Note also that, as indicated above, the
interface in this model is fixed. This corresponds to the fact that the system is a linearization around the trivial solution of a free boundary problem.
Set HΓ1 1(Ω1) 4 = nh|Ω1 h ∈ H 1 0(Ω) o and HΓ1 2(Ω2) 4 = nh|Ω2 h ∈ H 1 0(Ω) o . System (7.1) is
well-posed in the Hilbert space
H =4L2(Ω1)×HΓ12(Ω2)×L
2(Ω 2).
When Γ2 is a non-empty open subset of the boundary (or, more generally, of positive capacity),
inH the following norm is equivalent to the canonical one:
|f|H =h|f1|2L2(Ω 1)+|∇f2| 2 (L2(Ω 2))n+|f3| 2 L2(Ω 2) i1/2 , ∀f = (f1, f2, f3)∈H.
In this case the only stationary solution is the trivial one. This is due to the fact that Poincar´e inequality holds.
When Γ2, vanishes,| · |H is no longer a norm onH. In this case, there are non-trivial stationary
solutions of the system. Thus, the asymptotic behavior is more complex and one should rather expect the convergence of each individual trajectory to a specific stationary solution. To simplify the presentation in this section we assume that the capacity of Γ2 is positive.
The energy of system (7.1) is given by
E(t)=4E(y, z, zt)(t) =
1
2|(y(t), z(t), zt(t))|
2 H
and satisfies the dissipation law
(7.2) d
dtE(t) =−
Z
Ω1
|∇y|2dx.
Therefore, the energy of (7.1) decreases ast→ ∞.
In factE(t)→0 ast→ ∞, without any geometric conditions on the domains Ω1 and Ω2 (other
than the capacity of Γ2 being positive). However, due to the lack of compactness of the domain of
the generator of the underlying semigroup of system (7.1) for n≥2, one can not use directly the LaSalle’s invariance principle to prove this result. Instead, using the “relaxed invariance principle” ([178]), we conclude that y and zt tend to zero strongly in L2(Ω1) and L2(Ω2), respectively; while
z tends to zero weakly in HΓ12(Ω1) as t→ ∞. Then, we use the special structure of (7.1) and the
key energy dissipation law (7.2) to obtain the strong convergence of z inHΓ1
2(Ω1) ([198]).
Once the energy of each individual trajectory has been shown to tend to zero as t goes to ∞, we analyze the rate of decay. In particular, it is natural to analyze whether there is an uniform exponential decay rate, i. e. whether there exist two positive constantsC and α such that
(7.3) E(t)≤CE(0)e−αt, ∀t≥0
for every solution of (7.1).
According to the energy dissipation law (7.2), the uniform decay problem (7.3) is equivalent to showing that: there exist T >0 and C >0 such that every solution of (7.1) satisfies
(7.4) |(y0, z0, z1)|2H ≤C Z T 0 Z Ω1 |∇y|2dxdt, ∀(y0, z0, z1)∈H.
Inequality (7.4) can be viewed as an observability estimate for equation (7.1) with observation on the heat subdomain. In principle, whether it holds or not depends very strongly on how the two componentsyandzof the solution are coupled along the interface. Indeed, the right-hand side term
of (7.4) provides full information onyin Ω1 and, consequently, also on the interface. Because of the
continuity conditions on the interface this also yields information onzand its normal derivative on the interface. But how much of the energy ofzwe are able to obtain from this interface information has to be analyzed in detail. It depends on two facts. First it may depend in a very significant way on whether the interfaceγ controls geometrically the wave domain Ω2 or not. Second, on the
Sobolev norm of the interface trace information we recover zand its normal derivative.
Remark 7.1 This argument also shows the close connections of the problems of control and that of the exponential decay of solutions of damped systems. Both end up being reducible to an observability inequality. This is particularly clear for the wave equation with localized damping:
utt−∆u+ 1ωut= 0,
In this case the energy is given by
E(t) = 1 2 Z h |ut|2+|∇u|2 i dx,
and the energy dissipation law reads
dE(t)
dt =−
Z
ω
|ut|2dx.
The energy has an uniform exponential decay rate if and only if there exists some time T and constant C >0 such that
E(0)≤C Z T 0 Z ω |ut|2dxdt.
Moreover, this observability estimate holds for the dissipative equation satisfied by u if and only if it holds for the conservative wave equation
ϕtt−∆ϕ= 0.
Thus we see that exponential decay is equivalent to observability which, as we know from previous sections, is also equivalent to controllability. This establishes a clear connection between control- lability and stabilization. Here the argument has been developed for the wave equation but similar developments could be done for plate and Schr¨odinger equations and, more generallly, for conser- vative evolution equations.
The fact that the exponential decay is equivalent to an observability inequality is also important for nonlinear problems. We refer to [200] and [43] for the analysis of the stabilization of nonlinear wave equations.
Returning to the coupled heat-wave system, as indicated in [196], there is no uniform decay for solutions of (7.1) even in one space dimension. The analysis in [196] exhibits the existence of a hyperbolic-like spectral branch such that the energy of the eigenvectors is concentrated in the wave domain and the eigenvalues have an asymptotically vanishing real part. This is obviously incompatible with the exponential decay rate. The approach in [196], based on spectral analysis, does not apply to multidimensional situations. But the 1−d result in [196] is a warning in the sense that one may not expect (7.4) to hold.
Exponential decay property also fails in several space dimensions, as the 1−dspectral analysis suggests. To prove this fact one has to build a familiy of solutions of the coupled system whose energy is mainly concentrated in the wave domain. This has been done in [198] following [170] using Gaussian Beams ([164] and [140]) to construct approximate solutions of (7.1) which are highly concentrated along the generalized rays of the D’Alembert operator in the wave domain Ω2
and are almost completely reflected on the interface γ. As we mentioned above, in the particular case of polygonal domains with a flat interface, one can do a simpler construction using plane waves. This result on the lack of uniform exponential decay, which is valid for all geometric configu- rations, suggests that one can only expect a polynomial stability property of smooth solutions of (7.1) even under the Geometric Control Condition, i.e., when the heat domain where the damping of the system is active is such that all rays of Geometric Optics propagating in the wave domain meet the interface in an uniform time. To prove this, we need to derive a weakened observability inequality. This can be done by viewing the whole system as a perturbation of the wave equation in the whole domain Ω, an argument that was introduced in [170] for the simpler interface conditions.
These results are summarized in the following section.