3 DERIVATION OF THE MAIN ESTIMATE 3.1 Preliminary Notation

We will use throughout the standard denotations for the “energy” of the system (4):

Eφ(t) =

Under the assumptions made above on the geometry Ω, we can assume throughout that there exists a dense set of data corresponding to smooth (enough) solutions to the PDE (4). Indeed, ifΩ has a smooth boundary (the first part of Assumption (A2)), this assertion follows from classical elliptic and semigroup theory. On the other hand, ifΩ is either a parallelepiped (the second part of Assumption (A2)) or else satisfies Assumption (A1) (so that in particularΩ is convex)], then one can appeal to [10]). In this way, one can justify the computations to be done below. Also, we will use the standard denotations:

Q ₀ = (₀, T − ₀) × Ω;

Σ0 = (0, T ) × Γ0; Σ1= (0, T ) × Γ1.

In addition, with φ = [φ, φ_t] and ψ = [ψ, ψ_t], we will use below the standard deno-tation for terms that are “below the level of energy”; namely,

l.o.t.(φ, ψ) ≡ C φ, ψ

C([0,T ];H¹⁻
(Ω)×H⁻
(Ω)×H¹⁻
(Γ0)×H⁻
(Γ0)), for some constantC, where  > 0.

Finally, we will need throughout the cutoff functionξ(t) ∈ C₀^∞(R), which is defined by having for arbitrary₀> 0,

ξ(t) =

3.2 Step 1 (The Conservation Relation)

Multiplying the first equation of (4) byφt, the second byψt, and subsequently integrating in time and space, and integrating by parts we obtain

1

Applying the Neumann boundary condition in (4) to the first equation above and summing the two yields the expected conservation of the system; i.e.,

E(t) = E(s) for all 0 ≤ s, t ≤ T . In particular then,

E(s) =

φ0, ψ0²

H for all 0 ≤ s ≤ T . (9)

3.3 Step 2 (The Acoustic Wave Estimates)

Leth be a [C²(Ω)]ⁿ-vector field, which will be eventually specified. With thish, we apply the “classical” wave multipliers (see, e.g., [14], [23], [27], [28]). Multiplying theφ-wave equation of (4) byh · ∇φ, integrating in time and space and using the Neumann boundary condition, we have

Next, we consider again the first wave equation in (4), this time multiplying by the quantity φdiv(˜h), where ˜h(x) ∈ 

C²(Ω)_n

is arbitrary. Integrating in time and space, and invoking Green’s Theorem and the identity ∇%

φdiv(˜h)&

Using now the Neumann boundary condition forφ in (4), along with Sobolev Trace Theory, we have the following inequality for anyC²-vector field ˜h:

""

We will now consider the two cases, corresponding to Assumptions (A1) and (A2).

164 G. Avalos and I. Lasiecka

3.3.1 Case I. Assumption (A1) is in Force

Given the assumptions on bothΓ0andΓ1, it is shown in [19] that there exists a vector field h(x) = [h1(x), h2(x), ..., hn(x)] ∈ [C²(Ω)]ⁿsuch that

(i)h · ν = 0 on Γ1;

(ii) The matrixH(x), defined by

[H(x)]_ij =^∂h_∂x^i(x)_j , 1 ≤ i, j ≤ n,

satisfiesH(x) ≥ ρ0I on Ω, for some positive constant ρ0.

(13)

Applying this particularh to (10), we obtain ρ0

onΓ0, as well as the conservation relation (9), we obtain from this the majorization,

ρ₀ whereh is the vector field in (13).

Combining (14) and (12) (with ˜h = h therein) now yields (ρ0− ) In turn, the inequalities (12) (taking therein ˜h, which satisfies div(˜h) = 1) and (15) (taking therein > 0 small enough) give the intermediate estimate,

T −  0

Now in estimating the tangential derivative ^∂φ_∂τ"""

Γ0

on the right-hand side of (16), there is no appeal to classical Sobolev trace theory. Instead, we recall the following estimate, a product of microlocal machinery.

Lemma 4 (see [17], Lemma 7.2). Let 0 > 0 be arbitrarily small. Let w be a solution of the wave equation on (0, T ) × Ω, or more generally, any second-order hyperbolic equation with smooth space-dependent coefficients Then,with wc ≡ ξw (with ξ(t) being the cutoff function define in (8)), we have the estimate

T

where Γ∗is a smooth connected segment of boundary Γ.

Then, invoking Lemma 4 (with Γ_∗ = Γ₀ therein), so as to handle the tangential derivative in (16), gives the following: under Assumption (A1), one has the integral esti-mate for the energy of the component φ of (4),

T −  0

3.3.2 Case II. Assumption (A2) is in Force

In this case, sinceΓ0is flat, one can construct aradial vector field h(x) ∈ [C²(Ω)]ⁿsuch that

h(x) · ν = 0 on Γ0. (19)

(Indeed, ifx₀∈ Γ₀, then we can takeh(x) = x − x₀.) Applying this vector fieldh to the relation (10), we obtain the inequality,



166 G. Avalos and I. Lasiecka

Using|∇φ|²=%

∂φ∂ν

2+^∂φ_∂τ²&

onΓ, the conservation relation (9), and the fact that radial h is parallel toΓ0, we obtain

Γ0ψ_t^∂φ_∂τdtdΓ: again, Sobolev trace theory attaches no meaning to the boundary trace ^∂φ_∂τ"""

Γ0

for initial data of finite energy. Instead, we deal with this term by the following “sharp” trace regularity result:

Lemma 5 (see [18], p. 113, Corollary 3.4(b) and Theorem 3.3(a) [with α = β = ³₄ therein]). Let Γ0 be a fla portion of the boundary, and let w solve the following wave

equation: 

Then one has the following estimate:

∂w

2As explicitly stated in [18], Theorem (3.3)(a) mandates that Neumann boundary datag be smoother in space and time in order to obtain the requisite tangential regularity; namely, g ∈ H¹⁴((0, T ) × Γ0) (again Γ0 is flat). However, in the details of proof, it is evident that indeed continuously,g ∈ L²(0, T ; H¹⁴(Γ0)) ⇒ ∂w

∂τ L²(0, T ; H⁻¹⁴(Γ0)).

Applying Lemma 5 (takingg ≡ ψ_ttherein), along withab ≤ ₂^δa²+ _2δ¹b²(taking

Coupling (23) with (21) yields

1

In turn, combining this estimate with that in (12) (again with ˜h = h), we obtain

T −  0

A subsequent application of the tangential estimate (17) (withΓ∗= Γ1) produces the following: Under Assumption (A2), the φ-component of the adjoint system (4) obeys the integral estimate,

168 G. Avalos and I. Lasiecka

3.4 Step 3 (An Estimate forψ)

Applying the multiplierψ to the second wave equation in (4), and integrating in time and space yields

A rearrangement of terms, combined with a use of Sobolev trace theory, yields



3.5 Step 4 (Final Statement of Derived Estimate)

Combining (18) and (26) (in the case that Assumption (A1) holds true), and (25) and (26) (if Assumption (A2) holds true), we have the preliminary estimate for the energy:

Lemma 6 (a) Under Assumption (A1), the solution to the PDE (4) satisfie the following estimate for all 0> 0:

where constant C is independent of time T .

(b) Under Assumption (A2), the solution to the PDE (4) satisfie the following where constant C is independent of time T .

Recall that our aim is to attain the inequalities (5) (under Assumption (A2)), and (6) (under Assumption (A2)). We can handle the terms_{T − 0}

0 E(t)dt and E(T ) in (27) and (28) by the conservation relation (9), and the lower-order terms by a compactness-uniqueness argument. Hence, the “bad terms” appear only in the inequality (27); i.e.,

_{T − 0}

In [4], we show outright the derivation of the following sequence of supporting Proposi-tions:

Proposition 7 The ψ-component of the PDE (4) satisfie the following estimate for ar-bitary 1, ₂> 0: Proposition 8 The ψ-component of the PDE system (4) obeys the estimate:



170 G. Avalos and I. Lasiecka

Proposition 9 The φ-component of the solution to the system (4) obeys the following esti-mate:

The Proposition 9 above can be used to prove the following lemma:

Lemma 10 Let  > 0 be arbitrary. Then the ψ-component of the system (4) satisfie the inequality 

Σ0

ξ²ψ_ττ² dΣ₀dt ≤ C ψ²_H2(0,T ;L²(Γ0))+ l.o.t.(φ, ψ),

where ξ is the cutoff function define in (8).

Proof of Lemma 10: Squaring both sides of theψc-wave equation in (4), integrating in time and space, and applying Cauchy–Schwartz, we obtain the inequality



Now applying the trace estimate (31) gives



Taking_ito be small enough yields the estimate



This completes the proof of Lemma 10.

Since

φ_t|_Γ0 = ψττ − ψtt on(0, T ) × Γ0, Lemma 10 immediately gives the following necessary estimate:

Corollary 11 Let  > 0 be arbitrary. Then the φ-component of the system (4) satisfie the inequality 

Σ0

ξ²φt|²_Σ0dΣ0≤ C ψ²_H2(0,T ;L²(Γ0))+ l.o.t.(φ, ψ).

In document Advances in Dynamics and Control Nonlinear Systems in Aviation Aerospace Aeronautics and Astronautics (Page 173-182)