POLYNOMIAL STABILIZATION FOR THE WAVE EQUATION WITH CONVEX-SHAPED DAMPING

HAL Id: hal-03268472

https://hal.archives-ouvertes.fr/hal-03268472

Preprint submitted on 23 Jun 2021

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

POLYNOMIAL STABILIZATION FOR THE WAVE EQUATION WITH CONVEX-SHAPED DAMPING

Chenmin Sun

To cite this version:

Chenmin Sun. POLYNOMIAL STABILIZATION FOR THE WAVE EQUATION WITH CONVEX- SHAPED DAMPING. 2021. �hal-03268472�

POLYNOMIAL STABILIZATION FOR THE WAVE EQUATION WITH CONVEX-SHAPED DAMPING

CHENMIN SUN

Abstract. We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. We show that if the damping vanishes as a H¨older function |x|^β, and in addition, the boundary of the damped region is strictly convex, the wave is stable at ratet^−1+2β+72 , which is better than the known optimal decay ratet^−1+β+31 for strip-shaped dampings of the same H¨older regularity. This illustrates the fact that the energy decay rate depends not only on the order of vanishing of the damping, but also on the shape of the damped region. The main ingredient of the proof is the averaging method (normal form reduction) developed by Hitrick and Sj¨ostrand ([Hi1][Sj]).

1. Introduction

1.1. Background. Let (M, g) be a compact Riemannian manifold with the Beltrami-Laplace operator

∆. Consider the damped wave equation

( ∂_t²u−∆u+a(z)∂tu= 0, inR+×M,

(u, ∂tu)|_t=0 = (u0, u1), in M, (1.1) where a(z) ≥ 0 is the damping. The well-posedness of (1.1) is a consequence of the Lumer-Philips theorem and the maximal dissipative property of the generator

A=

0 Id

∆ −a(z)

(1.2) on the Hilbert space H:= H¹(M)×L²(M). For a solution (u, ∂tu) ∈H¹(M)×L²(M), the energy defined by

E[u](t) := 1

2k∇u(t)k²_L2(M)+1

2k∂_tu(t)k²_L2(M)

is decreasing in time:

d

dtE[u](t) =− Z

M

a|∂_tu|² ≤0.

A basic question is the decay rate of the energy as t→+∞.

It was proved by Rauch-Taylor [RaT] (∂M = ∅) and by Bardos-Lebeau-Rauch [BLR] (∂M 6= ∅) that, for continuous dampinga∈C(M), if the setω ={a >0}verifies the geometric control condition (GCC), then there exists α0 >0 such that the uniform stabilization holds:

E[u](t)≤E[u](0)e^−α0t, ∀t≥0. (1.3)

1

If (GCC) for ω = {a > 0} is not satisfied, there are very few cases that the uniform stabilization (1.3) holds (see [BG17] and [Zh])¹. Lebeau [Le93] constructed examples with arbitrary slowly decaying initial data in the energy spaceH¹(M)×L²(M). Nevertheless, if the initial data is more regular, say inH²(M)×H¹(M), the uniform decay rate log(1 +t) holds ([Le93]). Since then, intensive research activities focus on possible improvement of the logarithmic decay rate for regular initial data, in special geometric settings.

Beyond (GCC), the determination of better decay rate for special manifoldsM and special dampings depends on at least the following factors from the existing literature:

(a) The dynamical properties for the geodesic flow of the underlying manifoldM.

(b) The dimension of the trapped rays as well as relative positions between trapped rays and the boundary ∂{a >0}of the damped region.

(c) Regularity and the vanishing properties of the dampinganear ∂{a >0}.

It is known that the energy decay rate is linked to the averaged function along the geodesic flow ϕ_t: ρ7→ A_T(a)(ρ) := 1

T Z T

0

a◦ϕt(ρ)dt, ρ∈T^∗M.

Indeed, (GCC) is equivalent to the lower boundA_T(ρ)≥c₀ >0 for some T >0 large enough on the sphere bundle S^∗M. Roughly speaking, when the geodesic flow is “unstable”, one may improve the energy decay rate (see [No] for more detailed explanation and references therein). As an illustration of (a), when M is a compact hyperbolic surface, Jin [Ji] shows the exponential energy decay rate for regular data living in H²(M)×H¹(M). In this direction, we refer also [BuC],[Ch],[CSVW],[Riv] and references therein.

The polynomial decay rate is the intermediate situation between the logarithmic decay rates and the exponential decay rates, exhibited in less chaotic geometry like the flat torus and bounded domains (see [LiR][BH05][Ph07][AL14] and references therein), where the generalized geodesic flows are unstable.

We refer [LLe],[BZu] for other situations of polynomial stabilization, where the undamped region is a submanifold.

We point out that the factor (a) is almost decisive for the observability (and exact controllability) of wave and Schr¨odinger equations. Comparing with the observability for the wave equation where (GCC) is the only criteria (see [BLR] [BG97]), the stabilization problem is more complicated. Indeed, it was shown in [AL14] (Theorem 2.3) that the observability for the Schr¨odinger semigroup in some time T > 0 implies automatically that the damped wave is stable at rate t⁻¹². However, this decay rate is not optimal in general. On the two-dimensional torus, if the damping function is regular enough and vanishing nicely, the decay rate can be very close tot⁻¹ ( [BH05][AL14]). Even when the damping is the indicator of a vertical (or horizontal) strip, the optimal decay rate is known to bet⁻²³ ([St]). These results provide evidences of factors (b) and (c) mentioned previously. As explained in [AL14], the significant difference to the controllability problem is that, there is no general monotonicity property of the type: a1 ≤ a2 implies the decay rate associated to a2 is better (or worse) than the decay rate associated toa1.

1If the trapped rays are all grazing on the boundary ∂{a >0}, the uniform stabilization (1.3) can be characterized (see [BG17] and [Zh]) by relative positions of the grazing trapped rays anda >0.

In this article, we revisit the polynomial stabilization for wave equations on flat torus. Our main result reveals that, with the same vanishing order, the curvature of the boundary of the damped region also affects the energy decay rate of damped wave equations.

1.2. The main result. We concern the polynomial decay rate for (1.1) on the two-dimensional flat torus² M =T² :=R²/(2πZ)²:

( ∂_t²u−∆u+a(z)∂tu= 0, inR+×T²,

(u, ∂_tu)|_t=0 = (u₀, u₁), in T². (1.4) To present the main result, we introduce some definitions.

Definition 1.1. We say that (1.4) is stable at rate t^−α, if there exists C > 0, such that all the solution u with initial data (u₀, u₁)∈ H² :=H²(T²)×H¹(T²) satisfies

(E[u](t))¹² ≤Ct^−αk(u₀, u1)k_H2. We say that the ratet^−α is optimal, if moreover

lim sup

t→+∞

t^α sup

06=(u0,u1)∈H²

(E[u](t))¹² k(u₀, u1)k_H2

>0.

Next we introduce the class of damping that we will consider:

Definition 1.2. The function class D^m,k,σ(T^d) (with kσ <1) is defined by:

D^m,k,σ :={f ∈C^m(T^d) :|∂^αf|.α,σ |f|^1−|α|σ,∀|α| ≤k}.

Note that D^m,k,σ1 ⊂ D^m,k,σ2, if σ1 < σ2 and kσ2 <1. This class contains non-negative functions which vanish like H¨older functions. One typical example is

a₁(z) =b(z)(max{0,|z| −0.1})^σ1 ∈ D^m,m,σ(T^d),

where σ > _m¹, b∈C^m(T^d) and inf_T^db >0. The associated damped region is {z ∈T^d:a1(z) >0}= {z ∈T^d:|z|<0.1} is a disc. Another example is the strip-shaped dampinga2(z) =a2(x) such that {z∈T² :a₂(z)>0}:= (−0.1,0.1)_x×Ty and for some m≥4,

d^m

dx^ma2(x)≤0 nearx= 0.1 and d^m

dx^ma2(x)≥0 nearx=−0.1.

It was shown in Lemma 3.1 of [BH05] that a₂ ∈ D^m,m,m1(T²).

For a ∈ D^m,k,σ, we denote by Σ_a := ∂{a(z) > 0}. Let T²_A,B := R²/(2πA×2πB) be a general flat torus defined via the covering map π_A,B : R² 7→ T²_A,B. An open set ω ⊂ T²_A,B is said to be locally strictly convex, if each component ofπ⁻¹_A,B(ω)⊂R² is strictly convex, i.e. the boundary of each component of π_A,B⁻¹ (ω) is C², with strictly positive curvature, as a curve in R². Sometimes, we also say that the boundary is locally strictly convex.

Our main result is the following:

2We take the periodic to be 2π only for simplicity of statement of the main results. The main result remains valid for any general torusR²/(2πAZ×2πBZ),A, B >0.

Theorem 1.1. Let β >4, m ≥10. Assume that a≥0, a∈ D^m,2,β1 and the open set ω := {z∈T² : a(z) > 0} is locally strictly convex. Assume that a(z) is locally H¨older of order β near ∂ω, in the sense that there exists R₀ >1, such that

1 R0

dist(z, ∂ω)^β ≤a(z)≤R0dist(z, ∂ω)^β, for z∈ω near ∂ω.

Then the damped wave equation (1.4) is stable at rate t^−1+2β+72 .

Remark 1.3. As a comparison, if a(z) = a(x) depends only on one direction (supported on the vertical strip ω) and is locally H¨older of order β near ∂ω, the optimal stable rate is t^−1+β+31 (see [Kl][DKl]) which is worse than t^−1+2β+72 . Our result provides examples that, with the same local H¨older regularity, smaller damped regions better stabilize the wave equation. To the best knowledge of the author, Theorem 1.1also provides the first example where not only the vanishing order of the damping can affect the stable rate, but also the shape of the boundary of the damped region.

ω2

ω1

a1(z) = (0.1− |z|)^β₊, a2(z) = (0.5− |x|)^β₊

T²=R²/(2πZ)²

The dampinga₁ generates better decay rate thana₂

Remark 1.4. It would be interesting to investigate whether the stable ratet^−1+2β+72 in Theorem 1.1 is optimal.

Remark 1.5. It was shown in [AL14] that, when the damping satisfies|∇a| ≤Ca^1−β1 for large enough β, then (1.4) is stable at rate t^−1+β+44 (Theorem 2.6 of [AL14]). With an additional assumption on

|∇²a| ≤ Ca^1−β2, our proof of Theorem 1.1 essentially provides an alternative proof of this rougher stable rate. Indeed, we need one more condition for |∇²a| in order to perform the normal form reduction in the Section 5. Once reduced to the one-dimensional setting, we are able to apply the same argument of Burq-Hitrik [BH05].

Remark 1.6. For the reason of exhibition, we have used a contradiction argument and the notion of semiclassical defect measures in the proof of Theorem 1.1. Comparing to the argument of [AL14], we do not make use of second semiclassical measures.

Finally we give some microlocal interpretation of Theorem1.1. It is known that the decay rate of the damped wave equation is related to the time average along geodesics (see [No]). As the damping has conormal singularities at its boundary∂ωof the damped region, the decay rate depends more precisely on the reflected and transmitted energy of waves concentrated on trapped rays. If along the conormal direction, the damping is more regular, its interaction with transversal free waves is weaker (analog to the high-low frequency interaction), hence the transmission effect is stronger than the reflection, and consequently, the decay rate is better. When the boundary∂ωof the damed region is convex, the

average of the damping along any direction gains ¹₂ local H¨older regularity, near the vanishing points along the transversal direction (see Proposition2.4for details). This heuristic indicates that, with the same local H¨older regularity near the boundary (of the damped region), the convex-shaped damping has better stable rate for (1.4), than strip-shaped dampings (that are invariant along one direction).

1.3. Resolvent estimate. The proof of Theorem 1.1 relies on Borichev-Tomilov’s criteria of the polynomial semi-group decay rate and the corresponding resolvent estimate forA given in (1.2):

Proposition 1.7([BoT10]). We have Spec(A)∩iR=∅. Then, the following statements are equivalent:

(a)

(iλ− A)⁻¹

_L(H)≤C|λ|^α1 for allλ∈R,|λ| ≥1;

(b) The damped wave equation (1.1) is stable at rate t^−α.

By Proposition 1.7 (see also Proposition 2.4 of [AL14]), the proof of Theorem (1.1) is reduced to the following semiclassical resolvent estimate:

Theorem 1.2. Let β >4, m ≥10. Assume that a≥0, a∈ D^m,2,β1 and the open set ω := {z∈T² : a(z) > 0} is locally strictly convex. Assume that a(z) is locally H¨older of order β near ∂ω, in the sense that there exists R0 >1, such that

1

R₀dist(z, ∂ω)^β ≤a(z)≤R₀dist(z, ∂ω)^β, for z∈ω near ∂ω.

Then there exist h₀ ∈(0,1)and C₀>0, such that for all 0< h≤h₀, k(−h²∆−1 +iha(z))⁻¹k_L(L2(T²)) ≤C0h^{−2−2β+52} .

As a comparison, let us recall the main resolvent estimate in [DKl][Kl], corresponding to the optimal energy decay rate t^−1+β+31 when the damping a(z) depends only on x variable and is locally H¨older of orderγ:

Theorem 1.3 ([DKl][Kl]). Let γ ≥0. Assume that W =W(x) ≥0 and {W >0} is disjoint unions of vertical strips (αj, βj)x×Ty and{W ≥0} 6=T². Assume moreover that for each j∈ {1,· · · , l},

C1Vj(x)≤W(x)≤C2Vj(x) on(αj, βj), where V_j(x)>0 are continuous functions on (α_j, β_j), satisfying

V_j(x) =







(x−αj)^γ, αj < x < 3αj +βj

4 ,

(β_j−x)^γ, αj+ 3βj

4 < x < β_j.

(1.5) Then there exist h0 ∈(0,1)and C >0 such that for all 0< h < h0,

k(−h²∆−1 +ihW(x))⁻¹k_L(L2)≤Ch^−2−γ+21 .

Furthermore, the above resolvent estimate is optimal, in the sense that there exists quasi-modes (u_h)0<h≤h₀, such that

ku_hk_L2 = 1, k(−h²∆−1 +ihW(x))u_hk_L2(T²) =O(h^2+γ+21 ).

In the rest of the article, we will prove Theorem1.2. The proof is based on a contradiction argument.

This leads to the fact that the semiclassical measure associated to quasi-modes (uh)h>0 of order o(h^2+2β+52 ) is non-zero along a periodic direction on the phase space. We then derive a contradiction by showing that the restriction of semiclassical measure to this trapped direction is zero. This will be achieved in three major steps:

• In Section 4, using the positive commutator method, we show that the semiclassical measure corresponding to the transversal high frequency part of scale &h^{−12−2β+51} is zero.

• In Section 5, we deal with scales for transversal low frequencies. Using the averaging method, we transfer quasi-modes (u_h)_h>0 to new quasi-modes (v_h)_h>0, satisfying new equations that commute with the vertical derivative. This allows us to reduce the problem to the one- dimensional setting. This is the key step of the proof, for which we need several elementary properties of the averaging operator, presented in Section 2.

• In Section 6, we prove a one-dimensional resolvent estimate (Proposition 6.1).

At the end of this article, we add two appendices. In Appendix A, we reproduce the proof of Theorem 1.3in order to be self-contained and to fix some gaps in the paper of [DKl]. In Appendix B, we present several technical results about the semiclassical pseudo-differential calculus, needed in Section5.

Acknowledgment. The author is supported by the program: “Initiative d’excellence Paris Seine”

of CY Cergy-Paris Universit´e and the ANR grant ODA (ANR-18-CE40- 0020-01).

2. The averaging properties of functions

In this section, we prove several properties of the averaging operator, which will be used in Section 5 for normal form reductions. Given a directionv∈S¹, we define the averaging operator along v:

f 7→ A(f)_v(z) := lim

T→∞

1 T

Z T 0

f(z+tv)dt.

Lemma 2.1. Assume that v∈S¹. Then for anyf ∈ D^m,k,σ(T²) andf ≥0, A(f)v∈ D^m,k,σ(T²).

Proof. First we assume that the orbitt :7→ z+tv is periodic and denote by Tv its period. Then for any f ∈ D^m,k,σ,

A(f)v(z) = 1 T_v

Z Tv

0

f(z+tv)dt.

Since the functions7→ |s|^1−|α|σ is concave, by Jensen’s inequality we have 1

T_v Z Tv

0

|f(z+tv)|^1−|α|σdt≤ 1 T_v

Z Tv

0

f(z+tv)dt 1−|α|σ

. (2.1)

Indeed, if RTv

0 f(z+tv)dt= 0, then f(z+tv)≡0 for allt∈[0, Tv], and the inequality (2.1) is trivial.

Assume now thatX₀= _T¹

v

RTv

0 f(z+tv)dt >0, then for anyX ≥0, we have X^1−|α|σ ≤X₀^1−|α|σ+ (1− |α|σ)X₀^−|α|σ(X−X0).

Replacing the inequality above byX =f(z+tv) and averaging overt∈[0, T_v], we obtain (2.1). Since by definition,|∂^αf|.α,σ|f|^1−|α|σ for all|α| ≤k, we get

|∂^α(A(f)v)(z)|.α,σ |A(f)v(z)|^1−|α|σ.

Finally, if t7→z+tv is ergodic, then A(f)v(z) is independent ofz and the proof is trivial. The proof

of Lemma2.1is now complete.

By the triangle inequality, the following Lemma is immediate:

Lemma 2.2. Let v∈S¹. For any function f onT², there holds

|A(f)_v| ≤ A(|f|)_v.

Moreover, if f1, f2 are two non-negative functions such that f1 ≤f2, we have A(f₁)v ≤ A(f₂)v.

Lemma 2.3. Assume that f ∈ D^m,k,σ(T²) and f ≥0. Denote by F(x, y) :=

Z y

−π

(f(x, y⁰)− A(f)_e2(x))dy⁰, −π < y < π, where e2 = (0,1). Then for 0≤j≤k, we have

|F(x, y)| ≤4πA(f)e2(x), |∂_x^jF(x, y)| ≤4πA(f)^1−jσ_e2 . Moreover, for all j1 ≥0,1≤j2 ≤k−j1, we have

|∂_x^j1∂_y^j2F(x, y)| ≤ A(f)_e^1−(j₂ ^1+j2−1)σ(x) +f(x, y)^{1−(j1+j2−1)σ}. Proof. Sincef ≥0, the bound for |F(x, y)|is trivial. Taking derivatives, we get

∂_x^j1∂_y^j2F(x, y) =∂_y^j2 Z y

−π

(∂_x^j1f(x, y⁰)−∂_x^j1A(f)_e2(x))dy⁰. When j₂ = 0, the absolute vaule of the above quantity can be bounded by

2πA(f)^1−j_e2 ^1σ(x) + 2π∂_x^j1A(f)e2(x)≤4π(A(f)e2)^1−j1σ(x),

thanks to Lemma 2.1 and Lemma 2.2 and Jensen’s inequality (2.1). When j2 ≥1, by definition, we have

∂_x^j1∂_y^j2F(x, y) =∂_y^j2−1∂_x^j1f(x, y)−∂_y^j2−1∂^j_x¹A(f)e2(x).

Taking the absolute value and the proof of Lemma2.3 is complete.

Finally, we prove the key geometric proposition, applying us to improve the local H¨older regularity for averaged damping functions, provided that the original damped region is strictly convex:

Proposition 2.4. Assume that a ∈ D^m,k,σ(T²) such that a(z) ≥ 0 and Σ_a is a disjoint union of strictly convex curves. Assume moreover that there exists R >0 such that for every z∈ {a >0} near Σa,

R⁻¹·dist(z,Σa)^1σ ≤a(z)≤R·dist(z,Σa)^1σ.

Then for any periodic direction v ∈ S¹, we have A(a)_v ∈ D^m,k,σ+22σ , as a one-dimensional periodic function. Furthermore, there exists R_v >0, such that for every z ∈ {A(a)_v(z) > 0}near Σ_A(a)v, we have

R⁻¹_v dist(z,Σ_A(a)v)^1σ+12 ≤ A(a)_v(z)≤R_vdist(z,Σ_A(a)v)^1σ+12. (2.2) Proof. Without loss of generality, we assume thatv= e23 and assume that Ω :={a >0} hasl=l(v) connected components Ω1,· · ·,Ω_l such that the boundary Σa,j of each Ωj is strictly convex. We first consider the situation where l= 1. By translation invariance, we may assume that Ω₁ ={a₁ >0} is contained in the fundamental domain (−Kπ, Kπ)_x×(−M π, M π)_y. Then the functionA(a)_e2 can be identified as a function on Rx,

Since Ω1 ={a₁ >0} is strictly convex, each lineP_x of R², passing through (x,0) and parallel to e2

can intersect at most 2 points of the curve Σ_a,1. Consider the function x 7→ P(x) := mes(P_x∩Ω₁).

This function is continuous and is supported on a single interval I = (α, β) ⊂ (−Kπ, Kπ). Since A(a₁)(x) = 0 if P(x) = 0, the vanishing behavior of A(a₁) is determined when x is close to α and β.

Below we only analyze A(a₁)(x) for x ∈ [α, α+), since the analysis is similar for x near β. First we observe that P_α must be tangent at a pointz0 := (α, y0) to the curve Σa,1. For sufficiently small >0, we may parametrize the curve Σ_a,1nearz₀by the functionx=α+g(y) withg(y₀) =g⁰(y₀) = 0 and g⁰⁰(y₀) =c₀ >0, thanks to the convexity. Therefore, there exists aC¹ diffeomorphismY = Φ(y) from a neighborhood of y0 to a neighborhood ofY = 0 such that Φ(y0) = 0 and g(y) =Y². For each x∈(α, α+),P_x∩ N_a1 ={(x, l⁻(x)),(x, l⁺(x))}. We have

l⁺(x) = Φ⁻¹(√

x−α), l⁻(x) = Φ⁻¹(−√

x−α).

Ω1

x z0

α α+

(x, l⁻(x)) (x, l⁻(x))

Px

Averaging improves the local H¨older regularity

Since near z0, we have a(z) ∼dist(z,Σa) = dist(z,Σa,1) ∼(x−(α+g(y)))

1 σ

+. For x∈(α, α+), we have

A(a₁)(x) = 1 2Kπ

Z Kπ

−Kπ

a1(x, y)dy= 1 2Kπ

Z l⁺(x) l⁻(x)

a1(x, y)dy

= 1 2Kπ

Z

√x−α

−√ x−α

|x−α−Y²|^1σ|(Φ⁻¹)⁰(Y)|dY ∼_K,σ|x−α|^1σ+12.

3In general, by liftingT²toT²v, the covering mapT²v7→T²is isometric, hence each connected component of the lifted damped regions is still convex. See Subsection3.2for more details about the changing coordinates.

Finally, sincea₁∈ D^m,k,σ, forj ≤k,x∈(α, α+),

|∂_x^jA(a₁)(x)| ≤ 1 2Kπ

Z Kπ

−Kπ

|∂^j_x¹a1(x, y)|dy.K

Z l⁺(x) l⁻(x)

a^1−jσ₁ (x, y)dy .|x−α|^{1−jσσ +21} ∼(A(a₁)(x))^1−σ+22σj.

This implies thatA(a₁)∈ D^m,k,σ+22σ .

To complete the proof of Proposition 2.4, we need to deal with the situation wherel > 1. In this case, the supports ofA(a_j) may overlap. By linearity and the inequality

|∂_x^j0A(a₁+· · ·a_l)| ≤

l

X

j=1

|∂^j_x⁰A(a_j)| ≤C(a₁,· · ·, a_l)A(a₁+· · ·+a_l)^1−2σj

0 σ+2, we deduce thatA(a)∈ D^m,k,σ+22σ . It remains to show (2.2). We define

S+:=

j∈ {1,· · ·, l}: Z x0+

x0

A(a_j)(x)dx >0, ∀ >0 , S−:=

j∈ {1,· · ·, l}: Z x0

x0−

A(a_j)(x)dx >0, ∀ >0 .

Observe thatx₀ ∈Σ_A(a) if and only ifA(a_j)(x₀) = 0 for allj∈ {1,· · · , l}andS₊∪S−6=∅.Note that forj∈S±, we have

dist(x,ΣA(a))^12+σ1 = dist(x,ΣA(a_j))^21+σ1 ∼ A(a_j)(x), ∀x∓x0>0 nearx0,

and A(a_j) = 0, in a neighborhood of x0, for all j /∈S+∪S−. Summing over j ∈S+∪S−, we obtain

(2.2). The proof of Proposition 2.4is now complete.

3. Contradiction argument and the first microlocalization

3.1. A priori estimate and the contradiction argument. We will adapt basic conventions for notations in the semiclassical analysis ([Zw12]). Denote byP_h=−h²∆−1 and we fix the parameter σ = ¹_β throughout this article. We denote byδ_h the small parameter such that δ_h →0 as h→0. We denote by~=h¹²δ

1 2

h second semiclassical parameter. For the proof of Theorem1.2, we fixδ_h=h^2β+52 . Theorem1.2 is the consequence of the following key proposition:

Proposition 3.1. Let u_h be a sequence of quasi-modes of width h²δ_h with δ_h =h^2β+52 i.e.

(Ph+iha)uh =fh =o_L²(h²δh).

Then if u_h =O_L²(1), we have u_h =o_L²(1).

The proof of Proposition3.1will occupy the rest of this article. We argue by contradiction. Up to extracting a subsequence and renormalization, we may assume that

ku_hk_L2(T²)= 1. (3.1)

The following a priori estimate is simple:

Lemma 3.2. We have the following apriori estimates:

(a) ka^1/2u_hk_L2 =o(h¹²δ

1 2

h) =o(~).

(b) kh∇u_hk²_L2 − ku_hk²_L2 =o(h²δ_h).

Proof. Multiplying the equation (P_h+iha)u_h =f_h by u_h, and integrating by part, we get kh∇u_hk²_L2 − ku_hk²_L2+ih(auh, uh)_L² = (fh, uh)_L².

Taking the imaginary part and the real part, we obtain (a) and (b), with respectively.

Assume thatµ is the semi-classical defect measure associated to the sequence (uh)h>0, that is, for any symbol a∈C_c^∞(T^∗T²),

h→0lim(Op_h(a)u_h, u_h)_L² =hµ, ai. (3.2) Indeed, since the sequence (uh)h>0 is bounded L²(T²), there exist a subsequence, still denoted as (u_h)_h>0, and a Radon measure µ on T^∗T², such that (3.2) holds. For the proof of this existence of semi-classical measure, one may consult Chapter 5 of [Zw12].

Lemma 3.3. we have

supp(µ){(z, ζ)∈T^∗T²:|ζ|= 1} and µ|_ω×R2 = 0, where ω ={z∈T² :a >0}.

Proof. This property follows from the standard elliptic regularity which only requires quasi-mode for P_hof orderO_L²(h). The damping termihau_hcan be roughly treated as an errorO_L²(h). For example,

one can consult Theorem 5.4 of [Zw12] for a proof.

Letϕt be the geodesic flow onT^∗T². We recall the following invariant property of the semiclassical measure:

Lemma 3.4. The semiclassical measure µ is invariant by the flow ϕt, i.e.

ϕ^∗_tµ=µ.

Proof. This property holds true for quasi-mode of Ph of order o_L²(h). From (a) of Lemma 3.2, we have f_h −ihau_h = o_L²(h). The proof then follows from a standard propagation argument (see for

example Theorem 5.5 of [Zw12]).

3.2. Reducing to periodic trapped directions. We perform the change of coordinate, following [BZ12]. By identifying T²=R²/(2πZ)², we decomposeS¹ as rational directions

Q:={ζ ∈S¹ :ζ = (p, q)

pp²+q², (p, q)∈Z², gcd(p, q) = 1}

and irrational directions R := S¹\ Q. Since the orbit of an irrational direction is dense, by Lemma 3.3and Lemma 3.4, we have

µ=µ|_T2×Q = X

ζ0∈Q

µζ0.

It suffices to show that, for each ζ₀= √^(p0,q0)

p²₀+q₀² ∈ Q, the restricted measureµ_ζ0 is zero⁴. Denote by Λ₀, the rank 1 submodule of Z² generated by Ξ₀ = (p₀, q₀). Denote by

Λ^⊥₀ :={ζ ∈R² :ζ·Ξ0 = 0}

the dual of the submodule Λ0. Denote by

T²_Ξ0 := (RΛ0/(2πΛ0))×(Λ^⊥₀/(2πZ)²∩Λ^⊥₀).

Then we have a natural smooth covering mapπ_Ξ0 :T²_Ξ0 7→T² of degree p

p²₀+q₀². The pullback of a 2π×2π periodic function f satisfies

(π^∗_Ξ0f)(X+kτ, Y +lτ) = (π_Ξ^∗₀f)(X, Y), k, l∈Z,(X, Y)∈R², whereτ = 2πp

p²₀+q₀².

T²Ξ₀ Ξ₀= (3,−2)

Λ0

Λ^⊥₀

By pulling back to the torusT²_Ξ0, we can identify the sequence (u_h)⊂L²(T²) as (π_Ξ^∗

0u_h)⊂L²(T²_Ξ0) and in this new coordinate system, ζ0 = _|Ξ^Ξ0

0| = (0,1). The semi-classical defect measure µ on T^∗T² is the pushforward of the semi-classical measure associated to (π^∗_Ξ

0u_h). Since the period of the torus T²_Ξ0 has no influence of the analysis in the sequel⁵, we will still use the notation T² to stand forT²_Ξ0, the variables z = (x, y), ζ = (ξ, η) to stand for variables Z = (X, Y),Ξ on T^∗T_Ξ²

0, and assuming the period to be 2π for simplicity. The only thing that will change is that the pre-image of the damping π_Ξ⁻¹

0(ω) is now a disjoint union of p

p²₀+q₀² copies of ω on T²_Ξ0, and each component is still strictly convex. For this reason, in the hypothesis of Proposition2.4, we assume that the boundary of{a >0}

is made of disjoint unions of strictly convex curves.

4. Analysis of the transversal high frequencies

Recall thatζ₀ = (0,1), and our goal is to show thatµ1_ζ=ζ0 = 0. In this section, we deal with the transversal high frequencies of size O(~⁻¹) and use the positive commutator method to show these portions are propagated into the flowout of the damped region ω.

4In fact, we only need to consider finitely manyζ0 ∈ Q, since whenp²0+q²0 is large enough, the associated periodic direction is close to an irrational direction and the trajectory will eventually enterω.

5As we fix one periodic directionζ0 and consider the semi-classical limith→0, one does not need to worry about the fact that the period 2πp

p²₀+q₀² may be very large.

For the geodesic flow ϕ_t on T^∗T² and ζ ∈S¹, we denote byϕ_t(·, ζ) :T²→T² the projection of the flow mapϕt. By shifting the coordinate, we may assume

ω0:=I0×Ty ⊂ [

t∈[0,2π]

ϕt(·, ζ₀)(ω), where

I₀ = (−σ₀, σ₀)⊂π₁({z:a(z)≥c₀ >0}), for someσ₀< π 100

and π1 :T²7→Tx the canonical projection. Therefore, there exist 0 >0, c0>0 sufficiently small and T₀>0, such that for any |ζ|= 1, z₀ ∈ω₀,|ζ−ζ₀| ≤₀,

Z T0

0

(a◦ϕ_t)(z₀, ζ)dt≥c₀>0. (4.1) ω0

supp(a) ζ0= (0,1)

To simplify the notation, for any functionb, we denote by A(b)(x) =A(b)_ζ0(x) = 1

2π Z π

−π

b(x, y)dy.

To microlocalize the solution nearζ₀, we pickψ₀∈C_c^∞(R) and consider u¹_h :=ψ₀ ^{hD x}

0

u_h, then (Ph+iha)u¹_h =f_h¹ :=ψ0

hDx

0

fh−ih ψ0

hDx

0

, a uh. Letµ1 be the semiclassical measure ofu¹_h, thenµ1 =ψ ^ξ

0

µ.

Lemma 4.1. We have

ka¹²u¹_hk_L2 =o(h¹²δ

1 2

h), kf_h¹k_L2 =o(h²δ_h).

Proof. It suffices to show that f_h¹ =o_L²(h²δ_h), and the first assertion follows from the same proof of (a) of Lemma3.2. By the symbolic calculus,

i

ψ₀ hD_x 0

, a

=⁻¹₀ hOp_h ψ₀⁰ ξ 0

∂_xa

+O_L2(⁻²₀ h²).

From the pointwise inequality for the non-negative functiona:

|∇a(x)|² ≤2kak_W^2,∞a(x), (4.2) we have, for someC >0,

Ca−

⁻¹₀ ψ⁰₀ ξ ₀∂_xa

2≥0 onT^∗T². Therefore, by the sharp G˚arding inequality, we get

⁻¹₀ Op_h ψ₀⁰ ξ ₀

∂xa u_h

_L2 ≤C|(au_h, u_h)_L²|¹² +Ch¹²ku_hk_L2.

Together with (a) of Lemma3.2, this implies that

ih ψ0

hDx

0

, a uh

L² =o(h⁵²) =o(h²δh).

The proof of Lemma4.1 is complete.

Recall that ~=h¹²δ

1 2

h. Letψ∈C_c^∞(R), and consider

v_h =ψ(~D_x)u¹_h, w_h = (1−ψ(~D_x))u¹_h. (4.3) In this decomposition,w_h corresponds to the transversal high frequency part, whilev_h corresponds to the transversal low frequency part for which will be treated in next sections. Note that

(P_h+iha)v_h=ψ(~Dx)f_h¹−ih[ψ(~Dx), a]u¹_h =:r_1,h (4.4) and

(P_h+iha)w_h= (1−ψ(~Dx))f_h¹+ih[ψ(~Dx), a]u¹_h =:r_2,h. (4.5) We need to show that the commutator termh[ψ(~D_x), a]u¹_h can be viewed as the remainder:

Lemma 4.2. We have

kr_1,hk_L2 +kr_2,hk_L2 =o(h²δh) =o(h~²).

Consequently, from Lemma 3.2,

ka¹²v_hk_L2 +ka²¹w_hk_L2 =o(h¹²δ

1 2

h) =o(~).

Proof. According to the symbolic calculus,

i[ψ(~D_x), a] =~Op_~(ψ⁰(ξ)∂_xa) +C~²Op_~(∂²_ξψ·∂_x²a) +O_L(L2)(~³).

Using the fact that a ∈ D^m,2,σ and σ < ¹₄, we have a¹² ∈ C_c²(T²). Applying the special symbolic calculus LemmaB.4 (b) with κ=∂_x(a¹²), b₂ =a¹² and ϕ=ψ⁰, we have

1

2Op_~(ψ⁰(ξ)∂xa) =Op_~(ψ⁰(ξ)∂x(a¹²))a¹² −1

i~Op_~(ψ⁰⁰(ξ)∂x(a¹²)·∂x(a¹²)) +O_L(L2)(~²).

Applying LemmaB.4 (a) with κ=b1 =∂x(a¹²), ϕ=ψ⁰⁰, we have

−~

iOp_~(ψ⁰⁰(ξ)∂x(a¹²)·∂x(a¹²)) =−~

iOp_~(ψ⁰⁰(ξ)∂x(a¹²))∂x(a¹²) +O_L(L2)(~²).

Since ψis only a function of ξ and a¹²∂_xa∈L^∞, applying LemmaB.3, we have

Op_~ ψ⁰(ξ)∂xa a¹²

a¹²u¹_h

L²(T²)≤Cka²¹u¹_hk_L2 =o(~).

For the term Op_~(∂_ξ²ψ∂_x²a), since |∂_x²a|.a^1−2σ .a¹², by the sharp G˚arding inequality, Op_~(Ca− |ψ⁰⁰(ξ)∂_x²a|²)u¹_h, u¹_h

L² ≥ −C~ku¹_hk²_L2. Thus k~²Op_~(ψ⁰⁰(ξ)∂_x²a)u¹_hk_L2 =O(~

5

2). Therefore, by Caldr´on-Vaillancourt (Theorem B.1), we can write

ih[ψ(~D_x), a] =h~A_~a¹² +h~²B_~∂_x(a¹²) +O_L(L2)(h~³),

with A_~, B_~ bounded operators on L², uniformly in ~. Since |∂_x(a¹²)| .a^12−σ, by (a) of Lemma 3.2 and the interpolation, we get

kih[ψ(~D_x), a]u¹_hk_L2 =o(h~²) +O(h~³).

This completes the proof of Lemma 4.2.

Remark 4.3. Compared to [AL14] where the damping only satisfiesa∈W^k0,∞(T²) and|∇a|.a^1−σ, our assumption a ∈ D^m,2,σ is slightly stronger, in order to ensure the commutator of the damping term is still a remainder. Indeed, here we chose ~ =h¹²δ

1 2

h h¹² while in [AL14], the authors chose

~=h^α, α < ¹₃ (see the sentence after Proposition 7.2 of [AL14]). The use of the sharp G˚arding as in [AL14] would not get o(h~²) for the remaindersr1,h, r2,h. We also remark that a direct application of the Caldr´on-Vaillancourt theorem in the symbolic calculus requiresa¹² ∈W^3,∞. Since we assume only a∈ D^m,2,σ, (thusa¹² ∈W^2,∞), we need to exploit the special structure of the commutator [ψ(~D_x), a]

and apply the special symbolic calculus LemmaB.4.

Recall that ω₀ =I₀×Ty. Lemma 4.4. We have

kw_h1_ω0k_L2+kh∇w_h1_ω0k_L2 =O(h¹²), as h→0.

Proof. The proof follows from the classical propagation argument, using the geometric control condi- tion. Take small intervalsI₀⁰ ⊂Tx, I1 = (σ1, σ2)⊂Ty, such thatI0 ⊂I₀⁰ and ω1 =I₀⁰ ×I1 ⊂ {a≥δ0} for some δ0 > 0. For any z0 = (x0, y0) ∈ ω0, by the geometric control condition, there exist T₁ > 0, δ₁ > 0, δ₂ > 0, and the small neighborhood U = (x₀−δ₁, x₀ +δ₁)×(y₀−δ₁, y₀+δ₁) of z0, such that for all|ζ−ζ0| ≤0, z∈U, we have

z+sζ∈ω₁, s∈[T₁−δ₂, T₁+δ₂].

In particular, a(z+sζ) ≥δ0. Without loss of generality, we assume that π > σ2 > σ1 > y0+δ1 >

y0−δ1 >−π. Pick two cutoffs χ1(x), χ2 ≥ 0, supported in (x0−δ1, x0+δ1), (y0−δ1, y0+δ1) and equal to 1 on (x₀−δ₁/2, x₀+δ₁/2),(y₀ −δ₁/2, y₀ +δ₁/2), with respectively. Let χ₀ ∈ C_c^∞(R) be a cutoff near|ζ−ζ0| ≤0. For anys≥0, define the symbol

b_s(z, ζ) :=χ₀(ζ)·(χ₁⊗χ₂)◦ϕ−s(z, ζ) =χ₀(ζ)χ₁(x−sξ)χ₂(y−sη).

ω₀ ω1

U

ϕT₀(·, ζ)(U)⊂ω1

Direct computation yields d

ds(Op_h(b_s)w_h, w_h)_L2 =(Op_h(∂_sb_s)w_h, w_h)_L2 =−(Op_h(ζ· ∇_zb_s)w_h, w_h)_L2.

Integrating this equality froms= 0 to s=T₀,

(Op_h(bT1)wh, wh)_L²−(Op_h(b0)wh, wh)_L² =− Z T1

0

(Op_h(ζ· ∇_zbs)wh, wh)_L²ds. (4.6) Note that for fixed s∈[0, T1],

i

h[P_h,Op_h(b_s)] = 2Op_h(ζ· ∇_zb_s) +O_L(L2)(h), we have

(Op_h(b₀)w_h, w_h)_L² =(Op_h(b_T1)w_h, w_h)_L² + i 2h

Z _T1

0

([P_h,Op_h(b_s)]w_h, w_h)_L²+O(h). (4.7) Using the equation

Phwh =r2,h−ihawh, we have

1

h([Ph,Op_h(bs)]wh, wh)_L² =2i

h Im(Op_h(bs)wh, r2,h−ihawh)_L²

=o(h^1+δ) +O(1)ka¹²whk_L2ka¹²Op_h(bs)whk_L2

=O(h), (4.8)

where to the last step, we write

a¹²Op_h(bs) = Op_h(bs)a¹² + [a¹²,Op_h(bs)]

and use the last assertion of Lemma 4.2, as well as the symbolic calculus.

Finally, from the support property of b₀(z) =χ₀(ζ)χ₁(x)χ₂(y), we have a(z)χ0(ζ)≥δ0bT1(z, ζ).

Thus by the sharp G˚arding inequality,

(Op_h(b_T1)w_h, w_h)_L² ≤δ⁻¹₀ (Op_h(a(z)χ₀(ζ))w_h, w_h)_L² +O(h)≤Cδ₀⁻¹ka¹²w_hk²_L2 +O(h).

Combining this with (4.7),(4.8) and the last assertion of Lemma 4.2, we deduce that kw_h1Uk_L2 = O(h¹²), kh∇w_h1Uk_L2 = O(h¹²). By the partition of unity of ω0, we complete the proof of Lemma

4.4.

Now we are ready to prove the main result in this section, that is the transversal high frequency part is of ordero_L2(1):

Proposition 4.5. We have kw_hk_L2 =O(δ_h) =O(~h⁻¹²), as h→0.

Proof. We use the positive commutator method to detect the transversal propagation, similarly as in [BS19]. Recall thatω0 =I0×T and I0 = (−σ₀, σ0), σ0 < ₁₀₀^π . Take φ=φ(x) ∈ C^∞(Tx; [0,1]) such that:

supp(1−φ)⊂I₀, φ≡0 near[−σ0

2 ,σ0

2 ], supp(φ⁰)⊂I₀. Denote by

X(x) := (x+π)1_{−π≤x<−}^σ0

2 + (x−π)1^σ0 2 ≤x<π,