Jin Xu

Luo Jiawen

Wang Zhiqiang

Liang Zhenguo

REDUCIBILITY OF 1-D QUANTUM HARMONIC OSCILLATOR WITH A NEW UNBOUNDED OSCILLATORY PERTURBATIONS

, we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schrödinger equation

can be reduced in H 1 (R) to an autonomous system for most values of the frequency vector ω, where Λ ⊂ R \ {0}, |Λ| < ∞ and x := √ 1 + x 2 . The functions a k (θ) and b k (θ) are analytic on T n σ and µ ≥ 0 will be chosen according to the value of β. Comparing with [25], the novelty is that the phase functions of oscillatory integral are more degenerate when β > 1.

Introduction of the main results

1.1. Main theorem. Following [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] we continue to consider the reducibility for the time dependent Schrödinger equation i∂ t u = H ǫ (ωt)u, x ∈ R,

H ǫ := -∂ xx + x 2 + ǫX(x, ωt), (1.1)
where X(x, θ) = x µ k∈Λ a k (θ) sin(k|x| β ) + b k (θ) cos(k|x| β ) with Λ ⊂ R \ {0}, |Λ| < ∞ and x := √ 1 + x 2 . The functions a k (θ) and b k (θ) are analytic on T n σ = {a + bi ∈ C n /2πZ n : |b| < σ} with σ > 0 and β > 1 and µ ≥ 0 will be chosen in the following. We first introduce some functions and spaces. Hermite functions. The harmonic oscillator T = -∂ xx + x 2 has eigenfunctions (h m ) m≥1 , so called the Hermite functions, namely

T h m = (2m -1)h m , h m L 2 (R) = 1, m ≥ 1. (1.2)
Linear spaces. For s ≥ 0 denote by H s the domain of T s 2 endowed by the graph norm. For s < 0, the space H s is the dual of H -s . Particularly, for s ≥ 0 a integer we have

H s = {f ∈ L 2 (R) : x α ∂ β f ∈ L 2 (R), ∀ α, β ∈ N 0 , α + β ≤ s}.
We also define the complex weighted-ℓ 2 -space ℓ 2 s := {ξ = (ξ m ∈ C, m ≥ 1) : m≥1 m s |ξ m | 2 < ∞}. To a function u ∈ H s we associate the sequence ξ of its Hermite coefficients by the formula u = m≥1 ξ m h m (x). In the following we will identify the space H s with ℓ 2 s by endowing both space the norm

u H s = ξ s =   m≥1 m s |ξ m | 2   1 2 . Define l * = l(β, µ) =        1 4 β 6 -µ , 1 < β < 2, 1 4 2 9 -µ , β = 2, 1 4
β-2
4β-2 -µ , β > 2.

(1.3)

Then we can state our main theorem.

Theorem 1.1. Assume a k (θ) and b k (θ) are analytic on T n σ with σ > 0 and β > 1 and µ satisfies

0 ≤ µ <      β 6 , 1 < β < 2, 2 9 , β = 2, β-2 4β-2 , β > 2.
(1.4)

There exists ǫ * > 0 such that for all 0 ≤ ǫ < ǫ * there is a closed set D ǫ ⊂ D 0 = [0, 2π] n of asymptotically full measure such that for all ω ∈ D ǫ , the linear Schrödinger equation (1.1) reduces to a linear autonomous equation in H 1 . More precisely, for any ω ∈ D ǫ there exists a linear isomorphism Ψ ∞ ω,ǫ (θ) ∈ L(H s ′ ) with 0 ≤ s ′ ≤ 1, analytically dependent on θ ∈ T n σ/2 and unitary on L 2 (R), where Ψ ∞ ω,ǫ -Id ∈ L(H 0 , H 2l * )∩L(H s ′ ) and a bounded Hermitian operator Q ∈ L(H 1 ) such that t → u(t, •) ∈ H 1 satisfies (1.1) if and only if t → v(t, •) = Ψ ∞ ω,ǫ u(t, •) ∈ H 1 satisfies the autonomous equation

i∂ t v = -v xx + x 2 v + ǫQ(v),
furthermore, there are constants C, K > 0 such that

Meas(D 0 \ D ǫ ) ≤ Cǫ 3l * 2(2l * +5)(2l * +1) , Q L(H p ,H p+4l * ) + ∂ ω Q L(H p ,H p+4l * ) ≤ K, ω ∈ D ǫ , p ∈ N, Ψ ∞ ω,ǫ (θ) -Id L(H 0 ,H 2l * ) , Ψ ∞ ω,ǫ (θ) -Id L(H s ′ ) ≤ Cǫ 1 3 , (ω, θ) ∈ D ǫ × T n σ/2 .
Consequently, Theorem 1.1 follows in the considered range of parameters the H 1 norms of the solutions are all bounded forever and the spectrum of the corresponding operator is pure point. 1.2. Related results and a critical lemma. In the following we recall some relevant reducibility results. For 1-D quantum harmonic oscillators('QHO' for short) with periodic or quasi-periodic in time bounded perturbations see [START_REF] Combescure | The quantum stability problem for time-periodic perturbations of the harmonic oscillator[END_REF], [START_REF] Enss | Bound states and propagating states for time -dependent hamiltonians[END_REF] , [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF], [START_REF] Liang | Reducibility of 1-D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials[END_REF], [START_REF] Wang | Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay[END_REF] and [START_REF] Wang | Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations[END_REF] .

In [START_REF] Bambusi | Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods[END_REF] Bambusi and Graffi proved the reducibility of 1-D Schrödinger equation with an unbounded time quasiperiodic perturbation in which the potential grows at infinity like |x| 2l with a real l > 1 and the perturbation is bounded by 1 + |x| β with β < l -1. The reducibility in the limiting case β = l -1 was proved by Liu and Yuan in [START_REF] Liu | Spectrum for quantum duffing oscillator and small-divisor equation with large-variable coefficient[END_REF]. Recently, the results in [START_REF] Bambusi | Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods[END_REF] and [START_REF] Liu | Spectrum for quantum duffing oscillator and small-divisor equation with large-variable coefficient[END_REF] have been improved by Bambusi in [START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF][START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF], in which he firstly obtained the reducibility results for 1-D QHO with unbounded perturbations.

It seems that the reducibility method in [START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF][START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF] is hard to be applied for 1-D Schrödinger equations with the unbounded oscillatory perturbations(see remark 2.7 in [START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF]). The authors [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], [START_REF] Liang | Reducibility of 1-d Schrödinger equation with unbounded oscillation perturbations[END_REF] solved this problem by Langer's turning point and oscillatory integral estimates. We remark that the critical step in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] is to build up a decay estimate of the integral R x µ e ikx h m (x)h n (x)dx, in which the phase functions of oscillatory integral are φ mn (x

) := ζ m (x) -ζ n (x) + kx, where ζ m (x) = x Xm √ λ m -t 2 dt with X m = √ 2m -1 = √ λ m .
Comparing with [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], in this paper the phase functions Ψ mn (x) := ζ m (x) -ζ n (x) + kx β with β > 1 are more degenerate. For 1 < β ≤ 2, we use a similar method as [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF]. The most difficult part as [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] is the integral

Xm-X ν 2 m X 2 3 m
x µ e ikx β h m (x)h n (x)dx where ν 2 = 1 -β 3 for 1 < β < 2 and ν 2 = 5 9 when β = 2. We have to discuss different cases in order to obtain a suitable lower bound of the derivatives of the phase function. For β > 2 we find a new simple proof which follows from Corollary 3.2 in [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF], Lemma 6.1 and a straightforward computation. As Lemma 1.7 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] we have the following.

Lemma 1.2. Assume h m (x) satisfies (1.2). For any k = 0, R x µ e ik|x| β h m (x)h n (x)dx ≤ C • C k,β (mn) -l(β,µ) , m, n ≥ 1
for some absolute constant C > 0, where µ ≥ 0, β > 1, l(β, µ) defined in (1.3) and

C k,β =      |β(β -1)(β -2)k| -1 3 ∨ |k| -1 ∨ |k| 1 4-2β , 1 < β < 2, |k| -1 ∨ 1, β = 2, |βk| -1 ∨ 1, β > 2.
Remark 1.3. In fact l(1, µ) = 1 12 -µ 4 has been proved in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF].

In the end we review some relative results. Eliasson-Kuksin [START_REF] Eliasson | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF] initiated to prove the reducibility for PDEs in high dimension. See [START_REF] Grébert | On reducibility of quantum harmonic oscillator on R d with quasiperiodic in time potential[END_REF] and [START_REF] Liang | Reducibility of quantum harmonic oscillator on R d with differential and quasi-periodic in time potential[END_REF] for higher-dimensional QHO with bounded potential. The first reducibility result for n-D QHO was proved in [START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in ddimensions with polynomial time-dependent perturbation[END_REF] by Bambusi-Grébert-Maspero-Robert. Towards other PDEs with unbounded perturbations see the reducibility results by Montalto [START_REF] Montalto | A reducibility result for a class of linear wave equations on T d[END_REF] for linear wave equations on T d and Bambusi, Langella and Montalto [START_REF] Bambusi | Reducibility of non-resonant transport equation on with unbounded perturbations[END_REF] for transportation equations( [START_REF] Feola | Reducibility of first order linear operators on tori via Moser's theorem[END_REF]). Feola and Grébert [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF] set up a reducibility result for a linear Schrödinger equation on the sphere S n with unbounded potential( [START_REF] Feola | Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential[END_REF]).

The reducibility results usually imply the boundedness of Sobolev norms. Delort [START_REF] Delort | Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential[END_REF] constructed a t s/2 -polynomial growth for 1-D QHO with certain time periodic perturbation( [START_REF] Maspero | Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations[END_REF]). Basing on a Mourre estimate, Maspero [START_REF] Maspero | Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon[END_REF] proved similar results for 1-D QHO and half -wave equation on T and the instability is stable in some sense. For a polynomial periodic or quasi-periodic perturbations relative with 1-D QHO we refer to [START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in ddimensions with polynomial time-dependent perturbation[END_REF], [START_REF] Graffi | Absolute continuity of the Floquet spectrum for a nonlinearly forced harmonic oscillator[END_REF], [START_REF] Liang | 1-d quasi-periodic quantum harmonic oscillator with quadratic time-dependent perturbations: Reducibility and growth of Sobolev norms[END_REF] and [START_REF] Luo | Growth of Sobolev norms in 1-D Quantum harmonic oscillator with polynomial time quasi-periodic perturbation[END_REF]. For 2-D QHO with perturbation which is decaying in t, Faou-Raphaël [START_REF] Faou | On weakly turbulent solutions to the perturbed linear harmonic oscillator[END_REF] constructed a solution whose H 1 -norm presents logarithmic growth with t. For 2-D Schrödinger operator Thomann [START_REF] Thomann | Growth of Sobolev norms for linear Schrödinger operators[END_REF] constructed explicitly a traveling wave whose Sobolev norm presents polynomial growth with t, based on the study in [START_REF] Schwinte | Growth of Sobolev norms for coupled Lowest Landau Level equations[END_REF] for linear Lowest Landau equations(LLL) with a time-dependent potential. There are also many literatures, e.g. [START_REF] Bambusi | Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori[END_REF][START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF][START_REF] Berti | Long time dynamics of Schrödinger and wave equations on flat tori[END_REF][START_REF] Bourgain | Growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potentials[END_REF][START_REF] Bourgain | Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential[END_REF][START_REF] Fang | On growth of Sobolev norms in linear Schrödinger equations with time dependent Gevrey potentials[END_REF][START_REF] Maspero | On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms[END_REF][START_REF] Wang | Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations[END_REF], which are closely relative to the upper growth bound of the solution in Sobolev space.

Our article is organized as follows: in Section 2 we state the reducibility theorem, i.e. Theorem 2.1. In Section 3, through checking all the assumptions in Theorem 2.1 we prove Theorem 1.1. In Section 4 we prove Lemma 1.2 for 1 < β ≤ 2 and the case for β > 2 is delayed in Section 5. Some auxiliary lemmas are presented in the Appendix.

Notation: We use the notations N 0 = {0, 1, 2, • • • }, N = {1, 2, • • • }, T n = R n /2πZ n and T n σ = {a + bi ∈ C n /2πZ n : |b| < σ}.
For Hilbert spaces H 1 , H 2 we denote by L(H 1 , H 2 ) the space of bounded linear operators from H 1 to H 2 and write L(H 1 , H 1 ) as L(H 1 ) for simplicity.

A KAM theorem

Following [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF][START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF] we introduce the KAM Theorem from [START_REF] Liang | Reducibility of quantum harmonic oscillator on R d with differential and quasi-periodic in time potential[END_REF] especially for 1-D case.

(ab) α |A b a | < ∞.
We also denote M + α be the subspace of M α satisfying that an infinite

matrix A ∈ M + α if |A| α+ := sup a,b∈N (ab) α (1 + |a -b|)|A b a | < ∞.
In fact one can prove that for all α > 0, a matrix in M + α defines a bounded operator on ℓ 2 0 . However, when α ∈ (0, 1 2 ), we can't insure that M α ⊂ L(ℓ 2 0 , ℓ 2 s ) for any s ∈ R. This means that P x makes no sense when the perturbation operator P ∈ M α and x ∈ ℓ 2 0 . Fortunately, from Lemma 2.1 in [START_REF] Grébert | On reducibility of quantum harmonic oscillator on R d with quasiperiodic in time potential[END_REF] or Lemma 2.2 in [START_REF] Liang | Reducibility of quantum harmonic oscillator on R d with differential and quasi-periodic in time potential[END_REF] one can show

M α ⊂ L(ℓ 2 1 , ℓ 2 -1
) and thus the reducibility in H 1 can be built up in Theorem 1.1 instead of L 2 .

Parameters. In this paper ω will play the role of a parameter belonging to D 0 = [0, 2π] n . All the constructed maps will depend on ω with C 1 regularity. When a map is only defined on a Cantor subset of D 0 the regularity is understood in Whitney sense.

A class of quadratic Hamiltonians. Let D ⊂ D 0 , α > 0 and σ > 0. We denote by M α (D, σ) the set of mappings as

T n σ × D ∋ (θ, ω) → Q(θ, ω) ∈ M α which is real analytic on θ ∈ T n σ and C 1 continuous on ω ∈ D. And we endow this space with the norm [Q] D,σ α := sup ω∈D,|ℑθ|<σ |k|=0,1 |∂ k ω Q(θ, ω)| α .
The subspace of M α (D, σ) formed by

F (θ, ω) such that ∂ k ω F (θ, ω) ∈ M + α , |k| = 0, 1, is denoted by M + α (D, σ) and endowed with the norm [F ] D,σ α+ := sup ω∈D,|ℑθ|<σ |k|=0,1 |∂ k ω F (θ, ω)| α+ .
Besides, the subspace of M α (D, σ) that are independent of θ will be denoted by M α (D) and for

N ∈ M α (D), [N ] D α := sup ω∈D,|k|=0,1 |∂ k ω N (ω)| α .
2.2. The reducibility theorem. In this section we present an abstract reducibility theorem for a quadratic Hamiltonian quasiperiodic in time of the form

H(t, ξ, η) = ξ, N η + ǫ ξ, P (ωt)η , (ξ, η) ∈ X 1 ⊂ X 0 , (2.1)
and the corresponding Hamiltonian system

ξ = -iN ξ -iǫP T (ωt)ξ, η = iN η + iǫP (ωt)η, (2.2)
where N = diag{λ a , a ∈ N} satisfies the following spectrum assumptions: Hypothesis A1-Asymptotics. There exist positive constants c 0 , c 1 , c 2 such that

c 1 a ≥ λ a ≥ c 2 a and |λ a -λ b | ≥ c 0 |a -b|, ∀ a, b ∈ N.
Hypothesis A2-Second Melnikov condition in measure estimates. There exist positive constants α 1 , α 2 and c 3 such that the following holds: for each 0 < κ < 1 4 and K > 0 there exists a closed subset

D ′ = D ′ (κ, K) ⊂ D 0 with Meas(D 0 \D ′ ) ≤ c 3 K α1 κ α2 such that for all ω ∈ D ′ , k ∈ Z n with 0 < |k| ≤ K and a, b ∈ N we have |k • ω + λ a -λ b | ≥ κ(1 + |a -b|).
Then we can state our reducibility results.

Theorem 2.1. Given a nonautonomous Hamiltonian (2.1), we assume that (λ a ) a∈N satisfies Hypothesis A1-A2 and P (θ) ∈ M α (D 0 , σ) with α, σ > 0. Let γ 1 = max{α 1 , n + 3} and γ 2 = αα2 2αα2+5 , then there exists ǫ * > 0 such that for all 0 ≤ ǫ < ǫ * there are (i) a Cantor set

D ǫ ⊂ D 0 with Meas(D 0 \ D ǫ ) ≤ Cǫ 3δα 2α+1 for a δ ∈ (0, γ2 24 );
(ii) a C 1 family in ω ∈ D ǫ (in Whitney sense), linear, unitary, analytically dependent on θ ∈ T n σ/2

and symplectic coordinate transformation Φ ∞ ω (θ) :

X 0 → X 0 , (ω, θ) ∈ D ǫ × T n σ/2 , of the form (ξ + , η + ) → (ξ, η) = Φ ∞ ω (θ)(ξ + , η + ) = (M ω (θ)ξ + , M ω (θ)η + ), where Φ ∞ ω (θ) -Id satisfies for 0 ≤ s ′ ≤ 1 Φ ∞ ω (θ) -Id L(X0,X2α) , Φ ∞ ω (θ) -Id L(X s ′ ) ≤ Cǫ 1 3 ; (iii) a C 1 family of autonomous quadratic Hamiltonian in normal forms H ∞ (ξ + , η + ) = ξ + , N ∞ (ω)η + = m≥1 λ ∞ m ξ +,m η +,m , ω ∈ D ǫ , where N ∞ (ω) = diag{λ ∞ m , m ∈ N} is diagonal and is close to N in the sense of [N ∞ (ω) -N ] Dǫ α ≤ Cǫ, such that H(t, Φ ∞ ω (ωt)(ξ + , η + )) = H ∞ (ξ + , η + ), t ∈ R, (ξ + , η + ) ∈ X 1 , ω ∈ D ǫ .

Application to the quantum harmonic oscillator

In this section we will prove Theorem 1.1 by applying Theorem 2.1 to the original equation (1.1). Following the strategies in [START_REF] Eliasson | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF], we expand u on the Hermite basis (h m ) m≥1 as well as ū by the following formula

u = m≥1 ξ m h m , ū = m≥1 η m hm .
Therefore the equation (1.1) is equivalent to a nonautonomous Hamiltonian system

ξm = -i ∂H ∂ηm = -i(2m -1)ξ m -iǫ P T (ωt)ξ m , ηm = i ∂H ∂ξm = i(2m -1)η m + iǫ (P (ωt)η) m , m ≥ 1, (3.1)
where H(t, ξ, η) = ξ, N η + ǫ ξ, P (ωt)η , (ξ, η) ∈ X 1 ⊂ X 0 , and N = diag{2m -1, m ≥ 1} and

P n m (ωt) = k∈Λ a k (ωt) R x µ sin k|x| β h m (x)h n (x)dx + k∈Λ b k (ωt) R x µ cos k|x| β h m (x)h n (x)dx, (3.2)
where the frequencies ω ∈ D 0 = [0, 2π] n are the external parameters.

The spectrum assumptions can be easily checked by the following two lemmas.

Lemma 3.1. When λ a = 2a -1, a ∈ N, Hypothesis A1 holds true with c 0 = c 2 = 1 and c 1 = 2. Lemma 3.2. When λ a = 2a-1, a ∈ N, Hypothesis A2 holds true with α 1 = n+1, α 2 = 1, c 3 = c(n) and D 0 = [0, 2π] n , D ′ = {ω ∈ [0, 2π] n : |k • ω + j| ≥ κ(1 + |j|), ∀ j ∈ Z, k ∈ Z n \ {0}}.
The following lemma is a direct corollary of Lemma 1.2. More precisely, in new variables given in Theorem 2.1, (ξ, η) = (M ω ξ + , M ω η + ), system (3.1) is conjugated into an autonomous system of the form:

ξ+,a = -iλ ∞ a (ω)ξ +,a , η+,a = iλ ∞ a (ω)η +,a , a ∈ N.
Therefore the solution subject to the initial datum (ξ + (0), η + (0)) reads

(ξ + (t), η + (t)) = (e -itN∞ ξ + (0), e itN∞ η + (0)), t ∈ R,
where

N ∞ = diag{λ ∞ a , a ≥ 1}.
Then the solution of (1.1) with the initial datum

u 0 (x) = a≥1 ξ a (0)h a (x) ∈ H 1 is formulated by u(t, x) = a≥1 ξ a (t)h a (x) with ξ(t) = M ω (ωt)e -itN∞ M T ω (0)ξ(0)
, where we use the fact

M ω -1 = M T ω since M is unitary. Now we define the coordinate transformation Ψ ∞ ω (θ) by Ψ ∞ ω (θ)   a≥1 ξ a h a (x)   := a≥1 M T ω (θ)ξ a h a (x) = a≥1 ξ +,a h a (x).
Then we have u(t, x) satisfies (1.1) if and only if v(t, x) = Ψ ∞ ω (ωt)u(t, x) satisfies the autonomous

equation i∂ t v = -v xx + x 2 v + ǫQ(v), where ǫQ   a≥1 ξ a h a (x)   = a≥1 ((N ∞ -N 0 )ξ) a h a (x) = a≥1 (λ ∞ a -λ a )ξ a h a (x).
The rest estimates are standard (see Lemma 3.4 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] for the details).

4. Proof of Lemma 1.2 when 1 < β ≤ 2 For reader's convenience, we will use the notations in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF]. In the whole section we will always suppose µ ≥ 0 and don't point it out in the following lemmas.

The eigenfunction of the quantum oscillator operator

T is h n (x) = (n!2 n π 1 2 ) -1 2 e -1 2 x 2 H n (x)
, where H n (x) is the n -th Hermite polynomial. Since h n (x) is an even (or odd) function when n is odd (or even), we only need to estimate

+∞ 0 x µ e ikx β h m (x)h n (x)dx, 1 ≤ m ≤ n. (4.1)
By Lemma 4.4 and Remarks 4.5, 4.6 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], when m > m 0 ,

h m (x) = (λ m -x 2 ) -1 4 ( πζ m 2 ) 1 2 H (1) 1 3
(ζ m ) + (λ m -x 2 ) -1 4 ( πζ m 2 ) 1 2 H (1) 1 3 (ζ m )O( 1 λ m ) := ψ (m) 1 (x) + ψ (m) 2 (x),
where

ζ m (x) = x Xm λ m -t 2 dt with X 2 m = λ m (X m > 0). Otherwise, when m ≤ m 0 , then h m (x) = ψ (m) 1 (x) + ψ (m) 2 (x) for x > 2X m0 , where ψ (m) 1 (x) = (λ m -x 2 ) -1 4 ( πζm 2 ) 1 2 H (1) 1 3
(ζ m ) and

|ψ (m) 2 (x)| ≤ C x 2 |ψ (m) 1 (x)|.
Following the same strategies in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] we distinguish 3 cases to estimate (4. .

Proof. When x ≤ X 0 , from Hölder inequality and n, m < C 0 , we have

X0 0 x µ e ikx β h m (x)h n (x)dx ≤ X µ 0 ≤ C (mn) 1 4 ( β 6 -µ)
.

where X 0 is a positive constant depending on

C 0 only. When x > X 0 , |X 2 m -x 2 | -1 4 < 1, we have πζm 2 H (1) 1 3
(ζ m ) ≤ e -|ζm| by Lemma 5.4 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF]. By Lemma 5.5 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] we have

|ζ m | ≥ 2 √ 2 3 X 1 2 m (x -X m ) 3 2 ≥ x -X 0 for x > X 0 . Thus +∞ X0 x µ e ikx β h m (x)h n (x)dx ≤ +∞ X0
x µ e -2(x-X0) dx ≤ Ce 2X0 ≤ C

(mn)

1 4 ( β 6 -µ)
.

Lemma 4.2. For m ≤ m 0 and n ≥ C 0 and µ ≥ 0,

+∞ 0 x µ e ikx β h m (x)h n (x)dx ≤ C (mn) 1 4 ( β 6 -µ)
.

Proof. We divide the integral into two parts.

+∞ 0 x µ e ikx β h m (x)h n (x)dx = X 1 3 n 0 + +∞ X 1 3 n . Since x > 2X m0 , we have |h m (x)| ≤ 2(x 2 -X 2 m ) -1 4 | πζ m 2 H (1) 1 3 (ζ m )| ≤ 2e -|ζm| .
On the other hand, for x ∈ [0, X

1 3 n ], one has |h n (x)| ≤ C(X 2 n -x 2 ) -1 4 . Note 1 < β < 2, it follows X 1 3 n 0 x µ e ikx β h m (x)h n (x)dx ≤ C X 1 3 n 0 x µ (X 2 n -x 2 ) -1 4 dx ≤ CX -1 6 + µ 3 n ≤ C (mn) 1 4 ( β 6 -µ) . When x ≥ X 1 3
n ≥ 2X m0 , from Lemma 5.5 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], e -|ζm| ≤ e -C(x-Xm) . Note h n (x) L 2 = 1, from Hölder inequality,

+∞ X 1 3 n x µ e ikx β h m (x)h n (x)dx ≤ C +∞ X 1 3 n x 2µ e -Cx dx 1 2 ≤ e -CX 1 3 n .

4.2.

The estimate for Case III. In the following we will turn to the complicated case when m, n > m 0 . We divide the integral into two parts +∞ 0

x µ e ikx β h m (x)h n (x)dx = Xn 0 + +∞ Xn . We first go to the latter case +∞ Xn .

The integral on

[X n , +∞). Lemma 4.3. For m 0 < m ≤ n, +∞ Xn x µ e ikx β h m (x)h n (x)dx ≤ C m 1 12 -µ 4 n 1 12 -µ 4 ≤ C (mn) 1 4 ( β 6 -µ)
.

We first estimate the integral on [2X n , +∞]. The following result is clear from [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF]. x µ e ikx β h m (x)h n (x)dx ≤ e -Cn .

For the integral on [X n , 2X n ], we prove that Lemma 4.5. For m 0 < m ≤ n,

2Xn Xn x µ e ikx β h m (x)h n (x)dx ≤ C m 1 12 -µ 4 n 1 12 -µ 4 .
Proof. As [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], we only need to estimate the following integral I := 2Xn Xn

x µ e ikx β ψ (m)

1 (x)ψ (n)
1 (x)dx since the rest three ones are higher order. I can be divided into two parts as

|I| = 2Xn Xn+X 1 3 n + Xn+X 1 3 n Xn x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤CX µ n 2Xn Xn+X 1 3 n + Xn+X 1 3 n Xn ψ (m) 1 (x)ψ (n) 1 (x) dx.
From Lemma 5.5 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], when x ≥ X n + X

1 3 n , |ζ n | ≥ 2 √ 2 3 X 1 2 n (x -X n ) 3 2 ≥ 2 √ 2 3 X n . Thus 2Xn Xn+X 1 3 n ψ (m) 1 (x)ψ (n) 1 (x) dx ≤C 2Xn Xn+X 1 3 n (x 2 -λ m ) -1 4 (x 2 -λ n ) -1 4 e -|ζn| dx ≤Ce -2 √ 2 3 Xn 2Xn Xn+X 1 3 n (x 2 -λ n ) -1 2 dx ≤ Ce -2 √ 2 3 Xn .
For the second part,

Xn+X 1 3 n Xn ψ (m) 1 (x)ψ (n) 1 (x) dx ≤C Xn+X 1 3 n Xn (x 2 -λ m ) -1 4 (x 2 -λ n ) -1 4 dx ≤ C Xn+X 1 3 n Xn (x 2 -λ n ) -1 2 dx ≤CX -1 2 n Xn+X 1 3 n Xn (x -X n ) -1 2 dx ≤ CX -1 3 n . It follows |I| ≤ CX µ-1 3 n ≤ C m 1 12 - µ 4 n 1 12 - µ 4
.

Combining with the above two lemmas we finish Lemma 4.3.

In the following we will estimate the integral on [0, X n ], which is the most complicated case. Note m 0 < m ≤ n, the following two cases have to be considered respectively: I.

X n > 2X m ; II. X m ≤ X n ≤ 2X m .
4.2.2. the integral estimate on [0, X n ] when X n > 2X m . Our aim in this part is to build the following Lemma 4.6.

For k = 0, if X n > 2X m and 1 < β ≤ 2, then Xn 0 x µ e ikx β h m (x)h n (x)dx ≤ C(|k| ι ∨ 1) m 1 8 -µ 4 n 1 12 -µ 4
, where m 0 < m ≤ n and ι =

1 4-2β , 1 < β < 2, 0, β = 2.
As [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF] we will use the following notation in the remained parts. We denote

f m (x) = ∞ 0 e -t t -1 6 1 + it 2ζm -1 6 dt and f n (x) = ∞ 0 e -t t -1 6 1 + it 2ζn -1 6 dt. When x ∈ [0, X m ], from a straightforward computation we have ψ (m) 1 (x) = (X 2 m -x 2 ) -1 4 πζ m 2 H (1) 1 3 (ζ m ) = (X 2 m -x 2 ) -1 4 e i(ζm-π 6 -π 4 ) Γ 5 6 ∞ 0 e -t t -1 6 1 + it 2ζ m -1 6 dt = C(X 2 m -x 2 ) -1 4 e iζm(x) f m (x).
Similarly, ψ [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF] ). Thus, Corollary 4.7. For x ∈ [0, X m ) and m ≤ n,

(n) 1 (x) = C(X 2 n -x 2 ) -1 4 e -iζn(x) f n (x). For x ∈ [0, X m ], denote Ψ(x) = (X 2 m - x 2 ) -1 4 (X 2 n -x 2 ) -1 4 • f m (x)f n (x) and g(x) = (ζ n (x) -ζ m (x) -kx β ) ′ = X 2 n -x 2 -X 2 m -x 2 - kβx β-1 ,then Ψ ′ (x) = 1 2 x(X 2 m -x 2 ) -5 4 (X 2 n -x 2 ) -1 4 • f m (x)f n (x) + 1 2 x(X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -5 4 • f m (x)f n (x) +(X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 • f ′ m (x)f n (x) + f m (x)f ′ n (x) . When x ∈ [0, X m ], |f m (x)| ≤ Γ( 5 6 ) and |f n (x)| ≤ Γ( 5
|Ψ ′ (x)| ≤C x(X 2 m -x 2 ) -5 4 (X 2 n -x 2 ) -1 4 + x(X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -5 4 + (X 2 m -x 2 ) 1 4 (X 2 n -x 2 ) -1 4 X m (X m -x) 3 + (X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) 1 4 X n (X n -x) 3 =C J 1 + J 2 + J 3 + J 4 ≤ C(J 1 + J 3 ).
We first estimate the integral on [0, X m -X

-1 3 m ]. Lemma 4.8. For k = 0, 1 < β < 2, if X n > 2X m , then Xm-X -1 3 m 0 x µ e ikx β h m (x)h n (x)dx ≤ C(|k| 1 4-2β ∨ 1) m 1 8 -µ 4 n 1 8 -µ 4
, where m 0 < m ≤ n.

Proof. First we estimate the main term of the integral

Xm-X -1 3 m 0 x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx = C Xm-X -1 3 m 0
x µ e i(ζm-ζn+kx β ) Ψ(x)dx, by method of oscillating integral estimate, where

Ψ(x) = (X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 • f m (x)f n (x). We discuss two different cases. Case 1: k ≤ X 2-β n 8 . In this case, we have g(x) ≥ X 2 n -x 2 - X 2 n 4 -x 2 -βkX β-1 n ≥ X n 2 -2 • X 2-β n 8 X β-1 n ≥ X n 4 .
Thus, by Lemma 6.1,

Xm-X -1 3 m 0 e i ζm-ζn+kx β Xn Xn x µ Ψ(x)dx ≤CX -1 n   ( x µ Ψ) (X m -X -1 3 m ) + Xm-X -1 3 m 0 ( x µ Ψ) ′ (x) dx   ≤CX -1 n   X µ m Ψ(X m -X -1 3 m ) + Xm-X -1 3 m 0 2 x µ (J 1 + J 3 ) + µ x µ-1 x x |Ψ(x)| dx   ≤CX -1 n X µ m   Ψ(X m -X -1 3 m ) + Xm-X -1 3 m 0 (J 1 + J 3 ) dx   + CµX -1 n Xm-X -1 3 m 0 x µ-1 |Ψ(x)| dx. Clearly, X µ m Ψ(X m -X -1 3 m ) ≤ CX µ m X 2 m -(X m -X -1 3 m ) 2 -1 4 X 2 n -(X m -X -1 3 m ) 2 -1 4 ≤ CX -1 3 +µ m , and Xm-X -1 3 m 0 µ x µ-1 |Ψ(x)| dx ≤ C X 2 m -(X m -X -1 3 m ) 2 -1 4 X 2 n -(X m -X -1 3 m ) 2 -1 4 Xm 0 µx µ-1 dx ≤C X 2 m -(X m -X -1 3 m ) 2 -1 4 X 2 n -(X m -X -1 3 m ) 2 -1 4 X µ m ≤ CX -1 3 +µ m , together with Xm-X -1 3 m 0 J 1 dx ≤ C Xm-X -1 3 m 0 x(X 2 m -x 2 ) -5 4 (X 2 m -x 2 ) -1 4 dx ≤ CX -1 3 m ,
and

Xm-X -1 3 m 0 J 3 dx ≤ CX -1 m Xm-X -1 3 m 0 (X m -x) -3 dx ≤ CX -1 3 m .

So we obtain

Xm-X

-1 3 m 0 x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤ CX -1 3 +µ m X -1
n . Now we turn to remained three terms. Since m 0 < m ≤ n,

Xm-X -1 3 m 0 x µ e ikx β ψ (m) 2 (x)ψ (n) 1 (x)dx ≤ C Xm-X -1 3 m 0 X -2+µ m (X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 dx ≤ CX -3 2 +µ m X -1 2 n ≤ Cn -1 4 + µ 2 .
Similarly, when m 0 < m ≤ n, we have

Xm-X -1 3 m 0 x µ e ikx β ψ (m) 1 (x)ψ (n) 2 (x)dx ≤ Cn -1+ µ 2 and Xm-X -1 3 m 0 x µ e ikx β ψ (m) 2 (x)ψ (n) 2 (x)dx ≤ Cn -1+ µ 2 .
Thus,

Xm-X -1 3 m 0 x µ e ikx β h m (x)h n (x)dx ≤ C n 1 4 -µ 2 ≤ C m 1 8 -µ 4 n 1 8 -µ 4 , m 0 < m ≤ n. Case 2: k > X 2-β n 8 > 0. Since m ≤ n, we have 2n ≤ (8k) 2 2-β + 1. It follows that Xm-X -1 3 m 0 x µ e ikx β h m (x)h n (x)dx ≤ CX µ m ≤ CX µ m m 1 8 n 1 8 m 1 8 n 1 8 ≤ Ck 1 4-2β m 1 8 -µ 4 n 1 8 -µ 4 .
Combining with these two cases we finish the proof. Lemma 4.9.

For k = 0, if X n > 2X m , then Xm-X 2 3 m 0 x µ e ikx 2 h m (x)h n (x)dx ≤ C m 1 8 -µ 4 n 1 8 -µ 4
, where m 0 < m ≤ n.

Proof. We first estimate the main part of the integral. By the oscillating integral estimate,

Xm-X 2 3 m 0 x µ e ikx 2 ψ (m) 1 (x)ψ (n) 1 (x)dx = C Xm-X 2 3 m 0 x µ e i(ζm-ζn+kx 2 ) Ψ(x)dx, where Ψ(x) = (X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 • f m (x)f n (x). Since g ′′ (x) ≥ g ′′ (0) = 1 Xm -1 Xn ≥ 1 2 X -1 m , by Lemma 6.1, Xm-X 2 3 m 0 e i ζm-ζn +kx 2 Xn Xn x µ Ψ(x)dx ≤CX 1 3 m   ( x µ Ψ) (X m -X 2 3 m ) + Xm-X 2 3 m 0 ( x µ Ψ) ′ (x) dx   ≤CX 1 3 m   X µ m Ψ(X m -X 2 3 m ) + Xm-X 2 3 m 0 2 x µ (J 1 + J 3 ) + µ x µ-1 x x |Ψ(x)| dx   ≤CX 1 3 m X µ m   Ψ(X m -X 2 3 m ) + Xm-X 2 3 m 0 (J 1 + J 3 ) dx   + CµX 1 3 m Xm-X 2 3 m 0 x µ-1 |Ψ(x)| dx.
The estimate comes from three terms. Clearly, for x ∈ [0, X m -X

2 3
m ] we have

|Ψ(x)| ≤ C(X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 ≤ C(X m X n ) -1 4 (X m -x) -1 4 (X n -x) -1 4 ≤ CX -5 12 m X -1 2 n . It follows that X µ m Ψ(X m -X 2 3 m ) ≤ CX µ-5 12 m X -1 2 n , and
Xm-X 2 3 m 0 µ x µ-1 |Ψ(x)| dx ≤ CX -5 12 m X -1 2 n Xm 0 x µ-1 dx ≤ CX µ-5 12 m X -1 2 n , together with Xm-X 2 3 m 0 J 1 dx ≤ C Xm-X 2 3 m 0 x(X 2 m -x 2 ) -5 4 (X 2 n -x 2 ) -1 4 dx ≤ CX -5 12 m X -1 2 n ,
and

Xm-X 2 3 m 0 J 3 dx ≤ CX -3 4 m X -1 2 n Xm-X 2 3 m 0 (X m -x) -11 4 dx ≤ CX -5 12 m X -1 2 n , we obtain Xm-X 2 3 m 0 x µ e ikx 2 ψ (m) 1 (x)ψ (n) 1 (x)dx ≤ CX -1 12 +µ m X -1 2 n ≤ C(X m X n ) µ 2 -1
4 . The estimate of rest parts of the integral is similar with Lemma 4.8. Thus, Xm-X , where m 0 < m ≤ n and

ν 1 = -1/3, 1 < β < 2, 2/3, β = 2. Proof. First, Xm Xm-X ν 1 m x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤C Xm Xm-X ν 1 m x µ (X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 dx ≤CX -1 4 +µ m (X 2 n -X 2 m ) -1 4 Xm Xm-X ν 1 m (X m -x) -1 4 dx ≤CX -1 4 +µ m (X 2 n - X 2 n 4 ) -1 4 X 3 4 ν1 m ≤ CX -3 8 + 3 8 ν1+ µ 2 m X -3 8 + 3 8 ν1+ µ 2 n . Similarly, Xm Xm-X ν 1 m x µ e ikx β ψ (m) j1 (x)ψ (n) j2 (x)dx ≤ CX -3 8 + 3 8 ν1+ µ 2 m X -3 8 + 3 8 ν1+ µ 2 n
, j 1 , j 2 ∈ {1, 2}.

Thus we finish the proof.

Lemma 4.11. When X n > 2X m and 1 < β ≤ 2,

Xn Xm x µ e ikx β h m (x)h n (x)dx ≤ C m 1 8 -µ 4 n 1 12 -µ 4
, where m 0 < m ≤ n.

Proof. When X n > 2X m0 , X m + X -1 3 m ≤ Xn 2 + 1 ≤ 3 4 X n . It follows Xm+X -1 3 m Xm x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤CX µ m Xm+X -1 3 m Xm (x 2 -X 2 m ) -1 4 (X 2 n -x 2 ) -1 4 dx ≤CX -1 4 +µ m X 2 n -(X m + X -1 3 m ) 2 -1 4 Xm+X -1 3 m Xm (x -X m ) -1 4 dx ≤CX -1 2 + µ 2 m X -1 2 + µ 2 n . From X n > 2X m , we have X n -X -1 3 n ≥ 3
2 X m , together with Lemma 5.5 in [START_REF] Liang | Reducibility of 1-d quantum harmonic oscillator equation with unbounded oscillation perturbations[END_REF], thus

3 2 Xm Xm+X -1 3 m x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤CX µ m 3 2 Xm Xm+X -1 3 m (x 2 -X 2 m ) -1 4 (X 2 n -x 2 ) -1 4 e iζm dx ≤CX -1 4 +µ m X 2 n -(X n -X -1 3 n ) 2 -1 4 3 2 Xm Xm+X -1 3 m (x -X m ) -1 4 e -(x-Xm) dx ≤CX -1 4 +µ m X -1 6 n ∞ 0 t -1 4 e -t dt ≤ CX -1 4 + µ 2 m X -1 6 + µ 2 n . When x ≥ 3 2 X m , x -X m ≥ 1 3 x, it follows Xn-X -1 3 n 3 2 Xm x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤C Xn-X -1 3 n 3 2 Xm x µ (x 2 -X 2 m ) -1 4 (X 2 n -x 2 ) -1 4 e iζm dx ≤CX -1 4 m X 2 n -(X n -X -1 3 n ) 2 -1 4 Xn-X -1 3 n 3 2 Xm (x -X m ) -1 4 +µ e -(x-Xm) dx ≤CX -1 4 m X -1 6 n ∞ 0 t -1 4 +µ e -t dt ≤ CX -1 4 m X -1 6 n . Thus, Xn Xn-X -1 3 n x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤C Xn Xn-X -1 3 n x µ (x 2 -X 2 m ) -1 4 (X 2 n -x 2 ) -1 4 e iζm dx ≤C (X n -X -1 3 n ) 2 -X 2 m -1 4 X -1 4 n Xn Xn-X -1 3 n (X n -x) -1 4 x µ e -1 3 x dx ≤CX -1 2 m X -1 4 n Xn Xn-X -1 3 n (X n -x) -1 4 dx ≤CX -1 2 m X -1 2 n ≤ C(X m X n ) -1 2 + µ 2 .
Combining with all the above, we have

Xn Xm x µ e ikx β ψ (m) 1 (x)ψ (n) 1 (x)dx ≤ C m 1 8 - µ 4 n 1 12 - µ 4
. The rest estimates are similar as above.

Combining with Lemma 4.8, 4.9, 4.10, 4.11, we finish the proof of Lemma 4.6.

The integral estimate on

[0, X n ] when X m ≤ X n ≤ 2X m . One can split the integral into Xn 0 x µ e ikx β h m (x)h n (x)dx =   X 2 3 m 0 + Xm-X ν 2 m X 2 3 m + Xn Xm-X ν 2 m   x µ e ikx β h m (x)h n (x)dx,
and estimate them respectively, where

ν 2 = 1 -β 3 , 1 < β < 2, 5 9 , β = 2.
Our main aim in this part is to build the following two lemmas.

Lemma 4.12. For X m ≤ X n ≤ 2X m and k = 0,

Xn 0 x µ e ikx 2 h m (x)h n (x)dx ≤ C(|k| -1 ∨ 1) m 1 18 -µ 4 n 1 18 -µ 4 , where C > 0, m 0 < m ≤ n. Lemma 4.13. For X m ≤ X n ≤ 2X m , k = 0 and 1 < β < 2, Xn 0 x µ e ikx β h m (x)h n (x)dx ≤ C(|k| -1 ∨ |β(β -1)(β -2)k| -1 3 ∨ 1) m β 24 -µ 4 n β 24 -µ 4 , where C > 0, m 0 < m ≤ n.
From a straightforward computation we have

Lemma 4.14. For X m ≤ X n ≤ 2X m and 1 < β ≤ 2, X 2 3 m 0 x µ e ikx β h m (x)h n (x)dx ≤ C m 1 12 -µ 4 n 1 12 -µ 4 ,
where C > 0, m 0 < m ≤ n.

Next we estimate the integral on [X

2 3 m , X m -X 5 9
m ], for which we discuss different cases as the following.

Lemma 4.15. If X m ≤ X n ≤ 2X m , when k > 0 and 0 ≤ X 2 n -X 2 m ≤ kX 4 3 m , then Xm-X 5 9 m X 2 3 m x µ e ikx 2 h m (x)h n (x)dx ≤ Ck -1 2 m 7 36 -µ 4 n 7 36 -µ 4
, where m 0 < m ≤ n.

Proof. We first estimate

Xm-X 5 9 m X 2 3 m x µ e ikx 2 ψ (m) 1 (x)ψ (n) 1 (x)dx = C Xm-X 5 9 m X 2 3 m
x µ e i(ζm-ζn+kx 2 ) Ψ(x)dx .

Notice that

g ′ (x) ≤ kX 4 3 m X m 2X 14 9
m -X x µ e i 2 3k (ζm-ζn+kx

2 )• 3k 2 (X 2 m -x 2 ) -1 4 (X 2 n -x 2 ) -1 4 • f m (x)f n (x)dx ≤Ck -1 2   ( x µ Ψ)(X m -X 5 9
m ) +

Xm-X Notice that g(x) = X 2 n -x 2 -X 2 m -x 2 -βkx β-1 ≥ -kX Proof. First,

. Combining with the above two cases, we finish the proof. Lemma 4.12 and 4.13 follow directly by the lemmas in subsection 4.2.3. Combining with all the lemmas in this section we finish the proof of Lemma 1.2 for 1 < β ≤ 2.

Proof of Lemma 1.2 when β > 2

In the following we will suppose that m ≤ n without losing the generality. As the case 1 < β ≤ 2 we only need to estimate the integral on [0, ∞]. We first apply Theorem 3.1 in [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF] to obtain the integral estimates on [2X m , ∞) as follows.

Lemma 5.1. For µ ≥ 0, then we have ∞ 2Xm

x µ e ikx β h m (x)h n (x)dx ≤ C (mn) .

Proof. By Theorem 3.1 in [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF] we have ∞ 2Xm

x µ e ikx β h m (x)h n (x)dx

≤C ∞ 2Xm x -1 x µ+1 |h m (x)||h n (x)|dx ≤C x -1 1 2 L 2 (R) • x µ+1 h m (x) 1 2 L 2 ({x:|x|≥2Xm}) • h n L ∞ (R) ≤CX -1 6 m X -1 6 n ≤ C(mn) 1 4 (µ-1 3 ) . ≤C|kβ| -1 X -ν3(β-1)+1 n X µ m ≤ C|kβ| -1 (X m X n ) µ 2 X -β-2
4β-2 n ≤C|kβ| -1 (mn) .

Proof. The proof is similar as Lemma 5.2, we omit it.

Hence, combining the above four Lemmas we obtain Lemma 1.2 when β > 2.

Acknowledgements The authors were partially supported by Natural Science Foundation of Shanghai (19ZR1402400) and NSFC grant (12071083).

Appendix

In the following we will introduce two technical lemmas without proof. The first lemma provides an estimate of oscillatory integral. For more details, see [START_REF] Stein | Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals[END_REF]. Lemma 6.1. ( [START_REF] Stein | Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals[END_REF]) Suppose φ is real-valued and smooth in (A, B), ψ is complex-valued, and that |φ (k) (x)| ≥ 1 for all x ∈ (A, B). Then The second lemma shows that the L p -norm of the eigenfunction of harmonic oscillating operator can be controlled by its L 2 -norm. Lemma 6.2 ( [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF]). Sppose that h(x) is the eigenfunction of Herimite operator with the corresponding eigenvalue µ 2 . Then h L p ≤ µ ρ(p) h L 2 , where

ρ(p) = -( 1 2 -1 p ), 2 ≤ p < 4, -1 3 + 1 3 ( 1 2 -1 p ), 4 < p ≤ ∞.

Lemma 3 . 3 .

 33 Assume that a k (θ) and b k (θ) are analytic on T n σ for any nonzero k ∈ Λ with σ > 0 and β > 1 and µ satisfies (1.4), then there exists α = l(β, µ) > 0 such that the matrix functionP (θ) defined by (3.2) is analytic from T n σ into M α .Proof of Theorem 1.1. Expanding the Hermite basis (h m ) m≥1 , the Schrödinger equation (1.1) becomes Hamiltonian system (3.1), which is the form of equation (2.2) with λ a = 2a -1. By lemmas above, we can apply Theorem 2.1 to (3.1) with γ 1 = n + 3, γ 2 = α 2α+5 and δ = γ2 48 . This follows Theorem 1.1.

  1): I. m, n < C 0 := 2 8 m 3 0 ; II. m ≤ m 0 and n ≥ C 0 ; III. m, n > m 0 . 4.1. the estimates for Case I and Case II. Lemma 4.1. When n, m < C 0 , +∞ 0x µ e ikx β h m (x)h n (x)dx ≤ C

Lemma 4 . 4 .

 44 For m 0 < m ≤ n, +∞ 2Xn

x 4 ,

 4 µ e ikx 2 h m (x)h n (x)dx ≤ m 0 < m ≤ n. Lemma 4.10. If X n > 2X m and 1 < β ≤ 2, then Xm Xm-X ν 1 mx µ e ikx β h m (x)h n (x)dx ≤

  computation, g ′′ (x) ≥ 0. It follows |g ′ (x)| ≥ 3 2 k on x ∈ [X

  The estimates for the rest three terms are easier. In fact, by m > m 0 ,

Lemma 5 . 4 . 3 n 0 x

 5430 For X ν3 n < 2X m and β > 2, thenX ν µ e ikx β h m (x)h n (x)dx ≤ C

e

  iλφ(x) ψ(x)dx ≤ c k λ -1/k |ψ(B)| + B A |ψ ′ (x)|dx holds when: (i) k ≥ 2, or (ii) k = 1 and φ ′ (x) is monotonic.The bound c k is independent of φ, ψ and λ.

  2.1. Setting. Linear spaces. For p ≥ 0 we defineX p := ℓ 2 p × ℓ 2 p = {ζ = (ζ a = (ξ a , η a ) ∈ C 2 ) a∈N , ζ p < ∞} with ζ 2 p = a∈N a p (|ξ a | 2 + |η a | 2). We equip the space with the symplectic

	structure i	a∈N	dξ a ∧ η a .
	Infinite matrices. Denote by M α the set of infinite matrices A : N × N → C with the norm |A| α := sup a,b∈N

9 m

.

Lemma 4.20. For k = 0, X m ≤ X n ≤ 2X m and 1 < β ≤ 2, we have

Proof. For the integral on [X m , X n ], we discuss two different cases:

We split the integral into three parts. First,

Finally, from

Xn Xn-X -ν 2 n

x µ e ikx β ψ

Thus,

Next we consider the integral on [0, 2X m ]. Define ν 3 = 5β-4 2(β-1)(2β-1) ∈ (0, 1) when β > 2. In the following we will discuss two different cases depending on whether X ν3 n ≥ 2X m or not.

The case X ν3 n ≥ 2X m . In this case we directly estimate the rest integral on [0, 2X m ]. Lemma 5.2. For X ν3 n ≥ 2X m and β > 2, then 2Xm 0

x µ e ikx β h m (x)h n (x)dx ≤ C

(mn)

.

Proof. From Lemma 6.2, 2Xm 0

x µ e ikx β h m (x)h n (x)dx

The case X ν3 n < 2X m . In this case we split the rest integral into two parts as follows. Lemma 5.3. For k = 0, X ν3 n < 2X m and β > 2, then


domain: sdu phys

text:
Leonardo B Monachesi
email: lmonachesi@unrn.edu.ar

Fabio I Zyserman
email: zyserman@fcaglp.unlp.edu.ar

Laurence Jouniaux
email: l.jouniaux@unistra.fr

Arthur H Thompson

Electromagnetic/acoustic coupling in partially-saturated porous rocks: An extension of Pride's theory

Keywords: Electrokinetics, partially-saturated porous media, Electric double layer, Volume Averaging Method

In this paper a set of equations governing the electromagnetic/acoustic coupling in partially-saturated porous rocks in the low-frequency regime is derived. The equations are obtained by volume averaging of fundamental electromagnetic and mechanical equations valid at the porescale, following the same procedure as the one developed in the seminal paper of S. Pride for porous media where the fluid electrolyte fully saturates the pore space. In the present approach it is assumed that the porous rock is partially saturated with a wetting-fluid electrolyte (water)

and a non-wetting fluid (air). We also assume that an electromagnetic/mechanical coupling exists at the water-solid and water-air contact surfaces through adsorbed excess charges balanced by mobile ions in the water. The proposed approach is valid at the low-frequency regime, where capillary pressure perturbations can be safely neglected. The governing equations thus derived are similar to the ones obtained by Pride with the main difference that the various coefficients, including the electrokinetic coupling coefficient and electric conductivity appearing in the transport equations are functions of the water saturation and depend on electrical and topological properties of both electric double layers.

Introduction

The phenomenon of seismic-to-electromagnetic energy conversion along with its counterpart have been known for a long time by the geophysical community, as well as the existence of several theoretical models and experiments aiming to describe it, i.e., (Frenkel, 1944;[START_REF] Neev | Electrokinetic effects in fluid-saturated poroelastic media[END_REF][START_REF] Thompson | Geophysical applications of electrokinetic conversion[END_REF]. The work of [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], presenting a closed set of equations modeling the propagation of coupled mechanical and electromagnetic perturbations in an isotropic porous medium saturated with an electrolyte, created a long-standing wave of interest of many research groups around the world.

Consequently, laboratory work, field measurements and theoretical developments have followed, enlarging our comprehension of the nature of the involved phenomena and giving rise to further questions. Among them, the issue of extending Pride's theory to account for either partially saturated rocks or rocks fully saturated with immiscible fluids has been studied in several publications.

We give here an overview of several of their works, for a more insightful read we suggest the review [START_REF] Jouniaux | A review on electrokinetically induced seismoelectrics, electro-seismics, and seismo-magnetics for Earth sciences[END_REF] and the books [START_REF] Revil | The Seismoelectric Method: Theory and Application[END_REF]Grobbe et al, 2020). [START_REF] Haines | Seismoelectric data processing for surface surveys of shallow targets[END_REF]; [START_REF] Dupuis | Seismoelectric imaging of the vadose zone of a sand aquifer[END_REF] suggested that the full range of saturation has to be considered when performing seismoelectric studies of partially saturated regions, as for example the vadose zone, to get a better understanding of the electrokinetic phenomenon. As a way of testing the dependence of seismoelectric signals on saturation, [START_REF] Bordes | Seismoelectric measurements for the characterization of partially saturated porous media[END_REF] showed the absence of coseismic signal on completely dry rocks. [START_REF] Zyserman | Finite element modeling of SHTE and PSVTM electroseismics[END_REF] numerically studied the electroseismic response of an hydrocarbon reservoir considering the presence of gas and oil by proposing an effective fluid approach, but considered the electrokinetic coupling coefficient to be that of the full saturation scenario. An extension to a partially saturated medium was introduced [START_REF] Warden | Seismoelectric wave propagation numerical modeling in partially saturated materials[END_REF] based on a) the relation between the electrokinetic coupling coefficient and the streaming potential coefficient and b) different extensions of the latter to partial saturation scenarios existing in the literature. This approach has been employed in several later works, e.g., [START_REF] Zyserman | Borehole seismoelectric logging using a shear-wave source:Possible application to CO 2 disposal?[END_REF] numerically studied the seismoelectric response of a CO 2 geological deposition site, analyzing its dependence on water saturation. Laboratory experiments were performed by [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] on the seismoelectric response of partially saturated sand using effective fluids in the associated computations of transfer functions. Another model was proposed by [START_REF] Jardani | Seismoelectric couplings in a poroelastic material containing two immiscible fluid phases[END_REF] for seismoelectric conversions by considering two immiscible fluids taking into account the contribution of the electric double layer at the fluid-fluid interface. Applications for contaminated aquifer and vadose zone were also developed: [START_REF] Munch | Detection of Non-Aqueous Phase Liquids Contamination by SH-TE Seismoelectrics: a Computational Feasibility Study[END_REF] numerically modeled the seismoelectric response of an aquifer with different degrees of contamination due to the presence of (D)NAPLs, and [START_REF] Zyserman | Dependence of shear wave seismoelectrics on soil textures: a numerical study in the vadose zone[END_REF] studied the seismoelectric responses of the vadose zone when considering different soil textures and water saturations. Moreover the first analytic expression for the interface Electromagnetic/acoustic coupling in partially-saturated porous rocks response (IR) studying the SHTE response of a partially saturated medium overlying a fully saturated one was published by [START_REF] Monachesi | An analytical solution to assess the SH seismoelectric response of the vadose zone[END_REF]. More recently passive electromagnetic sources have been considered by [START_REF] Zyserman | Numerical modelling of passive electroseismic surveying[END_REF] to study the seismic responses of a hydrocarbon reservoir at different oil saturations.

Several authors analyzed the dependence on saturation of the streaming potential coefficient through laboratory studies and theoretical models [START_REF] Perrier | Characterization of electrical daily variations induced by capillary flow in the non-saturated zone[END_REF][START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF][START_REF] Revil | Streaming potentials in two-phase flow conditions[END_REF][START_REF] Linde | Streaming current generation in two-phase flow conditions[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Vinogradov | Multiphase streaming potential in sandstones saturated with gas/brine and oil/brine during drainage and imbibition[END_REF]. All these studies lead to a constant or monotonous decreasing behavior of the electric response when the water saturation diminishes. On the other hand, other studies [START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF][START_REF] Allègre | Modelling the streaming potential dependence on water content during drainage: 1. A 1D modelling of SP using finite element method[END_REF][START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF][START_REF] Fiorentino | Two-phase lattice boltzmann modelling of streaming potentials: influence of the gas-water interface on the electrokinetic coupling[END_REF] measured and predicted a non monotonous behavior for a diminishing water saturation; in the first of these works oil was the considered wetting fluid. In the last of these references, as it was previously suggested by [START_REF] Allègre | Self-potential response to drainage-imbibition cycles[END_REF][START_REF] Allègre | Influence of water pressure dynamics and fluid flow on the streaming-potential response for unsaturated conditions[END_REF], it was shown through a numerical lattice-Boltzmann procedure that indeed the polarization of the air-water interface does play an important role both in the amplitude of the electrokinetic coupling and in its non monotonic behavior. These results were reaffirmed by the analysis performed by [START_REF] Jouniaux | Saturation Dependence of the Streaming Potential Coefficient[END_REF] revisiting laboratory data published by different authors.

As a conclusion of the published research to the date, there is a need for models able to explain the electromagnetic/mechanical coupling for partiallysaturated porous rocks. There is a clear evidence that the electric double layer effect of the water-air interface must be taken into account in any proposed model.

In this work we derive a set of equations governing the coupled electromagnetic/acoustic phenomena in partially-saturated porous rocks valid in the low-frequency regime, where the wetting fluid (water) is assumed to be an ideal electrolyte and the non-wetting phase is air. We introduce two main assumptions in our derivation: the first one is the existence of an electrical double layer at the water-air interface [START_REF] Creux | Specific cation effects at the hydroxide-charged air/water interface[END_REF][START_REF] Yang | Analytical study of saturation effects on seismic vertical amplification of soil layer[END_REF] whose potential does not interact with the one generated by the electric double layer at the rock matrix-water interface. The second assumption is that, in the wave propagation frequency regime the capillary pressure perturbations are negligible, implying that pressure perturbations in both wetting and non-wetting fluids are the same [START_REF] Berryman | Bulk elastic wave propagation in partially saturated porous solids[END_REF][START_REF] Pride | Deriving the equations of motions for porous isotropic media[END_REF]. It is also assumed with [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] that the porous rock is homogeneous and isotropic at the macroscopic scale and that local variations in ion concentration are negligible. The presence of the non-wetting phase has a major role in the proposed model, not only because the electromagnetic/mechanical coupling is assumed to exist in both water-solid and water-air surfaces through adsorbed excess charges balanced by mobile ions in the water, but also due to its influence in the emerging electromagnetic and mechanical properties of the medium. As it is shown along this work, the various coefficients thus derived depend on both the relative fraction of water and air and on the topological features of the latter within the porous space. As a corollary, the equations for the flow-regime in partially saturated porous media are derived.

Electromagnetic and mechanical equations at the pore scale

Let us start with a pore-scale description of the considered scenario. Fig. 1A shows a schematic representation of a partially-saturated rock (adapted from [START_REF] Culligan | Interfacial area measurements for unsaturated flow through a porous medium[END_REF]). The solid grains are represented in gray, water in white and air in black. Note that water is assumed to be the wetting phase, and Electromagnetic/acoustic coupling in partially-saturated porous rocks as such, the surface limiting the water volume, S w , is always the same whatever the water saturation s w , with the exception of s w = 0 (case that will not be treated here). This means that the air never gets in contact with the solid grains. For very low water saturation there will be thin films of water surrounding the grains [START_REF] Culligan | Interfacial area measurements for unsaturated flow through a porous medium[END_REF]. On the other hand, the surface surrounding the non-wetting fluid, S nw , will be different for different water saturation values, and of course it will exist for s w < 1. Fig. 1B the free ion distributions within the electrolyte are conceptualized following the same approach given by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. The electric double layer structure is assumed to be present in both contact surfaces, so there will be two shear planes; S w between solid and wetting-phase, and S nw between wetting and non-wetting phases.

Both the solid and the non-wetting phase (air) are assumed to be electrically insulating while the wetting fluid (water) is assumed to be an electrolyte with L ionic species. Then, Maxwell's equations can be written as follows: For the solid phase (s)

∇ • B s = 0, (1)
∇ • D s = 0, (2)
∇ × E s = -Ḃs , (3)
∇ × H s = Ḋs , (4)
for the wetting phase (w)

∇ • B w = 0, (5)
∇ • D w = L l=1 ez l N l , (6)
∇ × E w = -Ḃw , (7)
∇ × H w = Ḋw + J w , (8)
and for the non-wetting phase (nw)

∇ • B nw = 0, (9)
∇ • D nw = 0, (10)
∇ × E nw = -Ḃnw , (11)
∇ × H nw = Ḋnw . (12)
J w in Eq. ( 8) represents the ionic-current density and is given by

J w = L l=1 ez l [-kT b l ∇N l + ez l b l N l E w + N l uw ] , ( 13
)
where uw is the wetting fluid velocity. In this expression, kT is the thermal energy, ez l represents the net charge and sign of each species-l ion, N l represents the density and b l the mobility.

The boundary conditions on the surface S w are

n • (B s -B w ) = 0, ( 14
) n • (D s -D w ) = Q w , ( 15
) n × (E s -E w ) = 0, ( 16
) n × (H s -H w ) = Q w us , (17)
n • J w = Qw , ( 18
)
where n is the unit vector normal to S w , directed from wetting phase to solid, us is the velocity of the solid (= uw on S w ) and Q w is the free charge per unit area of the wetting adsorbed layer. In the same way, the boundary conditions on the surface S nw are given by:

n • (B nw -B w ) = 0, ( 19
) n • (D nw -D w ) = Q nw , ( 20
) n × (E nw -E w ) = 0, ( 21
) n × (H nw -H w ) = Q nw unw , (22)
n • J w = Qnw . ( 23
)
Note that the surface S nw together with S w constitute the boundary of the volume occupied by the wetting phase, i.e., ∂V w = S w ∪ S nw . Finally we need to close the system with the constitutive relations

B ξ = µ 0 H ξ , (24)
D ξ = ϵ 0 κ ξ E ξ , (25)
where µ 0 is the vacuum magnetic permeability, ϵ 0 the vacuum electric permittivity and κ ξ , ξ = s, w, or nw is the corresponding dielectric constant. Now, the time dependence of any field variable A is written as

A(t) = A 0 + Re a(ω)e -iωt , (26)
where the first term represents the static-equilibrium field and the second one represents the deviation in the field produced by a harmonic perturbation.

Both Q 0 w and Q 0 nw are assumed to be uniform over the corresponding surfaces, and as a consequence E 0 w and N 0 l will be the only nonzero static fields [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. Solving Eqs. ( 6), ( 7), ( 8) and ( 13) it is possible to show that the electric potential near a shear plane can be approximated by an exponential distribution

Φ 0 (χ) = ζe -χ/d , ( 27
)
where χ is a local coordinate for the distance normal to the surface, d is the Debye length, defined as

1 d 2 = L l=1 (ez l ) 2 N l ϵ 0 κ w kT , ( 28
)
and ζ is the zeta potential being the static electric potential at the shear plane (i.e., S w or S nw ). Then, a different solution on each considered surface is expected to be obtained. If the Debye length associated with each shear plane is much smaller than any geometrical feature of the porous medium (even smaller than the thin water films thicknesses), then the electric potential at each shear plane can be treated separately, as two non-interacting electric potentials, as illustrated in Fig. 1C. This is just the "thin-double-layer" approximation adopted by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], employed here twice. Then we will have

Φ 0 w (χ) = ζ w e -χ/dw and Φ 0 nw (χ) = ζ nw e -χ/dnw , (29)
with their corresponding electric fields E 0 w,(w) = -∇Φ 0 w and E 0 w,(nw) = -∇Φ 0 nw . The zeta potentials ζ w and ζ nw are related to the respective charge per unit area of S w and S nw as follows

Q 0 w ≃ 2d w L l=1 ez l N l,(w) e -ez l ζw 2kT and Q 0 nw ≃ 2d nw L l=1 ez l N l,(nw) e -ez l ζnw 2kT . ( 30
)
Note that the bulk-ionic concentrations N l,(w) and N l,(nw) must satisfy N l = N l,(w) + N l,(nw) , and the respective Debye lengths are given by

1 d 2 w = L l=1 (ez l ) 2 N l,(w) ϵ 0 κ w kT and 1 d 2 nw = L l=1 (ez l ) 2 N l,(nw) ϵ 0 κ w kT . ( 31
)
Now we need to state the equations governing the disturbances. Preserving only the linear contributions, we obtain

∇ • b s = 0, (32)
∇ • d s = 0, (33)
∇ × e s = iωb s , (34)
∇ × h s = -iωd s , (35)
∇ • b w = 0, (36)
∇ • d w = L l=1 ez l n l , (37)
∇ × e w = iωb w , (38)
∇ × h w = -iωd w + j w , (39)
∇ • b nw = 0, (40)
∇ • d nw = 0, (41)
∇ × e nw = iωb nw , (42)
∇ × h nw = -iωd nw , (43)
where

j w = L l=1 ez l -kT b l ∇n l + ez l b l N 0 l e w + n l E 0 w + N 0 l uw . ( 44
)
The boundary conditions on S w are

n • (b s -b w ) = 0, (45) n • (d s -d w ) = 0, (46)
n × (e s -e w ) = 0, ( 47)

n × (h s -h w ) = Q 0 w us , (48)
n • j w = 0, (49)
and on S nw we have

n • (b nw -b w ) = 0, ( 50
) n • (d nw -d w ) = 0, (51)
n × (e nw -e w ) = 0, ( 52)

n × (h nw -h w ) = Q 0 nw unw , (53)
n • j w = 0. ( 54
)
The mechanical equations governing the conservation of linear momentum of the solid and both fluid phases are

-iωρ s us = ∇ • τ s , (55)
-iωρ w uw = ∇ • τ w + L l=1 ez l N 0 l e w + n l E 0 w , (56)
-iωρ nw unw = ∇ • τ nw , (57)
where the solid, wetting-phase and non-wetting phase stress tensors are respectively given by

τ s = K s ∇ • u s I + G s ∇u s + ∇u T s - 2 3 ∇ • u s I , (58)
τ w = K w ∇ • u w I -iωη w ∇u w + ∇u T w - 2 3 ∇ • u w I , (59)
τ nw = K nw ∇ • u nw I -iωη nw ∇u nw + ∇u T nw - 2 3 ∇ • u nw I . ( 60
)
In the expression for τ s , K s and G s are the bulk and shear modulus of the solid phase, while in τ w , K w is the wetting-phase bulk modulus and η w its viscosity. The corresponding parameters for the non-wetting phase are τ nw , K nw and η nw . The boundary conditions on S w are

n • (τ s -τ w ) = -Q 0 w e s , (61)
u s -u w = 0, (62)
and on

S nw n • (τ nw -τ w ) = -Q 0 nw e nw , (63)
u nw -u w = 0. ( 64
)
3 Volume average of the governing equations

In this section, the volume-averaging procedure given in [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] is applied

following the same assumptions. Let V A be the averaging volume. Then, the volume average of a microscopic field a ξ associated with the ξth phase is given by

⟨a ξ ⟩ = 1 V A V ξ a ξ dV, (65)
where V ξ is the volume occupied by the ξth phase within V A , then the following theorems will hold [START_REF] Slattery | Flow of viscoelastic fluids through porous media[END_REF]:

⟨∇a ξ ⟩ = ∇ ⟨a ξ ⟩ + 1 V A S ξ n ξ a ξ dS, ( 66
) ⟨∇ • a ξ ⟩ = ∇ • ⟨a ξ ⟩ + 1 V A S ξ n ξ • a ξ dS, ( 67
) ⟨∇ × a ξ ⟩ = ∇ × ⟨a ξ ⟩ + 1 V A S ξ n ξ × a ξ dS, (68)
where n ξ is the normal to S ξ = ∂V ξ , and is defined as

n w = n, (69)
n s = -n, (70)
n nw = -n, (71)
with n directed from the wetting-phase to the solid or from the wetting-phase to the non-wetting phase. It will be useful to define two other averages related to ⟨a ξ ⟩, the phase average and total average, which are respectively given by

āξ = ⟨∇a ξ ⟩ /φ ξ , (72)
Ā = ξ ⟨∇a ξ ⟩ = ξ φ ξ āξ , (73)
where φ ξ = V ξ /V A is the volume fraction of the ξth phase.

The electromagnetic equations governing the disturbances [Eqs.( 32)-( 44)] are now averaged following the same procedure given in [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

As a result one obtains:

∇ • B = 0, (74)
∇ • D = s w ϕ L l=1 ez l nl . ( 75
) ∇ × Ē = iω B. ( 76
) ∇ × H = -iω D + J . ( 77
)
with the current density given by the following approximation [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]):

J ≃ s w ϕ [J c + J s ] , (78)
where

J c = 1 V w Vw L l=1 (ez l ) 2 b l N 0 l e w dV, (79)
J s = 1 V w Vw L l=1 ez l N 0 l vdV, ( 80
) v = uw -us , (81)
being us the phase-averaged velocity of the solid phase. The current J c is the macroscopic conduction current density and J s is the macroscopic streaming current density.

If we define the bulk-wetting fluid conductivity and express it as the sum of the contributions of both diffuse layers

σ w = L l=1 (ez l ) 2 b l N l = L l=1 (ez l ) 2 b l (N l,(w) + N l,(nw) ), (82)
then, the conduction current can be written as follows

J c = σ w V w Vw e w dV + 1 V w Sw dS D 0 L l=1 (ez l ) 2 b l (N 0 l,(w) -N l,(w) ) e w dχ + 1 V w Snw dS D 0 L l=1 (ez l ) 2 b l (N 0 l,(nw) -N l,(nw) ) e w dχ. ( 83
)
The variable χ measures the normal distance from the surfaces S w and S nw (into the wetting fluid) and D = max ξ {D ξ }, ξ = w, nw, where D ξ represents the distance over which the charge excess N 0 l,(ξ) -N l,(ξ) associated with each diffuse layer is significant (a few Debye lengths).

In the same way we can express the streaming current as follows:

J s = 1 V w Vw L l=1 ez l N 0 l,(w) vdV + 1 V w Vw L l=1 ez l N 0 l,(nw) vdV. ( 84
)
Substituting Coulomb's law for the static field into the last equation

J s = ϵ 0 κ w V w Vw ∇ • E 0 w,(w) vdV + ϵ 0 κ w V w Vw ∇ • E 0 w,(nw) vdV, (85)
which for homogeneous media can be approximated as [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF])

J s = ϵ 0 κ w V w Sw dS D 0 ∇Φ 0 w • ∇vdχ + ϵ 0 κ w V w Snw dS D 0 ∇Φ 0 nw • ∇vdχ, (86)
The integrals appearing in Eq. ( 83) and Eq. ( 86) will be solved in the next Section, once the fields at the pore-scale are related to their corresponding macroscopic fields.

It remains to average the electromagnetic constitutive laws. Performing this task on d ξ = ϵ 0 κ ξ e ξ yields

D = ϵ 0 [κ s (1 -ϕ)ē s + κ w s w ϕē w + κ nw (1 -s w )ϕē nw ] . (87)
If we assume that the magnetic susceptibilities are negligible in the three phases, we simply have

B = µ 0 H. ( 88
)
Taking the volume average of mechanical equations ( 55)-( 57) we respectively obtain

-iωρ s ⟨ us ⟩ = ∇ • ⟨τ s ⟩ - 1 V A Sw n • τ s dS, (89)
-iωρ w ⟨ uw ⟩ = ∇ • ⟨τ w ⟩ + 1 V A Sw n • τ w dS + 1 V A Snw n • τ w dS + 1 V A V A L l=1 ez l N 0 l e f + n l E 0 w dV, (90)
-iωρ nw ⟨ unw ⟩ = ∇ • ⟨τ nw ⟩ - 1 V A Snw n • τ nw dS, (91)
Adding ( 89), ( 90) and ( 91), applying the BC's ( 61) and ( 63) and the definition of the phase average we get the following approximation [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF])

-iω (1 -ϕ)ρ s us + s w ϕρ w uw + (1 -s w )ϕρ nw unw ) ≃ ∇ • [(1 -ϕ)τ s + s w ϕτ w + (1 -s w )ϕτ nw )] (92)
If we define:

ẇw = s w ϕ uw -us , ( 93
) ẇnw = (1 -s w )ϕ unw -us , ( 94
) τB = (1 -ϕ)τ s + s w ϕτ w + (1 -s w )ϕτ nw , (95)
ρ B = (1 -ϕ)ρ s + s w ϕρ w + (1 -s w )ϕρ nw , (96)
we can then write:

∇τ B = -iω ρ B us + ρ w ẇw + ρ nw ẇnw , (97)
It is convenient to introduce the effective fluid filtration wf as

ρ f wf = ρ w ww + ρ nw wnw , (98)
where ρ f = s w ρ w + (1 -s w )ρ nw . Then Eq. ( 97) is written

∇τ B = -iω ρ B us + ρ f ẇf , (99)
The averaging of the fluid equations ( 90) and ( 91) is addressed in Section 4.

Finally, the stress-strain relations are volume averaged following [START_REF] Pride | Deriving the equations of motions for porous isotropic media[END_REF]. Assuming that the capillary pressure perturbations p c = p nw -p w are negligible, then p w = p nw = p and the averaged constitutive relations result [START_REF] Berryman | Bulk elastic wave propagation in partially saturated porous solids[END_REF][START_REF] Pride | Deriving the equations of motions for porous isotropic media[END_REF])

τB = (K c ∇ • ūs + C∇ • wf )I + G ∇ū s + ∇ū s T - 2 3 ∇ • ūs I , ( 100
) -p = C∇ • ūs + M ∇ • wf , (101)
where

K c = K m + ϕK f + (1 -ϕ)K s ∆ 1 + ∆ , (102)
C = K f + K s ∆ 1 + ∆ , (103)
M = 1 ϕ K f 1 + ∆ . ( 104
)
in this expressions,

∆ = K f ϕK 2 s [(1 -ϕ)K s -K m ] , (105)
and K m and G in Eq.( 100) are the bulk modulus and shear modulus of the solid matrix respectively, and K f is the effective bulk modulus of the fluid phase, obtained by a Wood mean, i.e.,

K f = (s w /K w + (1 -s w )/K nw ) -1 .

Boundary-value problems

As mentioned before, to complete the averaging procedure we need so solve the integrals appearing in the various derived expressions. In order to do so, any pore-scale field needs to be related to macroscopic fields. In this section, we follow Pride's approach by introducing a new disk-shaped averaging volume,

where the boundary-value problems governing the pore-scale fields will be stated and solved.

Volume averaging over a thin disk

Let us consider an imaginary volume with the shape of a thin disk within the porous rock, defined by two large plane-parallel faces of area A separated by a distance H. Assume that a macroscopic potential difference and a macroscopic pressure difference exist between the two faces. The macroscopic fields normal to the disk face will then be given by the corresponding difference divided by H. Let z be the direction normal to the disk. The potential or pressure differences will be assumed as the boundary values for the corresponding porescale fields. We assume with [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] that e s = -∇φ s , e w = -∇φ w and Electromagnetic/acoustic coupling in partially-saturated porous rocks

e nw = -∇φ nw . Then ẑ • Ē = - ∆ϕ H , (106)
where

∆ϕ = φ ξ (H) -φ ξ (0), ξ = s, w, nw, (107)
is the potential difference between the two flat faces [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

Pore-scale electric fields and ion-number-density deviations

As was demonstrated in [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] the ion-number-density deviations n l,(w) can be neglected whenever the following two conditions are met: (1) The electric double layer must be much thinner than the grain sizes, and (2) a large dielectric contrast must exist between the grains and the electrolyte. Under these assumptions, the electric-field in the fluid produced by the double layer at the solid-wetting interface verifies e w,(w) (r) = -∇φ w,(w

) (r) = -∇Γ w (r) ∆ϕ H . ( 108
)
where the field Γ w possess units of length and is defined to satisfy

∇ 2 Γ w = 0, ( 109
)
n • ∇Γ w = 0 on S w , (110)
Γ w =        H on z = H, 0 on z = 0. ( 111
)
Following the same procedure but considering the electric double layer at S nw it is possible to get, under the same assumptions as before e w,(nw

) (r) = -∇Γ nw (r) ∆ϕ H , (112)
where

∇ 2 Γ nw = 0, ( 113
)
n nw • ∇Γ nw = 0 on S nw , (114)
Γ nw =        H on z = H, 0 on z = 0. ( 115
)
The last derivation will be valid for all frequencies and for large dielectric contrasts between the non-wetting phase (air) and the electrolyte.

We can now volume-average the electric fields in the disk. Let's start with ēw :

ēw = - 1 V w Vw ∇ϕ w dV = - 1 V w s w ϕAH ẑ + Sw nΓ w (r)dS + Snw nΓ nw (r)dS ∆ϕ H , (116)
multiplying by ẑ and replacing -n nw = n in the second integral

ēw = 1 + ẑ V w Sw nΓ w (r)dS - ẑ V w Snw n nw Γ nw (r)dS Ē, (117)
which can be written as

ēw = 1 - V p V w + V p V w 1 + ẑ V p Sw nΓ w (r)dS + V nw V w - V nw V w 1 + ẑ V nw Snw n nw Γ nw (r)dS Ē, (118)
or, identifying the tortuosities of the porous space α ∞ and of the non-wetting phase α ∞,nw as

1 α ∞ = 1+ ẑ V p Sw nΓ w (r)dS, 1 α ∞,nw = 1+ ẑ V nw Snw n nw Γ nw (r)dS, (119) then, ēw = 1 s w 1 α ∞ - (1 -s w ) s w 1 α ∞,nw Ē = 1 α∞,w Ē. ( 120
)
At this point we have all the ingredients to compute the macroscopic conduction current, which is completed in the following Section. It remains to compute the constitutive electromagnetic equation relating D and Ē. To this end, we need to compute ēs and ēnw . Note that

ēs = - 1 V s Vs ∇φ s dV = - 1 V s (1 -ϕ)A∆ϕẑ + Sw n s φ s dS . (121)
Because φ s = φ w at S w [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF])

ēs = - 1 V s Vs ∇φ s dV = - 1 V s (1 -ϕ)A∆ϕẑ - Sw nφ w dS = 1 - ẑ V s Sw nΓ w (r)dS Ē, ( 122
) then ēs = 1 + V p V s - V p V s 1 + ẑ V p Sw nΓ w (r)dS Ē, (123)
or

ēs = 1 (1 -ϕ) - ϕ (1 -ϕ) 1 + ẑ V p Sw nΓ w (r)dS Ē. ( 124
)
Using the definition of α ∞ we get

ēs = 1 (1 -ϕ) - ϕ (1 -ϕ) 1 α ∞ Ē = 1 α∞,s Ē. ( 125
)
Following the same reasoning and under the same hypothesis regarding the dielectric contrast we proceed to compute the field ēnw :

ēnw = - 1 V nw Vnw ∇φ nw dV = - 1 V nw (1 -s w )ϕA∆ϕẑ + Snw n nw φ w dS = 1 + ẑ V nw Snw n nw Γ nw (r)dS Ē. ( 126
) Then ēnw = 1 α ∞,nw Ē. (127)
It is easy to verify that Ē = (1 -ϕ)ē s + ϕs w ēw + ϕ(1 -s w )ē nw as expected from the definition of Ē.

Replacing the average fields in Eq. ( 87):

D = ϵ 0 κ s (1 -ϕ) α∞,s + κ w s w ϕ α∞,w + κ nw (1 -s w )ϕ α ∞,nw Ē. (128)
This last equation, together with Eq.( 88) complete the volume averaged expressions for the electromagnetic constitutive relations.

Pore-scale flow fields

Let us start recalling Eq.( 56) for the wetting-fluid flow at the pore-scale.

Neglecting n l this equation can be written as

-iωρ w uw = ∇ • τ w + L l=1 ez l N 0 l e w , (129)
which using v reads

-iωρ w v = ∇ • τ w + iωρ w us + L l=1 ez l N 0 l e w . (130)
Assuming that the relative flow is incompressible and defining ∇p = ∇p wiωρ w us , we can state the following boundary-value problem within the averaging disk

η w ∇ 2 v + iωρ w v = ∇p - L l=1 ez l N 0 l e w , (131)
∇ • v = 0, ( 132
) v = 0 on S w and S nw , (133) p
=        ∆P = ẑ • ∇p w -iωρ w us H, z = H, 0, z = 0. (134)
Following [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], the solution for v and p can be separated into mechanically and electrically induced contributions.

v = v m + v e , (135)
p = p m + p e , (136)
where m stands for fields induced by pressure gradients ∆P/H, while e stands for fields induced by electric potential gradients ∆ϕ/H.

Proceeding as [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] it is possible to show that the electrical contribution for v e in the low frequency regime will be given by

v e0 = - ϵ 0 κ w η w Φ 0 w -ζ w ∇Γ w ∆ϕ H - ϵ 0 κ w η w Φ 0 nw -ζ nw ∇Γ nw ∆ϕ H . (137)
Note that this last equation can be interpreted as the sum of two contributions coming from the electrical effects from both double layers

v e0,(w) = - ϵ 0 κ w η w Φ 0 w -ζ w ∇Γ w ∆ϕ H , (138)
and

v e0,(nw) = - ϵ 0 κ w η w Φ 0 nw -ζ nw ∇Γ nw ∆ϕ H . ( 139
)
Following [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] the mechanical contributions at the low frequency regime are given by

v m0 (r) = g(r) η w ∆P H , (140)
and

p m0 (r) = h(r) ∆P H , (141)
where g(r) has the units of length squared and h(r) has the units of length, and verify:

∇ 2 g = ∇h, (142)
∇ • h = 0, ( 143
) g = 0, on S w ∪ S nw = ∂V w , (144)
h =        H, z = H, 0, z = 0. (145)
In general, n • ∇h = -n • ∇ × ∇ × g ̸ = 0 both on S w and on S nw . Then, h ̸ = Γ w and h ̸ = Γ nw . However, it is convenient to express h as h = Γ w + δh w when we are close to S w , or h = Γ nw + δh nw when we are close to S nw , and

δh ξ , ξ = w, nw satisfies ∇ 2 δh ξ = 0, ( 146
)
n • ∇δh ξ = -n • ∇ × ∇ × g on S ξ , ( 147
)
δh ξ =        0, z = H, 0, z = 0. (148)
Then, using the first expression of h we will be considering the pressure field near S w and using the second one, the pressure field near S nw . We can write these as follows:

η w ∇ 2 v m,(w) = ∇ δp m,(w) + Γ w ∆P H ( 149
)
and

η w ∇ 2 v m,(nw) = ∇ δp m,(nw) + Γ nw ∆P H , ( 150
)
where δp m,(ξ) = δh ξ ∆P/H. Given that in both equations the right-hand sides are provided by fields associated with different boundary layers, then we will have two mechanical velocities v m,(w) and v m,(nw) . Now we proceed to volume average the non-wetting fluid flow Eq. ( 57). If we define the relative non-wetting solid-phase flow vector v nw = unwus we may write:

-iωρ nw v nw = ∇ • τ nw + iωρ nw us . (151)
Again, if we assume that the local relative flow is incompressible and defining ∇p = ∇p nw -iωρ nw us , the following boundary-value problem can be stated within the averaging disk

η nw ∇ 2 v nw + iωρ nw v nw = ∇p, (152)
∇ • v nw = 0, ( 153
)
v nw = 0 on S nw = ∂V nw , ( 154
) p =        ∆P = ẑ • ∇p nw -iωρ nw us H, z = H, 0, z = 0. ( 155
)
This is a purely mechanical problem, and can be solved in the same way as the wetting-fluid case. First, note that in the low-frequency regime we can write

∇ 2 v nw = ∇p η nw , ( 156
)
∇ • v nw = 0, ( 157
)
v nw = 0, on S nw = ∂V nw , ( 158
) p =        ∆P, z = H, 0, z = 0. ( 159
)
The solution to this boundary-value problem is given by

v nw0 (r) = g nw (r) η nw ∆P H , (160)
and

p 0 (r) = h nw (r) ∆P H , (161)
where g nw (r) and h nw (r) verify:

∇ 2 g nw = ∇h nw , (162)
∇ • h nw = 0, (163)
g nw = 0, on S nw = ∂V nw , (164) Electromagnetic/acoustic coupling in partially-saturated porous rocks

h nw =        H, z = H, 0, z = 0. ( 165
)
The so-called Stokes-flow geometry fields g and g nw are the only ones needed to establish relations between the various macroscopic-transport coefficients.

At this stage all the low-frequency regime pore scale fields needed to obtain the macroscopic transport coefficients have been obtained so the final transport equations will be computed in the following Section.

5 Transport equations

Conduction current density J c

Replacing the expressions for the averaged field ēw and the local fields e w,(w)

and e w,(nw) in Eq.( 83) we obtain

J c = σ w α∞,w Ē - 1 V w Sw ∇Γ w (r)dS D 0 L l=1 (ez l ) 2 b l (N 0 l,(w) -N l,(w) ) ∆ϕ H dχ - 1 V w Snw ∇Γ nw (r)dS D 0 L l=1 (ez l ) 2 b l (N 0 l,(nw) -N l,(nw) ) ∆ϕ H dχ. (166) Now, if we define: 2 Λ = α ∞ V p Sw ẑ • ∇Γ w (r)dS, ( 167
) 2 Λ nw = α ∞,nw V nw Snw ẑ • ∇Γ nw (r)dS, ( 168
)
and

C em,(w) = D 0 L l=1 (ez l ) 2 b l (N 0 l,(w) -N l,(w) ) dχ, (169)
C em,(nw) = D 0 L l=1 (ez l ) 2 b l (N 0 l,(nw) -N l,(nw) ) dχ, (170)
then, we have:

J c = σ w α∞,w + 2C em,(w) s w α ∞ Λ + (1 -s w ) s w 2C em,(nw) α ∞,nw Λ nw Ē. (171)
Λ is a fundamental porous material geometry parameter and is a measure of the weighted volume-to-surface ratio of the porous media. Therefore it has units of length. Similarly, Λ nw is the corresponding parameter of the non-wetting phase, and of course it will depend on the water saturation s w . The quantities C em,(w) and C em,(nw) are the conductances associated with electromigration of double layer ions. Following [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] the Debye approximation allows to obtain the following expressions

C em,(w) ≃ 2d w L l=1 (ez l ) 2 b l N l,(w) e -ez l ζw 2kT -1 , (172)
C em,(nw) ≃ 2d nw L l=1 (ez l ) 2 b l N l,(nw) e -ez l ζnw 2kT -1 . (173)

Streaming current density J s

Consider the average streaming current density given by Eq. (86):

J s = ϵ 0 κ w V w Sw dS D 0 ∇Φ 0 w • ∇vdχ + ϵ 0 κ w V w Snw dS D 0 ∇Φ 0 nw • ∇vdχ, (174)
where ∇Φ 0 ξ • ∇v = (∂Φ 0 ξ /∂χ)(∂v/∂χ), ξ = w, nw. Given the separation into an electrically induced field v e and a mechanically induced field v m , we also have J s = J se + J sm . Focusing on the electrical contribution, replacing the expressions for v e0,(w) and v e0,(nw) from ( 138) and (139) in the first and Electromagnetic/acoustic coupling in partially-saturated porous rocks second integrals of ( 174), respectively, we obtain

J se = - (ϵ 0 κ w ) 2 η w V w Sw ∇Γ w dS D 0 (∇Φ 0 w ) 2 dχ ∆ϕ H + Snw ∇Γ nw dS D 0 (∇Φ 0 nw ) 2 dχ ∆ϕ H . (175)
Then, from the definitions of Λ and Λ nw , we get

J se = (ϵ 0 κ w ) 2 η w 2 s w α ∞ Λ D 0 (∇Φ 0 w ) 2 dχ + 2(1 -s w ) s w α ∞,nw Λ nw D 0 (∇Φ 0 nw ) 2 dχ Ē, (176)
or

J se = 2C os,(w) s w α ∞ Λ + 2(1 -s w )C os,(nw) s w α ∞,nw Λ nw Ē, (177)
where

C os,(w) = (ϵ 0 κ w ) 2 η w D 0 (∇Φ 0 w ) 2 dχ, (178)
C os,(nw) = (ϵ 0 κ w ) 2 η w D 0 (∇Φ 0 nw ) 2 dχ. (179)
The parameters C os,(w) and C os,(nw) are the conductances due to electrically induced streaming of the excess double-layer ions. Following [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] they can be estimated as

C os,(w) = (ϵ 0 κ w ) 2 η w 4kT d w ϵ 0 κ w L l=1 N l,(w) e -ez l ζw 2kT -1 , (180)
C os,(nw) = (ϵ 0 κ w ) 2 η w 4kT d nw ϵ 0 κ w L l=1 N l,(nw) e -ez l ζnw 2kT -1 . (181)
These last expressions are valid in the low-frequency approximation [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

The mechanically induced streaming current density is now derived. Using the following identity

∇Φ 0 ξ • ∇v m = ∇ • (Φ 0 ξ ∇v m ) -Φ 0 ξ ∇ 2 v m (182)
and the fact that ∇ 2 v m (unlike ∇ 2 v e ) is approximately constant across the thin double layer in comparison with Φ 0 ξ , we have

J sm = ϵ 0 κ w V w Sw dS D 0 ∇ • (Φ 0 w ∇v m ) -Φ 0 w ∇ 2 v m dχ + ϵ 0 κ w V w Snw dS D 0 ∇ • (Φ 0 nw ∇v m ) -Φ 0 nw ∇ 2 v m dχ, (183)
then

J sm = ϵ 0 κ w ζ w V w Sw n • ∇v m - D 0 Φ 0 w ζ w dχ ∇ 2 v m dS + ϵ 0 κ w ζ nw V w Snw n • ∇v m - D 0 Φ 0 nw ζ nw dχ ∇ 2 v m dS, (184)
or, introducing dw and dnw defined as

dξ = D 0 Φ 0 ξ (χ) ζ ξ dχ, ξ = w, nw, (185)
we can write

J sm = ϵ 0 κ w ζ w V w Sw n • ∇v m -dw ∇ 2 v m dS + ϵ 0 κ w ζ nw V w Snw n • ∇v m -dnw ∇ 2 v m dS, (186)
Now, using the expressions for v m0 :

J sm0 = - ϵ 0 κ w ζ w η w ẑ V w • Sw n • ∇g -dw ∇h dS -∇p w + iωρ w us - ϵ 0 κ w ζ nw η w ẑ V w • Snw n • ∇g -dnw ∇h dS -∇p w + iωρ w us . (187)
Then, multiplying by s w ϕ we get

s w ϕJ sm0 = L m0 -∇p w + iωρ w us , (188)
where

L m0 = -ϕ ϵ 0 κ w ζ w η w ẑ V p • Sw n • ∇g -dw ∇h dS -ϕ(1 -s w ) ϵ 0 κ w ζ nw η w ẑ V nw • Snw n • ∇g -dnw ∇h dS. (189)
It is important to note here that in the case of full water saturation, the second term in the last equation is identically zero, not only because of the (1 -s w ) factor but also because due to the fact the water-air interface does not exist, i.e., ζ nw = 0. However, the first term will still be present because of the solid-water double layer effects. In this case, Eq. ( 189) coincides with the L m0 coefficient derived by Pride (1994, Eq. 212).

Relative flows v and vnw

Consider first the mechanically induced wetting-fluid flow. In the limit of low frequencies,

vm0 = 1 V w Vw v m0 (r)dV = 1 η w - 1 V w Vw ẑ • gdV -∇p w + iωρ w us , (190) then,
s w ϕv m0 = 1 η w - 1 V A Vw ẑ • gdV -∇p w + iωρ w us , (191)
or,

s w ϕv m0 = k 0,w η w -∇p w + iωρ w us , (192)
where

k 0,w = - 1 V A Vw ẑ • gdV. ( 193
)
The parameter k 0,w is the DC permeability of the wetting phase.

Next, the low-frequency electrically induced flow is integrated

ve0 = 1 V w Vw v e0 dV = ϵ 0 κ w η w ẑ V w • Vw Φ 0 w -ζ w ∇Γ w dV + Vw Φ 0 nw -ζ nw ∇Γ nw dV Ē. (194)
Performing the integrals and multiplying by s w ϕ the following expression is obtained

s w ϕv e0 = L e0 Ē, (195)
where

L e0 = -ϕ ϵ 0 κ w η w ζ w 1 α ∞ -(1 -s w ) - 2 dw α ∞ Λ + ζ nw 1 - (1 -s w ) α ∞,nw - 2 dnw (1 -s w ) α ∞,nw Λ nw . ( 196
)
Again, for fully-water saturated porous medium, the second term will not be present, and the first term will coincide with the coefficient L e0 derived by Pride (1994, Eq. 227). Electromagnetic/acoustic coupling in partially-saturated porous rocks

Note that from Eqs.( 171) and (177) we have

s w ϕ(J ce + J se ) = σ Ē, (197)
where

σ = s w ϕσ w α∞,w + 2ϕ(C em,(w) + C os,(w) ) α ∞ Λ + (1 -s w ) 2ϕ(C em,(nw) + C os,(nw) ) α ∞,nw Λ nw , (198)
In the limit of full water saturation, the third term in this last equation will not be present. Noting that α∞,w tends to α ∞ when s w tends to 1, the electric conductivity will coincide with the corresponding one derived by Pride (1994, Eq. 242).

Finally, let us consider the non-wetting fluid flow average. From Eq. ( 160)

vnw0 = 1 V nw Vnw v nw0 (r)dV = 1 η nw - 1 V nw Vnw ẑ • g nw dV -∇p nw + iωρ nw us , (199) then,
(1

-s w )ϕv nw0 = 1 η nw - 1 V A Vnw ẑ • g nw dV -∇p nw + iωρ nw us , (200) or, (1 -s w )ϕv nw0 = k 0,nw η nw -∇p nw + iωρ nw us , (201)
where

k 0,nw = - 1 V A Vnw ẑ • g nw dV, (202)
is the DC permeability of the non-wetting phase.

Summary of this section

By adding Eqs.( 192) and (195) we have

s w ϕv 0 = s w ϕ(v m0 + ve0 ) = k 0,w η w -∇p w + iωρ w us + L e0 Ē, (203)
and recalling that -iω ww = ẇw = s w ϕv the transport equation for the relative wetting-fluid velocity reads:

-iω ww = L e0 Ē + k 0,w η w -∇p w + iωρ w us , (204)
with L e0 given by ( 196) and k 0,w by ( 202). From Eq. ( 201), recalling that -iω wnw = (1 -s w )ϕv nw we can write the equation for the relative nonwetting fluid velocity as

-iω wnw = k 0,nw η nw -∇p nw + iωρ nw us . (205)
Adding ( 188) and (197) we have the final expression for the second transport equation:

J = σ Ē + L m0 -∇p w + iωρ w us , (206)
with L m0 given by ( 189) and σ by (198).

Finally, Onsager reciprocity L e0 = L m0 can be demonstrated in the same way as [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], by splitting L m0 and L e0 in their contributions from each double layer.

6 Final equations Note that under the assumption of negligible perturbations of capillary pressure (p c = 0), both pressure perturbations verify p w = p nw = p. Introducing the effective fluid w f (Eq. 98), Eqs. ( 204) and ( 205) can be placed in Eq. ( 98), which divided by ρ f reads

-iω wf = ρ w ρ f L ps 0 Ē - ρ w ρ f k 0,w η w + ρ nw ρ f k 0,nw η nw ∇p +iω ρ 2 w ρ f k 0,w η w + ρ 2 nw ρ f k 0,nw η nw us . (207)
Also, it is worthwhile to remark here that at p c = 0, the mechanical constitutive relations given by ( 100) and (101) will hold. Removing all the overbars appearing in the averaged variables, then from ( 76), ( 77), ( 88), ( 99), ( 100), ( 101), ( 128), ( 206) and ( 207) we get:

∇ × E = iωB, (208)
∇ × H = -iωD + J , ( 209
)
∇τ B = ω 2 [ρ B u s + ρ f w f ] , (210)
J = σ ps E + L ps 0 -∇p + ω 2 ρ w u s , (211)
-iωw f = ρ w ρ f L ps 0 E - ρ w ρ f k 0,w η w + ρ nw ρ f k 0,nw η nw ∇p +ω 2 ρ 2 w ρ f k 0,w η w + ρ 2 nw ρ f k 0,nw η nw u s , (212)
D = ϵ 0 κ s (1 -ϕ) α∞,s + κ w s w ϕ α∞,w + κ nw (1 -s w )ϕ α ∞,nw E, (213)
B = µ 0 H, (214)
τ B = (K c ∇ • u s + C∇ • w f )I + G ∇u s + ∇u s T - 2 3 ∇ • u s I , ( 215
) -p = C∇ • u s + M ∇ • w f , (216)
where the partially-saturated electrokinetic coupling coefficient is given by

L ps 0 = -ϕ ϵ 0 κ w η w ζ w 1 α ∞ -(1 -s w ) - 2 dw α ∞ Λ +ζ nw 1 - (1 -s w ) α ∞,nw - 2 dnw (1 -s w ) α ∞,nw Λ nw , (217)
and the partially-saturated electric conductivity is expressed as

σ ps = s w ϕσ w α∞,w + 2ϕ(C em,(w) + C os,(w) ) α ∞ Λ + (1 -s w ) 2ϕ(C em,(nw) + C os,(nw) ) α ∞,nw Λ nw . (218)
This is the final set of equations governing the coupled electromagnetic/acoustic wave propagation. Note that when s w → 1, Pride's final equations are recovered: when

s w → 1, then ρ f → ρ w , w f → w w , ρ B → (1 -ϕ)ρ s + ϕρ w and K f → K w .
In this limiting case we also have that L ps 0 → L 0 and σ ps → σ as was demonstrated in the previous section. Also note that κ 0,nw → 0 and κ 0,w → κ 0 , being κ 0 the DC permeability of the porous medium. Finally, given that from Eqs. ( 120) and ( 125) we respectively have

lim sw→1 1 α∞,w = 1 α ∞ and 1 α∞,s = 1 (1 -ϕ) - ϕ (1 -ϕ) 1 α ∞ , ( 219
) Electromagnetic/acoustic coupling in partially-saturated porous rocks in the fully water saturation case, the final equations ( 208)-( 216) are identical to Pride (1994, Eqs. 248-256) as expected. Note that if the frequencies are low enough that it is safe to neglect the inertial terms in Eqs. ( 210)-( 212), we obtain a set of equations valid to deal with electromagnetic/mechanical coupling for the so-called consolidation problems in poromechanics [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid: I. low frequency range[END_REF].

Governing equations for flow regime

In this section we consider different situations which allow for simplified versions of the governing equations given above, allowing to address problems of electromagnetic/mechanical coupling during fluid flow in partially-saturated porous rocks. Let us start considering that the applied pressure gradients are steady in time, and that we are in a position to consider the Quasi-Stationary Conduction (QSC) approximation of Maxwell's equations [START_REF] Rapetti | On quasi-static models hidden in Maxwell's equations[END_REF], that is, we can neglect all explicit time dependence in them. In this case, we can assume that

E = -∇Φ. (220)
If the solid frame can be assumed to be rigid as it is usual in groundwater flow problems [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF], the governing equations reduce to the system

∇ • J = 0, ( 221
) ∇ • ( ẇw + ẇnw ) = 0, ( 222
) J = -σ ps ∇Φ -L ps 0 ∇p w , ( 223
) ẇw = -L ps 0 ∇Φ - k 0,w η w ∇p w , ( 224
) ẇnw = - k 0,nw η nw ∇p nw , (225)
p c = p nw -p w . (226)
Note that Eqs. ( 221) and ( 222) state the conservation of charges and flow, respectively. Also notice that the capillary relation Eq. ( 226) has to be taken into account. This is usually dealt with by introducing a capillary-pressure function depending on saturation [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]. This set of equations describe the coupled electric/mechanic steady two-phase flow in a partially saturated porous rock.

In the case were quasistatic perturbations are applied to the porous rock, Eqs. ( 220) and ( 221) can still be considered to be valid [START_REF] Haines | Seismoelectric numerical modeling on a grid[END_REF][START_REF] Monachesi | An analytical study of seismoelectric signals produced by 1D mesoscopic heterogeneites[END_REF][START_REF] Rosas-Carbajal | Seismoelectric Signals Producedc by Mesoscopic Heterogeneities[END_REF]. However, Eq. ( 222)

no longer holds, and should be replaced by a set of constitutive relations appropriate for the case where p c ̸ = 0 (see for example [START_REF] Santos | Static and dynamic behavior of a porous solid saturated by a two-phase fluid[END_REF]). This case will be addressed in a forthcoming paper.

If, on the other hand, capillary pressure perturbations can be assumed to be zero, and also we are dealing with highly consolidated rocks (K m ≫ K f ) such that ∇ • u s can be neglected, then Eq. ( 222) should be replaced by ṗ = -M ∇ • ẇf and the quasistatic equations can be written as

∇ • J = 0, (227)
∇ • ẇf = - ṗ M , (228)
J = -σ ps ∇Φ -L ps 0 ∇p, (229)

ẇf = ρ w ρ f L ps 0 ∇Φ - ρ w ρ f k 0,w η w + ρ nw ρ f k 0,nw η nw ∇p. (230)
In the case of full water saturation, following the same procedure as before, it is easy to show that Eqs. ( 227)-( 230) are coincident with the corresponding ones in [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. Electromagnetic/acoustic coupling in partially-saturated porous rocks

Another particular set of equations is obtained when the non-wetting fluid is assumed to be stagnant (p nw = 0 and w nw = 0). This case finds useful applications in the study of partially-saturated steady water flow in the vadose zone, where the non-wetting phase (air) is in direct contact with the atmosphere.

Therefore its pressure perturbations p nw are zero (air at constant atmospheric pressure). Then, for the capillary pressure we have p c = -p w , which is the reason why this case is also known as capillary flow [START_REF] Perrier | Characterization of electrical daily variations induced by capillary flow in the non-saturated zone[END_REF]:

∇ • J = 0, (231)
In some applications, body forces originated in gravity effects play an important role, so they must be included in this set of equations. This is accomplished by simply replacing ∇p c by (∇p c + ρ w g), where g is the acceleration of gravity. In the case of full water saturation, following the same procedure as above, it is easy to show that Eqs. ( 231)-( 234) are coincident with Pride (1994, Eqs. 258-261).

If quasistatic perturbations and highly consolidated rocks are considered in this capillary flow regime, Eq. ( 232) should be replaced by ṗc = M ∇ • ẇw , where

M = K w s w ϕ + Kw K 2 s [(1 -ϕ)K s -K m ] . ( 235
)
This coefficient is obtained upon the volume average of the mechanical constitutive relations in partially-saturated rocks where the non-wetting fluid is assumed stagnant (see Appendix A).

Conclusions

We have derived a low-frequency extension of Pride's theory accounting for the case of partially-saturated homogeneous and isotropic porous rocks, where the wetting fluid is assumed to be an ideal electrolyte and the non-wetting fluid is air. The main hypotheses on deriving the governing equations for coupled electromagnetic/acoustic wave propagation are the existence of an additional electric double layer at the water-air interface and that both electric double layers don't interact, which would be the case when the corresponding Debye lengths are smaller than any other geometrical feature of the porous rock. This condition is likely to be met for any saturation condition, with the exception of very low water saturation, where thin-water films could be small enough to make the later hypothesis not valid. Also, both air and solid phases are assumed to be electrical insulators and to have high dielectric contrasts when compared with the wetting phase. Moreover, in our derivation the ion number density perturbations were neglected together with the capillary pressure perturbations in wave propagation frequency regime. For a given value of water saturation, the deduced governing equations show that both the electrokinetic coupling coefficient and the electric conductivity have contributions from the water-air electric double layer and also depend on water saturation and topological properties of the partially-saturated porous rock. We've also shown that in the limit of full water saturation the final equations coincide with Pride's model, as expected. We have also obtained simplified versions of the proposed governing equations valid for flow regime. The derived electrokinetic coupling L ps 0 and electrical conductivity σ ps will allow to model and interpret the experimental observations of the related streaming potential coefficient in unsaturated conditions taking into account the importance of the water/air interface.

6. 1

 1 Governing equations for coupled electromagnetic/acoustic wave propagation In this section, all the derived electromagnetic, mechanical and transport equations are gathered. The so-called electrokinetic coupling coefficient is simply noted by L ps 0 in what follows, where the supraindex ps express the fact that the coefficient is valid for a partially-saturated porous rock. The same notation is employed for the electric conductivity (198), i.e. σ ps .

∇

  • ẇw = 0, (232) J = -σ ps ∇Φ + L ps 0 ∇p c ,

  The electric potential distribution near each shear plane Sw and Snw is represented, where ζw, dw and ζnw, dnw are their respective zeta potentials and Debye lengths. The main assumption is that both Debye lengths are much smaller than any geometrical feature of the porous space, including the thickness of the thin water films δ.

					shows how
	A	B	S w	δ		S nw
		solid grain			air
		C
		Electric potential	ζ w	d w	d nw	ζ nw
	Fig. 1 A. Representation of partially-saturated porous rock (adapted from Culligan et al
	(2004)). Solid grains, water (wetting phase) and air (non-wetting phase) are represented in
	gray, white and black, respectively, B. Amplified view of a thin water film of thickness δ. The
	charge distribution is represented at both interfaces and the location of the corresponding
	shear planes are indicated; C.

Acknowledgments. L.M. and F.Z. acknowledge support from FONCYT through grant PICT 2019-03220. F.Z. acknowledges support from CONICET through grant PIP 112-201501-00192. L.J., L.M. and F.Z. acknowledge support from CNRS/INSU through the PICS SEISMOFLUID.

Declarations

Not applicable

Appendix A Mechanical constitutive relations in the case of stagnant non-wetting fluid If the non-wetting fluid is assumed to be stagnant (p nw = 0, w nw = 0 and p w = -p c ) then the volume average of the stress-strain relations are taken without the consideration of Eq. ( 60). Following [START_REF] Pride | Deriving the equations of motions for porous isotropic media[END_REF], taking into account that the wetting-fluid occupies a volume s w ϕ we obtain

where

In these expressions,

From Eq. (A2), if the solid displacement is negligible, then taking the first time derivative we have ṗc = M ∇ • ẇw .