Feng Cheng
email: fengcheng@hubu.edu.cn

Chao-Jiang Xu
email: xuchaojiang@nuaa.edu.cn

ON THE RADIUS OF SPATIAL ANALYTICITY FOR THE INVISCID BOUSSINESQ EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 35Q35, 76B03, 76W05 Boussinesq equations, Analyticity, Radius of analyticity

In this paper, we study the problem of analyticity of smooth solutions of the inviscid Boussinesq equations. If the initial datum is real-analytic, the solution remains real-analytic on the existence interval. By an inductive method we can obtain lower bounds on the radius of spatial analyticity of the smooth solution.

Introduction

In this paper, we consider the following multi-dimensional invisicd Boussinesq equation on the torus T d ,

         ∂ t u + (u • ∇)u + ∇p = θe d , ∂ t θ + (u • ∇)θ = 0, div u = 0,
u(x, 0) = u 0 (x), θ(x, 0) = θ 0 (x), (1.1) with div u 0 = 0. Here, u = (u 1 , . . . , u d ) is the velocity field, p the scalar pressure, and θ the scalar density. e d denotes the vertical unit vector (0, . . . , 0, 1). The Boussinesq systems play an important role in geophysical fluids such as atmospheric fronts and oceanic circulation (see, e.g., [START_REF] Guo | Spectral method for solving two-dimensional Newton-Boussinesq equation[END_REF][START_REF] Majda | Introduction to PDEs and Waves for the Atomsphere and Ocean[END_REF][START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF]). Moreover, the Boussinesq systems are important for the study of the Rayleigh-Benard convection, see [START_REF] Constantin | Infinite Prandtl number convection[END_REF][START_REF] Doering | Applied Analysis of the Navier-Stokes Equations[END_REF].

Besides the physical importance, the invisicd Boussinesq equations can also be viewed as simplified model compared with the Euler equation. In the case d = 2, the 2D inviscid Boussinesq equations share some key features with the 3D Euler equations such as the vortex stretching mechanism. It was also pointed out in [START_REF] Majda | Vorticity and incompressible flow[END_REF] that the 2D invisicid Boussinesq equations are identical to the Euler equations for the 3D axisymmetric swirling flows outside the symmetric axis.

The inviscid Boussinesq equations have been studied by many authors through the years, for instance [START_REF] Chae | Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations[END_REF][START_REF] Chae | Local existence and blow-up criterion for the Boussinesq equations[END_REF][START_REF] Shu | Small scale structures on Boussinesq convection[END_REF][START_REF] Guo | Spectral method for solving two-dimensional Newton-Boussinesq equation[END_REF][START_REF] Liu | Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces[END_REF][START_REF] Taniuchi | A note on the blow-up criterion for the inviscid 2D Boussinesq equations[END_REF][START_REF] Xu | Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms[END_REF][START_REF] Yuan | Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces[END_REF]. Specially, Chae and Nam [START_REF] Chae | Local existence and blow-up criterion for the Boussinesq equations[END_REF] studied local existence and uniqueness of the inviscid Boussinesq equation and some blow-up criterion in the Sobolev space, Yuan [START_REF] Yuan | Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces[END_REF] and Liu et al. [START_REF] Liu | Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces[END_REF] in the Besov space, Chae and Kim [START_REF] Chae | Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations[END_REF] and Cui et al. [START_REF] Cui | Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces[END_REF] in the Hölder spaces, Xiang and Yan [START_REF] Xiang | On the well-posedness of the Boussinesq equation in the Triebel-Lizorkin-Lorentz spaces[END_REF] in the Triebel-Lizorkin-Lorentz spaces. It was remarked that the global regularity for the inviscid Boussinesq equations even in two dimensions is a challenging open problem in mathematical fluid mechanics.

In this paper, we are concerned with the analyticity of smooth solutions of the inviscid Boussinesq equations (1.1). The analyticity of the solution for Euler equations in the space variables, for analytic initial data is an important issue, studied in [START_REF] Alinhac | Propagation de l'analyticité des solutions de systèmes hyperboliques non-linéaires[END_REF][START_REF] Benachour | Analyticité des solutions périodiques de l'équation d'Euler en trois dimensions[END_REF][START_REF] Bardos | Domaine d'analyticité des solutions de l'équation d'Euler dans un ouvert de R n[END_REF][START_REF] Bardos | Analyticité des solutions périodiques de l'équation d'Euler en deux dimensions[END_REF][START_REF] Kukavica | On the radius of analyticity of solutions to the three-dimensional Euler equations[END_REF][START_REF] Kukavica | The domain of analyticity of solutions to the three dimensional Euler equations in half space[END_REF][START_REF] Levermore | Analyticity of solutions for a generalized Euler equation[END_REF]. In particular, Kukavica and Vicol [START_REF] Kukavica | On the radius of analyticity of solutions to the three-dimensional Euler equations[END_REF] studied the analyticity of solutions for the Euler equations and obtained that the radius of analyticity τ (t) of any analytic solution u(t, x) has a lower bound

τ (t) ≥ C(1 + t) -2 exp -C 0 t 0 ∇u(s, •) L ∞ ds (1.2)
for a constant C 0 > 0 depending on the dimension and C > 0 depending on the norm of the initial datum in some finite order Sobolev space. The same authors in [START_REF] Kukavica | The domain of analyticity of solutions to the three dimensional Euler equations in half space[END_REF] obtained a better lower bound for τ (t) for the Euler equations in a half space replacing (1 + t) -2 by (1 + t) -1 in (1.2). In [START_REF] Cheng | On the Gevrey regularity of solutions to the 3D ideal MHD equations[END_REF], we have investigated the Gevrey analyticity of the smooth solution for the ideal MHD equations following the method of [START_REF] Kukavica | On the radius of analyticity of solutions to the three-dimensional Euler equations[END_REF]. The approach used in [START_REF] Kukavica | On the radius of analyticity of solutions to the three-dimensional Euler equations[END_REF][START_REF] Kukavica | The domain of analyticity of solutions to the three dimensional Euler equations in half space[END_REF][START_REF] Levermore | Analyticity of solutions for a generalized Euler equation[END_REF][START_REF] Cheng | On the Gevrey regularity of solutions to the 3D ideal MHD equations[END_REF] relies on the energy method in infinite order Gevrey-Sobolev spaces. Recently, Cappiello and Nicola [START_REF] Cappiello | Some remarks on the radius of spatial analyticity for the Euler equations[END_REF] developed a new inductive method to simplify the proof of [START_REF] Kukavica | On the radius of analyticity of solutions to the three-dimensional Euler equations[END_REF][START_REF] Kukavica | The domain of analyticity of solutions to the three dimensional Euler equations in half space[END_REF]. In this paper, we shall apply this inductive method to study the analyticity of smooth solution for the inviscid Boussinesq equations. The main additional difficulty arise from the estimate of the weak coupling term u • ∇θ.

The paper is organized as follows. In Section 2, we will give some notations and state our main results. In Section 3, we first recall some known results and then give some lemmas which are needed to prove the main Theorem. In Section 4, we finish the proof of Theorem 2.1.

Notations and Main Theorem

In this section we will give some notations and function spaces which will be used throughout the following arguments. Throughout the paper, C denotes a generic constant which may vary from line to line. Since we work on the torus T d throughout the paper, we shall write the function space L 2 or H k to represnt the functions that are squre integrable or squre integrable up to k-th derivative without mentioning the domain T d .

Let v = (v 1 , . . . , v d ) be a vector function, we say that v ∈ L 2 which means

v i ∈ L 2 for each 1 ≤ i ≤ d. We denote the L 2 norm of v by v L 2 = 1≤i≤d v i 2 L 2 . Let ρ be a scalar function, we say the pare (v, ρ) ∈ L 2 if v, ρ ∈ L 2 . We denote the L 2 norm of the pare (v, ρ) to be (v, ρ) L 2 = v 2 L 2 + ρ 2 L 2 . Denote •,
• to be the inner product in L 2 either for vector function or scalar function.

In [START_REF] Lions | Problemès aux limites non homogènes et applications[END_REF], it is stated that a smooth function f is uniformly analytic in T d if there exist M, τ > 0 such that

∂ α f L ∞ ≤ M |α|! τ |α| , (2.1)
for all multi-indices α = (α 1 , . . . , α d ) ∈ N d 0 , where |α| = α 1 + . . . + α d . The supremum of the constant τ > 0 in (2.1) is called the radius of the analyticity of f . Notice that we can also replace the L ∞ norm with a Sobolev norm H k , k ≥ 0.

Let n ≥ 0 be an integer and α, β ∈ N d be multi-indices, then the sequence

M n = n! (n + 1) 2 satisfies β<α α β M |α-β| M |β|+1 ≤ C|α|M |α| , (2.2)
for all multi-index α, β ∈ N d and some universal constant C, for proof please refer to [START_REF] Alinhac | Propagation de l'analyticité des solutions de systèmes hyperboliques non-linéaires[END_REF].

With these notations, we can state our main results.

Theorem 2.1. Let k > d 2 + 1, (u 0 , θ 0 ) be analytic in T d , satisfying div u 0 = 0 and ∂ α (u 0 , θ 0 ) H k ≤ BA |α|-1 |α|!/(|α| + 1) 2 , α ∈ N d , (2.3)
for some B ≥ 9 4 (u 0 , θ 0 ) H 2k+1 and A ≥ 1. Let u(t, x), θ(t, x) be the corresponding H k maximal solution of the inviscid Boussinesq equations (1.1), with the initial datum (u 0 , θ 0 ). Then u(t, x), θ(t, x) is analytic, and there exists constants C 0 , C 1 > 0, depending only on k and d, such that the radius of analyticity satisfying

τ (t) ≥ 1 A(1 + C 1 Bt) exp -C 0 t 0 1 + ∇u(s, •) L ∞ + ∇θ(s, •) L ∞ ds . (2.4)
Remark 2.1. In the case θ = 0, Theorem 2.1 recovers the result of Kukavica and Vicol [START_REF] Kukavica | On the radius of analyticity of solutions to the three-dimensional Euler equations[END_REF] and Cappiello and Nicola [START_REF] Cappiello | Some remarks on the radius of spatial analyticity for the Euler equations[END_REF] for the incompressible Euler equation.

Remark 2.2. When the dimension d = 2, the blow-up criterion proved by Chae and Nam in [START_REF] Chae | Local existence and blow-up criterion for the Boussinesq equations[END_REF] stated that the solution remains smooth up to T as long as

T 0 ∇θ(•, s) L ∞ ds < ∞.
So, it will be very interesting if the quantity

t 0 1 + ∇u(•, s) L ∞ + ∇θ(•, s) L ∞ ds
in the lower bound of the radius of analytic solution in (2.4) can be replaced by t 0 ∇θ(•, s) L ∞ ds.

The estimate of the Sobolev norm

In order to prove the main Theorem 2.1, we recall the following results about the local existence and uniqueness of H k -solution of the inviscid Boussinesq equations (1.1) which is a proposition in [START_REF] Wang | Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain[END_REF] for d = 3. Theorem 3.1 (Wang-Xie, Proposition 1.2 of [START_REF] Wang | Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain[END_REF]). If u 0 (x), θ 0 (x) ∈ H 3 (Ω) and u 0 (x) satisfies the divergence-free condition, then there exists T 2 > 0 such that the inviscid problem (1.1) admits a unique solution

u, θ ∈ C 0, T 2 ; H 3 (Ω) ∩ C 1 0, T 2 ; H 2 (Ω) .
The domain considered in [START_REF] Wang | Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain[END_REF] is a boundaed domain with smooth boundary conditions and this case can be naturally extended to periodic domain with periodic boundary conditions. The proof is due to the argument in [START_REF] Chae | Generic solvability of the axisymmetric 3D Euler equations and the 2D Boussinesq equations[END_REF] and [START_REF] Li | Boundary value problems for quasilinear hyperbolic system[END_REF]. Since it is standard, we omit the details here. When the dimension d = 2, Chae and Nam [START_REF] Chae | Local existence and blow-up criterion for the Boussinesq equations[END_REF] also proved the local existence and blow-up criterion.

In order to prove the main Theorem, we will need the following Lemma.

Lemma 3.2. Let d = 2, 3, and k > d 2 + 1 be fixed. Let (u, θ) ∈ C(0, T 1 ; H k ) be the corresponding maximal H k -solution of (1.1) with intial data (u 0 , θ 0 ) ∈ H k and u 0 satisfies the divergence-free condition and periodic boundary condition, then

∀ 0 ≤ t < T 1 , u(t, •), θ(t, •) H k ≤ (u 0 , θ 0 ) H k × exp C 0 t 0 1 + ∇u(s, •) L ∞ + ∇θ(s, •) L ∞ ds , (3.1)
where the constant C 0 depending on the dimension and k.

Proof. Since the initial data (u 0 , θ 0 ) ∈ H k satisfies ∇ • u 0 = 0 and the periodic boundary conditions, the local existence of the H k -solution (u, θ) is already known. We here only need to show the

H k energy estimate (3.1). Let α ∈ N d satisfies 0 ≤ |α| = α 1 + α 2 + . . . + α d ≤ k.
We first apply the ∂ α on both sides of the first equation of (1.1) and then take the L 2 -inner product with ∂ α u with both sides, which gives 1 2

d dt ∂ α u 2 L 2 + ∂ α (u • ∇u), ∂ α u = ∂ α (θe d ), ∂ α u , (3.2)
where the pressure term ∇∂ α p, ∂u = 0 is due to the fact u is divergence free and the domain considered here is a periodic domain.

We then apply the ∂ α on both sides of the second equation of (1.1) and take the L 2 inner product with ∂ α θ on both sides, which gives 1 2

d dt ∂ α θ 2 L 2 + ∂ α (u • ∇θ), ∂ α θ = 0. (3.3)
Now adding (3.2) with (3.3) and taking summation over 0 ≤ |α| ≤ k, we can obtain 1 2

d dt (u, θ) 2 H k = 0≤|α|≤k ∂ α (θe d ), ∂ α u - 0≤|α|≤k ∂ α (u • ∇u), ∂ α u - 0≤|α|≤k ∂ α (u • ∇θ), ∂ α θ . (3.4)
The Hölder inequality implies that

0≤|α|≤k ∂ α (θe d ), ∂ α u ≤ θ H k u H k . (3.5)
Notice that u is divergence free, by use of the Sobolev inequality which can be found in [START_REF] Majda | Vorticity and incompressible flow[END_REF] we can obtain

0≤|α|≤k ∂ α (u • ∇u), ∂ α u ≤ C ∇u L ∞ u 2 H k , (3.6)
where the constant C depends on k and the dimension d. In the same way, we can obtain

0≤|α|≤k ∂ α (u • ∇θ), ∂ α θ ≤ C ∇u L ∞ u H k + ∇θ L ∞ u H k θ H k , (3.7)
where the constant C also depends on k and the space dimension d. Substituting (3.5), (3.6) and (3.7) into (3.4), we obtain

1 2 d dt (u, θ) 2 H k ≤ C ∇u L ∞ + ∇θ L ∞ (u, θ) 2 H k + C (u, θ) 2 H k ≤ C 1 + ∇u L ∞ + ∇θ L ∞ (u, θ) 2 H k , (3.8)
where C is some constant depending on k, d. By the Gronwall inequality, (3.1) is then proved.

Lemma 3.2 tells us that the solution for the inviscid Boussinesq equation has the same Sobolev regularity as the initial data. In the following Lemma we will show that if the initial data (u 0 , θ 0 ) ∈ H k for arbitrary k ≥ 3, there exists an interval [0, T ] uniformly with respect to k such that the unique smooth solution (u, θ) ∈ L ∞ ([0, T ]; H k ). Lemma 3.3. Let (u, θ) be the H 3 -solution of the inviscid Boussinesq equation (1.1) on the time interval [0, T ], with initial data (u 0 , θ 0 ) ∈ H 3 . Then for all k ≥ 3, if (u 0 , θ 0 ) ∈ H k , the corresponding solution (u, θ) satisfies

(u, θ) ∈ L ∞ [0, T ], H k for all k ≥ 3.
Proof. We claim that for every 3 ≤ m ≤ k there exist a constant C m such that

sup 0≤s≤T u(•, s), θ(•, s) H m ≤ C m .
For m = 3 the statement follows from our assumption. Take 4 ≤ m ≤ k and suppose that the statement is true for m -1, i. e.

sup 0≤s≤T u(•, s), θ(•, s) H m-1 ≤ C m-1 .
We take the H m inner product of the first equation of (1.1) with u and take the H m inner product of the second equation of (1.1) with θ, which gives

1 2 d dt (u, θ) 2 H m = 0≤|α|≤m ∂ α (θe d ), ∂ α u - 0≤|α|≤m ∂ α (u • ∇u), ∂ α u - 0≤|α|≤m ∂ α (u • ∇θ), ∂ α θ .
For 4 ≤ m ≤ k, by (3.8) we obtain

d dt (u, θ) H m ≤ C 1 + ∇u L ∞ + ∇θ L ∞ (u, θ) H m ,
where C is a constant depending on m, d. Then the Gronwall inequality yields

(u(•, t), θ(•, t)) H m ≤ (u 0 , θ 0 ) H m × exp C t 0 1 + ∇u(•, s) L ∞ + ∇θ(•, s) L ∞ ds . (3.9)
By Sobolev embedding inequality, we have

∇u L ∞ ≤ C ′ u H 3 , ∇θ L ∞ ≤ C ′ θ H 3 ,
for some constant C ′ . Then from (3.9) and the assumption, we have

(u(•, t), θ(•, t)) H m ≤ (u 0 , θ 0 ) H m × exp C t 0 (1 + C ′ u(•, s) H 3 + C ′ θ(•, s) H m )ds ≤ (u 0 , θ 0 ) H m exp C(1 + 2C ′ C 3 )t .
So it easily follows that

sup 0≤s≤T u(•, s), θ(•, s) H m ≤ C m ,
which proves the Lemma by induction.

Remark 3.1. In Lemma 3.3, the uniform lifespan [0, T ] of the Sobolev solution is independent of the Sobolev order k which allows us to take k → ∞. In other words, if the initial datum (u

0 , θ 0 ) is C ∞ , then solution (u(t, x), θ(t, x)) is also C ∞ for almost every t ∈ [0, T ].

Proof of Theorem 2.1

In this Section, we will give the proof of the main theorem.

Proof of Theorem 2.1. By Lemma 3.3, we know that the solution (u, θ) is smooth, since (u 0 , θ 0 ) is. Now we claim that for all |α| = N > 2 we have

(∂ α u(t, •), ∂ α θ(t, •)) H k M |α| ≤ 2BA N -1 (1 + C 1 Bt) N -2 × exp C 0 (N -1) t 0 (1 + ∇u(s) L ∞ + ∇θ(s) L ∞ )ds , (4.1)
where C 0 , C 1 are positive constants depending only on k and d. We set

E N [(u, θ)(t)] = sup |α|=N ∂ α (u(t, •), θ(t, •)) H k M |α| .
To prove the claim, we proceed by induction on N . The result is true for N = 2 by (3.1) with notice that k + 2 < 2k + 1 and B ≥ 9 4 (u 0 , θ 0 ) H 2k+1 , A ≥ 1. Hence, let N ≥ 3 and assume (4.1) holds for multi-indices α of length 2 ≤ |α| ≤ N -1 and prove it for |α| = N .

For |α| = N, |γ| ≤ k, we first apply ∂ α+γ on both sides of the first and the second equation of (1.1) and then take the L 2 -inner product with ∂ α+γ u and ∂ α+γ θ respectivily, which gives 1 2

d dt ∂ α+γ (u, θ) 2 L 2 + ∂ α+γ (u • ∇u), ∂ α+γ u + ∂ α+γ (u • ∇θ), ∂ α+γ θ = ∂ α+γ (θe d ), ∂ α+γ u . (4.2)
Denote Lw = L u w := u • ∇w and

I 1 = [∂ α+γ , L]u := ∂ α+γ (u • ∇u) -u • ∇∂ α+γ u, I 2 = [∂ α+γ , L]θ := ∂ α+γ (u • ∇θ) -u • ∇∂ α+γ θ, I 3 = (∂ α+γ θ)e d .
Taking summation with 0 ≤ |γ| ≤ k in (4.2), we have by the Cauchy-Schwartz inequality

d dt ∂ α u(t, •), θ(t, •) H k ≤ 0≤|γ|≤k I 1 L 2 + 0≤|γ|≤k I 2 L 2 + ∂ α (u, θ) H k , (4.3)
where we used the the standard argument in [pp. 47-48, [START_REF] Rauch | Hyperbolic Partial Differential Equations and Geometric Optics[END_REF]]. It remains to estimate I 1 and I 2 . Note that by the Leibniz rule, we can expand the expression of I 1 as follows

I 1 = β≤α α β ∂ γ (∂ α-β u • ∇∂ β u) = β≤α α β δ≤γ, |β|+|δ|<|α|+|γ| γ δ ∂ α-β+γ-δ u • ∇∂ β+δ u,
where the restriction |β|+|δ| < |α|+|γ| is due to the fact that u•∇∂ α+γ u, ∂ α+γ u = 0 because u is divergence free. By [START_REF] Cappiello | Some remarks on the radius of spatial analyticity for the Euler equations[END_REF], we have

I 1 L 2 ≤ β≤α α β δ≤γ, |β|+|δ|<|α|+|γ| γ δ ∂ α-β+γ-δ u • ∇∂ β+δ u L 2 ≤ C N M N ∇u L ∞ E N [(u, θ)] + N M N B 2 A N -1 × exp C 0 (N -1) t 0 (1 + ∇u(s) L ∞ + ∇θ(s) L ∞ )ds (1 + C 1 Bt) N -3 .
We then follow the ideal of [START_REF] Cappiello | Some remarks on the radius of spatial analyticity for the Euler equations[END_REF] to estimate I 2 . Similarly, we can expand I 2 as

I 2 = β≤α α β ∂ γ (∂ α-β u • ∇∂ β θ) = β≤α α β δ≤γ, |β|+|δ|<|α|+|γ| γ δ ∂ α-β+γ-δ u • ∇∂ β+δ θ. (4.4)
Then we divide the summation of the right of (4.4) into three parts

I 2 = I 21 + I 22 + I 23 ,
where

I 21 = β≤α 0 =|β|≤|α|-2 α β δ≤γ, |β|+|δ|<|α|+|γ| γ δ ∂ α-β+γ-δ u • ∇∂ β+δ θ, I 22 = β=α α α |δ|=|γ|-1 γ δ ∂ γ-δ u • ∇∂ α+δ θ + |β|=|α|-1 α β δ=γ γ γ ∂ α-β u • ∇∂ β+γ θ + β=0 α β δ=0 γ δ ∂ α+γ u • ∇θ,
and

I 23 = β=α α α |δ|≤|γ|-2 γ δ ∂ γ-δ u • ∇∂ α+δ θ + |β|=|α|-1 α β |δ|≤|γ|-1 γ γ ∂ α-β u • ∇∂ β+γ θ + β=0 α β δ =0 γ δ ∂ α+γ u • ∇θ.
Estimation of I 21 : With the fact that H k is an algbra if k > d 2 + 1 and |γ| ≤ k, we have

I 21 L 2 ≤ β≤α 0 =|β|≤|α|-2 α β δ≤γ, |β|+|δ|<|α|+|γ| γ δ ∂ α-β+γ-δ u • ∇∂ β+δ θ L 2 ≤ C β≤α 0 =|β|≤|α|-2 α β δ≤γ, |β|+|δ|<|α|+|γ| γ δ ∂ α-β u H k ∇∂ β θ H k .
∂ α-β u H k ≤ M |α-β| 2BA |α-β|-1 × exp C 0 (|α -β| -1) t 0 (1 + ∇u(s) L ∞ + ∇θ(s) L ∞ )ds (1 + C 1 Bt) |α-β|-2
(4.6) and

∇∂ β θ H k ≤ M |β|+1 2BA |β| × exp C 0 (|β|) t 0 (1 + ∇u(s) L ∞ + ∇θ(s) L ∞ )ds (1 + C 1 Bt) |β|-1 . (4.7)
Substituting (4.6) and (4.7) into (4.5) and employing (2.2), we obtain

I 21 L 2 ≤C β≤α 0 =|β|≤|α|-2 α β M |α-β| M |β|+1 B 2 A N -1 × exp C 0 (N -1) t 0 (1 + ∇u(s) L ∞ + ∇θ(s) L ∞ )ds (1 + C 1 Bt) N -3 ≤CN M N B 2 A N -1 × exp C 0 (N -1) t 0 (1 + ∇u(s) L ∞ + ∇θ(s) L ∞ )ds (1 + C 1 Bt) N -3 .
Estimation of I 22 : In a similar way, we rewrite I 22 as

I 22 = β=α α α |δ|=|γ|-1 γ δ ∂ γ-δ u • ∇∂ α+δ θ + |β|=|α|-1 α β δ=γ γ γ ∂ α-β u • ∇∂ β+γ θ + β=0 α β δ=0 γ δ ∂ α+γ u • ∇θ = R 21 + R 23 + R 23 .
For R 21 , we obtain

R 21 L 2 ≤ β=α α α |δ|=|γ|-1 γ δ ∂ γ-δ u • ∇∂ α+δ θ L 2 ≤ β=α α α |δ|=|γ|-1 γ δ ∂ γ-δ u • ∇∂ α+δ θ L 2 ≤ C ∇u L ∞ ∂ α θ H k . (4.8)
for some constant C depending on k.

For R 22 , we have the following estimate

R 22 ≤ |β|=|α|-1 α β δ=γ γ γ ∂ α-β u • ∇∂ β+γ θ L 2 ≤ |β|=|α|-1 N δ=γ γ γ ∂ α-β u • ∇∂ β+γ θ L 2 ≤ CdN M N ∇u L ∞ E N [θ].
(4.9)

Then for R 23 , we have

R 23 L 2 ≤ β=0 α β δ=0 γ δ ∂ α+γ u • ∇θ L 2 ≤ ∇θ L ∞ ∂ α u H k . (4.10)
Summing up the estimates of (4.8), (4.9) and (4.10), we obtain

I 22 L 2 ≤ C ∇u L ∞ ∂ α θ H k + ∇θ L ∞ ∂ α u H k + dN M N ∇u L ∞ E N (θ) .
Estimate of I 23 : We divide I 23 as follows

I 23 = β=α α α |δ|≤|γ|-2 γ δ ∂ γ-δ u • ∇∂ α+δ θ + |β|=|α|-1 α β |δ|≤|γ|-1 γ γ ∂ α-β+γ-δ u • ∇∂ β+γ θ + β=0 α β δ =0 γ δ ∂ α+γ-δ u • ∇∂ δ θ = R 31 + R 32 + R 33 .
For R 31 , we have

R 31 L 2 ≤ C β=α α α |δ|≤|γ|-2 γ δ ∂ γ-δ u • ∇∂ α+δ θ L 2 ≤ C ∂ γ-δ u L ∞ ∇∂ α+δ θ L 2 ≤ C ∂ γ-δ u H k ∂ α θ H k-1 ≤ C ∂ γ-δ (u, θ) H k ∂ α (u, θ) H k-1 ≤ C (u 0 , θ 0 ) H 2k exp C 0 t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds sup j:αj ≥1 ∂ α-ej (u, θ) H k ≤ C (u 0 , θ 0 ) H 2k exp C 0 t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds × M N -1 BA N -2 exp C 0 (N -2) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds (1 + C 1 Bt) N -3 ≤ CM N -1 B 2 A N -2 exp C 0 (N -1) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds (1 + C 1 Bt) N -3 , (4.11
) where we used the fact |γ -δ| + k ≤ 2k in (4.11), the inductive hypothesis (4.1), and the fact that B > (u 0 , θ 0 ) H 2k+1 . For R 32 , we have

R 32 L 2 ≤ |β|=|α|-1 α β |δ|≤|γ|-1 γ γ ∂ α-β+γ-δ u • ∇∂ β+γ θ L 2 ≤ |α| ∂ α-β+γ-δ u L ∞ ∇∂ β+δ θ L 2 ≤ C|α| ∂ γ-δ u H k ∂ β θ H k ≤ C|α| ∂ γ-δ (u, θ) H k ∂ β (u, θ) H k ≤ CN M N -1 (u 0 , θ 0 ) H 2k exp[C 0 t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds] × BA N -2 exp C 0 (N -2) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds (1 + C 1 Bt) N -3 ≤ CM N -1 B 2 A N -2 exp C 0 (N -1) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds (1 + C 1 Bt) N -3 .
(4.12) For R 33 , in a similar way we have (4.15) We can take C 0 ≥ C in (4.15), so that Grownwall inequality [Lemma 2.1 in [START_REF] Cappiello | Some remarks on the radius of spatial analyticity for the Euler equations[END_REF]] gives

R 33 L 2 ≤ β=0 α β δ =0 γ δ ∂ α+γ-δ u • ∇∂ δ θ L 2 ≤ C ∂ α u H k-1 ∇∂ δ θ H k ≤ CM N -1 B 2 A N -2 exp C 0 (N -1) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds (1 + C 1 Bt) N -3 .
E N [(u, θ)(t)] ≤ exp C 0 (N -1) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds × E N [(u, θ)(0)] + CN B 2 A N -1 t 0 (1 + C 1 Bs) N -3 ds ≤ exp C 0 (N -1) t 0 (1 + ∇u L ∞ + ∇θ L ∞ )ds × E N [(u, θ)(0)] + CN C 1 (N -2) BA N -1 (1 + C 1 Bt) N -2 .
Note that we have E N [(u, θ)(0)] ≤ BA N -1 by the assumption (2.3). If we choose C 1 = 3C, so that 3C C1 = 1, since A ≥ 1, N ≥ 3 we have

E N [(u, θ)(0)] + CN C 1 (N -2) BA N -1 (1 + C 1 Bt) N -2 ≤ BA N -1 + 3C C 1 BA N -1 (1 + C 1 Bt) N -2 ≤ 2BA N -1 (1 + C 1 Bt) N -2 ,
and we obtain exactly (4.1) for |α| = N . Then the theorem is proved.

(4. 5 )

 5 Noting that 2 ≤ |α -β| ≤ N -1 and 2 ≤ |β| + 1 ≤ N -1, the hypothesis (4.1) for 2 ≤ |α| ≤ N -1 indicates that

(4. 13 )I 2 L 2 ≤ 1 + 0 ( 1 + 0 C(N - 1 )( 1 + 0 CN A N - 1 B 2 ( 1 + C 1 3 × 0 ( 1 +

 132210101101211301 Summing up (4.11),(4.12) and (4.13), we obtain CN M N ( ∇uL ∞ + ∇θ L ∞ )E N [(u, θ)] + CN M N B 2 A N -1 exp C 0 (N -1) t 0 ∇u(s) L ∞ + ∇θ(s) L ∞ ds .Combining the estimates I 1 and I 2 , we have from (4.3)d dt ∂ α (u, θ)(t) H k M |α| ≤ C(N -1)(1 + ∇u(t) L ∞ + ∇θ(t) L ∞ )E N [(u, θ)(t)] + CN A N -1 B 2 exp C 0 (N -1) t ∇u(s) L ∞ + ∇θ(s) L ∞ )ds × (1 + C 1 Bt) N -3 ,(4.14) where we used the Cauchy-Schwartz inequality and the fact N N -1 ≤ 3 2 if N ≥ 3 and the constant C > 0 depending only on the dimension d and k. Now we integrate (4.14) from 0 to t and take the supremum on |α| = N . We obtainE N [(u, θ)(t)] ≤ E N [(u 0 , θ 0 )] + t ∇u(t) L ∞ + ∇θ(t) L ∞ )E N [(u, θ)(s)]ds + t Bs) N -exp C 0 (N -1)s ∇u(ℓ) L ∞ + ∇θ(ℓ) L ∞ )dℓ ds.

Acknowledgements. The research of the second author is supported partially by "The Fundamental Research Funds for Central Universities of China".


Seonho Seok
email: seonho.seok@u-psud.fr

Hyungdal Park

Wonsuk Choi

Ockchul Kim

Jinseok Kim

Analysis of Thin Film Parylene-Metal-Parylene Device Based on Mechanical Tensile Strength Measurement

Keywords: FEM (Finite Element Method), Parylene, Neural probe, Implantable

This paper presents an FEM analysis and experiment of parylene-metal-parylene flexible substrate for implantable medical devices. Tensile strength measurement of the parylene-metal-parylene has been carried out and corresponding FEM modeling and simulation has been done to understand its mechanical behaviour. Besides, frequently encountered metal delamination on parylene substrate has been studied based on cohesive zone model of interface between the two materials.

I. INTRODUCTION

Many research efforts have been made to develop and improve of the prosthetic hands and arms for the amputees, and in recent years much progress has been observed in the development of life-like robotic hands and the means of controlling them with greater degree of freedom. For this purpose, micro-electro-mechanical systems (MEMS) technlogies have been used to fabricate neural inteface probes [START_REF] Najafi | Scaling limitation on silicon multichannel recoding probe[END_REF][START_REF] Campell | A silicon-based, three-dimensional neural interface: manufacturing processes for an intracortical electrode array[END_REF][START_REF] Takeuchi | 3D flexible multichannel neural probe array[END_REF]. However, existing MEMS-based neural electrodes would have limitation on neural interface due to its material characteristics. Thus, flexible neural electrodes have been recently proposed to minimize mechanical mismatch between the electrode and tissue after the electrode's implantation for a stable long-term recording. To this sense, peripheral neural interface (PNI) devices have been appeared to retrieve and send neural signals directly from and to the residual or existing peripheral nerves in this field [START_REF] Kim | Novel Neural Interface Electrode Array for the Peripheral Nerve[END_REF]. Recently, thin film flexible polymeric devices are being used for measuring nerve impulse from the central or peripheral nerve systems. While thin-film polymeric devices are flexible and biocompatible, they are prone to delamination and carry concerns about their mechanical robustness [START_REF] Lee | Mechanical properties of thin-film Parylenemetal-Parylen devices[END_REF]. A nanopillar array created by plasma etching could be used to enhance adhesion among different materials in the parylene-metal-parylene system [START_REF] Xie | Improving adhesion strength between layers of an implantable parylene-C electrode[END_REF]. ALD (Atomic layer deposition) Al 2 O 3 combined with the silane adhesion promoter A-174 would increase adhesion force between two parylene films [START_REF] Lee | Mechanical properties of thin-film Parylenemetal-Parylen devices[END_REF]. Parylene material has been shown that mechanical properties can be maintained after stored in PBS (Phosphate Buffered Saline) solution up to 12 months [START_REF] Oliva | On the use of Parylene C polymer as substrate for periphral nerve electrode[END_REF]. Therefore, mechanical or chemical treatments of the interface between polymer substrate and metal film is frequently required [START_REF] Chang | Adhesion-enhancing surface treatments for parylene deposition[END_REF].

In this paper, mechanical characterization of parylene based peripheral neural device has been presented. FEM models and simulation have been used to comprehend its mechanical behaviour and the effect of metal adhesion form on possible metal delamination on parylene substrate. For example, stress distribution of a polymeric neural probe winding a nerve has been presented in Fig. 1. There could be a critical failure near the probe edge due to applied axial force during implantation or operation. Note that the stress distribution on the neural probe depends on the way how it is winded about the nerve. Therefore, it is highly required to characterize mechanical properties of the thin-film probe device. Neural probe based on parylene typically has multi-layer structure consisting of top parylene, center metal electrode, and bottom parylene. Thus, a parylene-metal-parylene substrate has been fabricated to extract mechanical parameters required for FEM modeling. Fig. 2 shows the conceptual drawing of the test sample. The metal lines in between the parylene layers would be used as resistance monitor during tensile loading test. After metal electrode patterning, it has been observed that some electrodes have been delaminted from the parylene substrate as shown in Fig. 5. The interface between metal electrode and parylene has the lowest adhesion compared with other interfaces in parylene-metal-parylene system [START_REF] Xie | Improving adhesion strength between layers of an implantable parylene-C electrode[END_REF]. As maximal adhesion at metal-parylene interface is less than 2.5 MPa even with CF4 etch, it is included in FEM model of the metal delamination from parylene substrate [START_REF] Xie | Improving adhesion strength between layers of an implantable parylene-C electrode[END_REF]. Displacement load of 500 µm has been applied into one end, which the other end is fixed. Fig. 11 shows stress distribution of gold electrode and parylene substrate. Maximal stress is found near the fixed end which is corresponding to the test experiment as some cracked parylene near the fractured side of the test sample has been found as shown in Fig. 10. The test parylene sample has fractured at applied strain of 1.7 % as shown in Fig. 12. Strain at break of 1.7 % is too small compared with typical parylene mechanical properties [START_REF]Parylene properties chart for parylene N, C, D[END_REF]. Simulation result also shows tensile strength of the parylene material is reached when 10 % strain is applied. Note that tensile strength of the parylene-C material is 69 MPa [START_REF]Parylene properties chart for parylene N, C, D[END_REF]. Delamination of the metal has been studied based on CZM (Cohesive Zone Model) defining as parameters such as maximum contact normal stress and critical fracture energy [START_REF] Seok | A Theoretical Study on Post-it-like Debonding Process for BCB Cap Transfer Packaging Based on FEM Simulation[END_REF][START_REF] Seok | Fabrication and Modeling of Nitride Thin Film Encapsulation Based on Anti-Adhesion-Assisted Transfer Technique and Nitride/BCB Bilayer Wrinkling[END_REF]. Metal stress is chosen as input parameter together with CZM interface parameters. Fig. 13 shows deformed metal film due to delamination from parylene substrate and the deformed profile along with a side of the metal when metal film has residual stress of 230 MPa. Therefore, interface between metal film and parylene substrate should be carefully cared to achieve good adhesion.

III. CONCLUSION

Mechanical characteristics of parylen-metal-parylene system for implantable neural probe has been evaluated by using tensile strength measurement. Mean Young's modulus is 2.67 GPa similar value to typical parylene material, while strain at break is estimated 1.7 % which much lower than the expected value. Thus, more studies are needed for such lower elongation at break. Metal delamination on parylene substrate has been found and adhesion force of the metal has been studied based on FEM modeling and simulation. Residual stress of the metal could be a reason of the metal delamination even if parylene surface has been modified to enhance interface energy.

Fig. 1

 1 Fig. 1 An example of stress distribution of a neural probe winding a nerve

Fig. 2

 2 Fig. 2 Conceptual drawing of test sample for parylene-metal-parylene substrate

Fig. 3 Fig. 4

 34 Fig. 3 Fabrication process of the test parylene sample

Fig. 5

 5 Fig. 5 Metal delamination on parylene substrate

  Measure profile Fig. 6 Structure of the parylene-metal-parylene substrate for mechanical characterization A. Mechanical characterization To find mechanical properties of the parylene material and its mechanical behaviour, a tensile strength test device has been fabricated as shown in Fig. 7. The test device has size of 45 mm in length, 1.5 mm in width and 10 µm in thickness and two gold lines having 330 nm in thickness and 200 µm in width are placed in between the two paylene layers. The tensile strength test with Shimadzu EZ-S machine has been carried out as shown in Fig. 8. Force is applied to one end of the test device with speed of 1 mm per minute while the other end is fixed. Elongation length of the parylene test sample is recorded as a function of the applied force and then stressstrain curves are extracted as shown in Fig. 10. The extracted mechanical parameters of the test parylene-metal-parylene are Young's modulus of 2.67 GPa and Yield strength of 30.7 MPa. Note that the extracted mechanical parameters are averaged value of 3 test samples. Fig. 11 shows the fractured test parylene sample.

Fig. 7

 7 Fig. 7 Test device for tensile strength test

Fig. 11

 11 Fig. 11 Stress distribution during tensile strength test