Anna Jezierska

Caroline Chaux

Jean-Christophe Pesquet

Hugues Talbot

An EM approach for Poisson-Gaussian noise modeling

come

INTRODUCTION

In many real world problems, data are corrupted by random noise. Although the physical properties of acquisition systems often lead us to consider a specific probabilistic model for the noise, its parameters are usually unknown. For many simple probability distributions (e.g. Gaussian, Poisson,...), standard estimators are available, such as those provided by the Maximum Likelihood (ML), moment estimates or order statistics. When the noise is Poisson distributed, the Anscombe transform [START_REF] Anscombe | The transformation, of Poisson, binomial and negative-binomial data[END_REF] can be also applied so that the noise can be considered as approximately following a Gaussian distribution. However, at low count, the Anscombe transform introduces a significant bias. There exist stabilization methods for more complex forms of noise, but noise parameters must still be known.

When the noise takes a more complicated form such as a combination of Gaussian and Poisson distributions, and the signal-to-noise ratio is low, specific algorithms need to be designed. Such scenarios occur for example in, astronomy [START_REF] Benvenuto | The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise[END_REF], medical imaging [START_REF] Nichols | Spatiotemporal reconstruction of list-mode PET data[END_REF], microscopy imaging [START_REF] Paul | Automatic noise quantification for confocal fluorescence microscopy images[END_REF] but also in MACROscopy imaging.

In the literature, the noise estimation problem has been investigated either from a single or from several signal realizations. The estimation from a single realization is an underdetermined problem. Therefore some prior knowledge concerning the signal of interest must be included into the model. Parametric estimation problems for a Poisson plus zero-mean Gaussian noise from a single image were addressed, for example in [START_REF] Foi | Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data[END_REF][START_REF] Liu | Noise estimation from a single image[END_REF]. Since in [START_REF] Foi | Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data[END_REF] the authors focus on CCD camera applications, the expected signal-to-noise ratio is relatively high and the Anscombe transform can be success- fully applied. Similarly, the method derived in [START_REF] Liu | Noise estimation from a single image[END_REF] was developed for CCD camera images where the assumption of a zero-mean Gaussian component is well founded. Estimators based on several realizations [START_REF] Healey | Radiometric CCD camera calibration and noise estimation[END_REF] are more reliable and they do not necessarily require prior information about the target signal. Moreover, in many practical applications such as microscopy, several signal acquisitions are feasible (possibly through a calibration process).

The literature on parameter estimation for a Poisson plus Gaussian noise with nonzero mean, especially when a low level signal is expected, is very limited. Among the few contributions dealing with this problem, the author in [START_REF] Zhang | Contributions à la microscopie à fluorescence en imagerie biologique : modélisation de la PSF, restauration d'images et détection super-résolutive[END_REF] proposes a cumulant-based approach, whereas in [START_REF] Boulanger | Non-parametric regression for patch-based fluorescence microscopy image sequence denoising[END_REF][START_REF] Delpretti | Multiframe sure-let denoising of timelapse fluorescence microscopy images[END_REF], the authors make use of the Anscombe transform and a regression based approach. These algorithms can compute noise parameters for denoising procedures, in which they are normally assumed to be known [START_REF] Benvenuto | The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise[END_REF][START_REF] Luisier | Image denoising in mixed Poisson-Gaussian noise[END_REF].

In this work, we propose a multivariate parameter estimation method when the noise is assumed to be a combination of Poisson and Gaussian components. More precisely, we aim at performing an accurate estimation of the scale parameter α of the Poisson component , the mean c and the variance σ 2 of the Gaussian one. We first point out the limitations of an ML approach. This leads us to develop an Expectation-Maximization (EM) algorithm, the numerical implementation of which is discussed. Although the EM algorithm is a popular solution in statistical signal processing (see [START_REF] Stein | Detection of random signals in Gaussian mixture noise[END_REF] for Gaussian mixtures or [START_REF] Fessler | Space-alternating generalized expectationmaximization algorithm[END_REF] for Poisson noise), its use in the present context appears to be new. One of the crucial step in the proposed approach is the initialization of the EM iterative procedure. This initialization is realized by an accurate cumulant-based method.

The remaining of the paper is organised as follows: In Section 2 we present the considered model and introduce the notation used in this work. In Section 3, we briefly discuss ML estimation difficulties. Our algorithm is then described in Section 4. Finally, simulations are made in Section 5 showing the good performance and the robustness of the proposed approach.

PROBLEM

We consider data (u s ) 1≤s≤S where s corresponds to a location index (e.g. locating pixel (x, y) in 2D or (x, y, z) in 3D), which are corrupted by a Poisson-Gaussian noise, and for which we observe T realizations. Each realization will be indexed by t ∈ {1, . . . , T }, which can be a time index.

Such a framework leads us to the following model:

(∀s ∈ {1, . . . , S})(∀t ∈ {1, . . . , T }) R s,t = αQ s,t + N s,t (1)
where

Q s,t ∼ P(u s ), N s,t ∼ N (c, σ 2 ), α ∈ R is a scaling
parameter, (u s ) 1≤s≤S is the "clean" signal, and c ∈ R (resp. σ > 0) is the mean (resp. standard deviation) of the Gaussian noise.

The problem is then to estimate u = (u s ) 1≤s≤S , α, c and σ from the available observation field r = (r s,t ) 1≤s≤S,1≤t≤T , which is a realization of a random field R = (R s,t ) 1≤s≤S,1≤t≤T . We have thus S + 3 parameters to estimate.

In the following, it is assumed that u is deterministic and that Q = (Q s,t ) 1≤s≤S,1≤t≤T and N = (N s,t ) 1≤s≤S,1≤t≤T are mutually independent random fields. In addition, the components of N (resp. Q) are assumed to be independent.

MAXIMUM LIKELIHOOD ESTIMATOR

The ML estimate of the parameters is defined as

( u, α, c, σ ) = argmax (u,α,c,σ ) f R (r | u, α, c, σ ).
(

) 2
where f R (• | u, α, c, σ ) is the probability density function (pdf) of R. The ML estimator is known to usually have better statistical performance than moment estimates [START_REF] Van Trees | Detection, Estimation, and Modulation Theory[END_REF].

Let p R,Q (•, • | u, α, c, σ ) denote the mixed continuous- discrete probability distribution of (R, Q). By using Bayes rule, we get p R,Q (r, q | u, α, c, σ ) = f R|Q=q (r | u, α, c, σ )P(Q = q | u) = f N (r -αq | c, σ )P(Q = q | u) (3)
where

f R|Q=q (• | u, α, c, σ ) is the conditional pdf of R know- ing that Q = q and f N (• | c, σ ) is the pdf of N.
The desired likelihood (using the independence assumption for the components of N (resp. Q)) thus takes the following form:

f R (r | u, α, c, σ ) = ∑ q∈N ST p R,Q (r, q | u, α, c, σ ) = 1 (2π) ST /2 σ ST S ∏ s=1 exp(-Tu s ) T ∏ t=1 +∞ ∑ q s,t =1 exp - (r s,t -αq s,t -c) 2 2σ 2 u q s,t s q s,t ! .
The likelihood takes a rather intricate form, which makes the computation of the ML estimator quite difficult.

PROPOSED ALGORITHM

Expectation-Maximization approach

A possible way of circumventing the aforementioned difficulty consists of resorting to an EM algorithm. In this case, R is viewed as an incomplete random vector that must be completed by another vector. We propose here to consider that the completed vector is (R, Q). For conciseness, let us define θ = (u, α, c, σ ). The EM algorithm is given by the following iteration:

(∀n ∈ N) θ (n+1) = argmax θ J(θ | θ (n) ), (4)
where 2 (5)

J(θ | θ (n) ) = E Q|R=r,θ (n) [ln p R,Q (R, Q | θ )]. According to (3), we have -ln p R,Q (R, Q | θ ) = 1 2σ 2 S ∑ s=1 T ∑ t=1 (R s,t -αQ s,t -c)
+ ST 2 ln(2πσ 2 ) + T S ∑ s=1 u s - S ∑ s=1 ln u s T ∑ t=1 Q s,t + S ∑ s=1 T ∑ t=1 ln(Q s,t !).
By dropping the terms that are independent of θ , we see that the EM algorithm reduces to

(∀n ∈ N) θ (n+1) = argmin θ J(θ | θ (n) ), (6)
where

J(θ | θ (n) ) = 1 2σ 2 S ∑ s=1 T ∑ t=1 E Q|R=r,θ (n) [(r s,t -αQ s,t -c) 2 ] + ST 2 ln(σ 2 ) + T S ∑ s=1 u s - S ∑ s=1 ln u s T ∑ t=1 E Q|R=r,θ (n) [Q s,t ]. (7)
This leads us to the following iterative solution: for every n,

(∀s ∈ {1, . . . , S}) u (n+1) s = 1 T T ∑ t=1 E Q|R=r,θ (n) [Q s,t ] (8)
ST ∑ s,t E Q|R=r,θ (n) [Q s,t ] ∑ s,t E Q|R=r,θ (n) [Q s,t ] ∑ s,t E Q|R=r,θ (n) [Q 2 s,t ] c (n+1) α (n+1) = ∑ s,t r s,t ∑ s,t r s,t E Q|R=r,θ (n) [Q s,t ] (9) (σ 2 ) (n+1) = (10) 1 ST S ∑ s=1 T ∑ t=1 r s,t r s,t -α (n+1) E Q|R=r,θ (n) [Q s,t ] -c (n+1) .
So, provided that we are able to compute

E Q|R=r,θ (n) [Q s,t ] and E Q|R=r,θ (n) [Q 2 s,t ]
, the implementation of the EM algorithm is quite simple.

Let us now turn our attention to the computation of the required conditional mean values. For every (t, s) ∈ {1, . . . , T } × {1, . . . , S}, we have

E Q|R=r,θ (n) [Q s,t ] = +∞ ∑ q s,t =1 q s,t P(Q s,t = q s,t | R = r, θ (n) ). (11)
In addition,

P(Q s,t = q s,t | R = r, θ (n) ) = p R s,t ,Q s,t (r s,t , q s,t | θ (n) ) f R s,t (r s,t | θ (n) ) . (12)
Using again (3), this implies

E Q|R=r,θ (n) [Q s,t ] = ζ (n) s,t η (n) s,t , (13)
where

ζ (n) s,t = +∞ ∑ q s,t =1 exp - (r s,t -α (n) q s,t -c (n) ) 2 2(σ 2 ) (n) (u (n) s ) q s,t (q s,t -1)! (14) η (n) s,t = +∞ ∑ q s,t =0 exp - (r s,t -α (n) q s,t -c (n) ) 2 2(σ 2 ) (n) (u (n) s ) q s,t q s,t ! . (15)
Similarly, we have

E Q|R=r,θ (n) [Q 2 s,t ] = ξ (n) s,t η (n) s,t , (16)
where

ξ (n) s,t = +∞ ∑ q s,t =1 q s,t exp - (r s,t -α (n) q s,t -c (n) ) 2 2(σ 2 ) (n) (u (n) s ) q s,t (q s,t -1)! .
(17) In these formulas, q s,t acts as a summation index. As we can only perform finite summations, one can stop the summing at convergence. Alternatively, for improved efficiency we have derived lower and upper bounds for q s,t present in ( 14), ( 15) and ( 17). The bounds are functions of r s,t , α (n) , c (n) , (σ 2 ) (n) and u (n) s . The lower bound is denoted by q (n) s,t and upper bound by q (n) s,t . Due to the lack of space, this point will be discussed in an expanded version of this paper.

Initialization

In Section 4.1, an EM algorithm for Poisson-Gaussian noise parameter identification was derived. As with many implementations of the EM algorithm, an important consideration is how to initialize θ . The problem is now to find appropriate initial values of α (1) , c (1) , (σ 2 ) (1) and u (1) . We propose a moment based approach to set these values. Due to the mutual independence assumption,

κ n [R s,t ] = α n κ n [Q s,t ] + κ n [N s,t ] (18)
where κ n [A] designates the cumulant of order n of some random variable A. Using classical results about cumulants, we obtain:

• mean value:

κ 1 [R s,t ] = E[R s,t ] = αu s + c (19) • variance: κ 2 [R s,t ] = Var[R s,t ] = α 2 u s + σ 2 (20) • higher-order cumulants: n ≥ 3, κ n [R s,t ] = α n u s . (21) Let E[r s,t ] = 1 T ∑ T t ′ =1
r s,t ′ and let similar sample estimates Var[r s,t ] and κ 3 [r s,t ] be used for the other cumulants. Different procedures may be derived from (19), (20), and (21) in order to estimate θ , but they are not equally reliable. For example according to our observations,

κ 4 [r s,t ]
κ 3 [r s,t ] does not provide a very good estimate of α. Instead, we propose to use:

α (1) = S ∑ S s=1 E[r s,t ] Var[r s,t ] -∑ S s=1 E[r s,t ] ∑ S s=1 Var[r s,t ] S ∑ S s=1 ( E[r s,t ]) 2 -∑ S s=1 E[r s,t ] 2 , (22)
which is quite accurate, as only first and second order statistics are used. However, (σ 2 ) (1) cannot be computed in a similar manner and third order cumulants need to be considered. One of the possibilities is to compute (σ 2 ) (1) as median Var[r s,t ] -(α (1) ) -1 κ 3 [r s,t ] , but it can be shown that the cumulant estimate becomes sensitive when T is small or when u s takes large values. To account for this latter problem, we propose the following weighted least squares estimate of σ 2 :

(σ 2 ) (1) = ∑ s∈I Var[r s,t ] -6 Var[r s,t ] -(α (1) ) -1 κ 3 [r s,t ] ∑ s∈I Var[r s,t ] -6 (23) where I = s ∈ {1, . . . , S} | Var[r s,t ] -(α (1) ) -1 κ 3 [r s,t ] ≥ 0 .
Finally, the initialization is completed by:

c (1) = 1 S S ∑ s=1 E[r s,t ] -(α (1) ) -1 Var[r s,t ] + (σ 2 ) (1)
α (1) , (24) and (∀s ∈ {1, . . . , S}) u ) ).

(1) s = 1 α (1) ( E[r s,t ] -c ( 1
(25)

Overview of the proposed method

The tools introduced in Sections 4.1 and 4.2 constitute the main two ingredients of our estimation method, the pseudocode of which is summarized in Algorithm 1.

The iterative process stops when the maximum number of iterations N is reached, or if max

i=α,σ 2 ,c i (n+1) -i (n) ≤ δ ,
where δ > 0 is some tolerance.

SIMULATION EXAMPLES

The experimental results presented here aim at providing information about the performance of the proposed algorithm under different working conditions. In particular, in Section 5.1 the influence of the values of parameters c, α, σ 2 and T is studied. The proposed initialization is shown to be accurate for T > 200 and values of c, α and σ 2 neither too low nor too high. EM algorithm is investigated for smaller data set, when its benefits in terms of accuracy are more significant. Finally, an application of the proposed algorithm to real data is demonstrated (Section 5.2). The noise parameters are identified based on sequences of images corrupted with Poisson-Gaussian noise.

Validation of the proposed approach

We evaluate the proposed algorithms using S randomly generated u s values uniformly distributed over [0, 100[. This range was chosen in order to show the performance of our algorithm in the conditions when the Anscombe transform is less reliable. Signal R s,t is generated according to (1) for different set of parameter values for θ and T . Poisson and Gaussian noises are simulated using random number generators as proposed in Park et al. [START_REF] Park | Random number generators: Good ones are hard to find[END_REF].

Identification accuracy is evaluated in terms of relative absolute error, defined as:

err = σ 2 -σ 2 σ 2 + α- α α + c-ĉ c 3 ( 26
)
Algorithm 1 Proposed algorithm.

Initialization:

Compute α (1) using ( 22) Find I Compute (σ 2 ) (1) using (23) Compute c (1) using (24) Compute u (1) using (25) Set θ (1) = u (1) , α (1) , c (1) , (σ 2 ) (1) EM Algorithm:

For n = 1 . . . N                                              Set ζ (n) s,t = 0, ξ (n) s,t = 0, κ (n) s,t = 1 η (n) s,t = exp - (r s,t -c (n) ) 2 2(σ 2 ) (n) u (n) s,t
Compute q (n) s,t and q

(n) s,t
For q s,t = 1 . . . q (n) s,t -1 (see ( 14), ( 15), ( 17) )

κ (n) s,t ← κ (n) s,t u (n) s,t q s,t
For q s,t = q (n) s,t . . . q

(n) s,t             κ (n) s,t ← κ (n) s,t u (n) s,t q s,t χ (n) s,t = κ (n) s,t exp - (r s,t -α (n) q s,t -c (n) ) 2 2(σ 2 ) (n) ζ (n) s,t ← ζ (n) s,t + χ (n) s,t q s,t ξ (n) s,t ← ξ (n) s,t + χ (n) s,t q 2 s,t η (n) s,t ← η (n) s,t + χ (n) s,t Update E Q|R=r,θ (n) [Q s,t ] using (13) Update E Q|R=r,θ (n) [Q 2
s,t ] using (16) Update u (n+1) s using (8) Update α (n+1) and c (n+1) using ( 9) Update (σ 2 ) (n+1) using ( 10)

θ (n+1) = u (n+1) , α (n+1) , c (n+1) , (σ 2 ) (n+1)
where the estimates are denoted with a hat (e.g. σ ).

The bias and standard deviation of estimated parameters computed from 100 different noise realizations are presented in Tables 1, 2, 3 and 4. Moreover the mean error (err) over all 100 realizations is given. The following points are highlighted through these results:

• The reliability of cumulant-based approach increases with T (see Table 1); • the mean of estimated parameters σ 2 , ĉ, and α does not strongly depend on S, but the standard deviation of the estimates does (see Table 1); • the estimates provided by the cumulant-based approach for T ≥ 500 are very accurate (see Table 1); • parameter α is estimated very accurately with reduced dependence on parameters S and T ; • the initialization estimate is subject to higher errors for very high values of σ 2 (see Table 2); • this estimate is subject to higher errors for low values of c (see Table 3); • its accuracy also decreases for very low or very high values of α (see Table 4).

For T ≤ 200, the results obtained by the cumulant-based 2). Note that in this case, the initial signal-tonoise ratio is very low. One can also observe that the EM estimates for T = 200 are quite precise as the estimation error is only 5%.

Unsupervised image denoising

One possible application of our algorithm is the calibration of optical measurement systems, which we simulate in the following experiment. We created a time lapse sequences consisting of 40 images with resolution 100 × 100, each corrupted with Poisson-Gaussian noise characterized by σ 2 = 416, c = 10 and α = 50. This corresponds to an initial SNR value of 0.17 dB. Again, the challenge here stems from the fact that the value of T is low (40). One can observe some remaining noise in the result provided by our cumulantbased method (Fig. 1(c)), which is no longer visible in the EM result (Fig. 1(d)). This is also verified by inspecting SNR values, which are equal to 28. the fact that not only noise parameters α, σ 2 and c are well reconstructed but so are the image intensity values u s .

CONCLUSION

We have proposed a new EM-based approach dealing with Poisson plus Gaussian noise parameters estimation problems. We have shown that the proposed method leads to accurate results. We have also proposed an improved cumulantbased estimation method, which we used to initialize the EM algorithm. The improvement resulting from the EM iterations is especially significant when the number of realizations is small. As a side result, it allows us to obtain a good estimation of the original data when the noise parameters are unknown.

Figure 1 :

 1 Figure 1: (a,b,c,d) illustrate original image, its noisy version (scaled with parameter α), cumulant-based method and EM results, respectively.

Table 1 :

 1 Identified noise parameter versus S and T (c = 10, α = 10 and σ 2 = 100).

	Param.					Proposed initialization
	S	T	bias	σ 2	std	bias	ĉ	std	bias	α	std	err
		50	-14.83	20.29	-1.74 5.07	-0.21	0.14	0.22
	1024	100 200 500	-8.58 -11.69 -1.93	13.86 10.13 6.24	-0.54 4.04 1.18 2.7 -0.26 1.61	-0.09 -0.04 -0.02	0.11 0.04 0.04	0.16 0.14 0.06
	1000	-1.16	4.55	-0.24		1.1	-0.01	0.03	0.04
		50	-20.86	13.05	-1.62 2.61	-0.19	0.06	0.16
	4096	100 500	-10.77 -2.19	6.91 2.80	-0.88 1.76 -0.31 0.80	-0.10 -0.02	0.05 0.02	0.09 0.03
	1000	-0.99	2.13	-0.15 0.49	-0.01	0.01	0.02
	Param.				Proposed Initialization
	σ 2		bias	σ 2	std		bias	ĉ		std	bias	α	std	err
	0.25		-0.14		0.03	-0.11		5.15	-0.19 0.13	0.18
	25		-3.59		3.89	-0.18		4.19	-0.19 1.14	0.19
	400		-11.45		84.32	-0.84		10.53	-0.22 0.15	0.34

Table 2 :

 2 Identified noise parameter versus σ 2 (c = 10, α = 10, S = 1024 and T = 50).approach need to be further improved by EM. Table5provides some numerical results. Here the bias and standard deviation of estimated parameters computed from 20 different noise realizations are presented. One can observe that EM algorithm offers significant improvements in terms of accuracy. Note that, all the presented tests were performed under difficult conditions for cumulant-based method.

	The follow-

Table 5 :

 5 3 dB for the cumulantbased method and 35.6 dB for EM. This example illustrates Expectation-Maximization algorithm performance under difficult conditions for cumulant-based method.

				Param.					Proposed EM algorithm			Error
	No.	S	T	σ 2	c	α	bias	σ 2	std	bias	ĉ	std	bias	α	std	err cum	err EM
	1	1024	50	100 10	10	-1.86		14.02	-0.35		1.79	-0.21		0.05	0.22	0.11
	2	1024 100	100 10	10	0.54		10.92	0.27		1.38	-0.08		0.02	0.16	0.08
	3	1024 200	100 10	10	-0.91		7.89	0.09		0.88	-0.04		0.03	0.14	0.05
	4	1024	50	100	5	10	1.30		13.16	0.08		1.61	-0.20		0.05	0.33	0.15
	5	1024	50	25	10	1	-0.28		0.50	0.00		0.48	-0.02		0.01	0.64	0.03
	6	1024	50	400 10	10	7.51		45.24	0.71		5.13	-0.22		0.07	0.34	0.21

Table 3 :

 3 Identified noise parameter versus c (σ 2 = 100, α = 10, S = 1024 and T = 50).

	Param.				Proposed Initialization
	α	bias	σ 2	std	bias	ĉ	std	bias	α	std	err
	1	12.99		6.17	13.73		6.29	-0.02		0.02	0.64
	5	-2.93		5.69	-0.25		2.83	-0.09		0.07	0.15
	50	1.07		56.4	-3.55		25.7	-1.10		0.67	0.81

Table 4 :

 4 Identified noise parameter versus α (σ 2 = 25, c = 10, S = 1024 and T = 50).


Lucie Baudouin
email: lucie.baudouin@laas.fr

Eduardo Cerpa
email: eduardo.cerpa@usm.cl

Emmanuelle Crépeau
email: emmanuelle.crepeau@math.uvsq.fr

Alberto Mercado
email: alberto.mercado@usm.cl

Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation

Keywords: Inverse problem, Kuramoto-Sivashinsky equation, Carleman estimate AMS subject classifications: 35R30, 35K55

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

We focus in this paper on an inverse problem that consists in the determination of a coefficient in a partial differential equation (pde) from the partial knowledge of a given single solution of the equation. This class of problems (single-measurement coefficient inverse problems) was investigated using Carleman estimates for the first time in [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] by Bukhgeȋm and Klibanov. See [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF], [START_REF] Klibanov | Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data[END_REF] and the recent book [START_REF] Beilina | Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems[END_REF] for details about the so-called Bukhgeȋm-Klibanov method. This method was initially used to prove uniqueness for inverse problems (i.e. that each measurement corresponds to only one coefficient) from local Carleman estimates (estimates valid for solutions with compact support in the interior of the domain), as in [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF].

Regarding the continuity of the inverse problem of recovering the source term, the first Lipschitz stability result for a multidimensional wave equation was obtained by Puel and Yamamoto [START_REF] Puel | On a global estimate in a linear inverse hyperbolic problem[END_REF] using the uniqueness result and a compactness-uniqueness argument.

1 Global Carleman estimates (valid for solutions considered in the whole domain and satisfying boundary conditions) were applied to parabolic equations by first time in [START_REF] Yu | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF], where Lipschitz stability of an inverse problem is established. Since then, this type of inverse problems for parabolic equations has received a large amount of attention. The primary difference with respect to hyperbolic inverse problems is that parabolic problems are not time-reversible: therefore, an additional measurement must be added if that method is applied. As one can read in the discussion of the introduction of [START_REF] Yu | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF], the knowledge of the full-state of the solution for some positive time is required. Proving the Lipschitz stability without this assumption, which is usually needed when global Carleman inequalities are used, is still an open problem. Nevertheless, there are some uniqueness results with less assumptions on the measurements, that can be found in the litterature, such as [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] or some other inversion method in [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF][START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF].

Recent results regarding linear parabolic problems can be found in [START_REF] Benabdallah | Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation[END_REF] (discontinuous coefficient), [START_REF] Cristofol | Inverse problems for a 2 × 2 reaction-diffusion system using a Carleman estimate with one observation[END_REF] (systems), [START_REF] Ignat | Inverse Problem for the heat equation and the Schrodinger equation on a tree[END_REF] (network) and the references therein. In [START_REF] Boulakia | Some inverse stability results for the bistable reaction-diffusion equation using Carleman inequalities[END_REF][START_REF] Egger | Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation[END_REF][START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF], nonlinear parabolic equations were even considered.

Among others pde's coefficient/source inverse problems where Carleman estimates have been used we can mention, without being exhaustive, logarithmic stability [START_REF] Bellassoued | Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation[END_REF], Calderón problem [START_REF] Santos Ferreira | Limiting Carleman weights and anisotropic inverse problems[END_REF] or Schrödinger equation [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF].

In this paper, we consider a 1D nonlinear fourth-order parabolic equation called Kuramoto-Sivashinsky (K-S) equation. This equation was proposed independently by Kuramoto and Tsuzuki [START_REF] Kuramoto | On the formation of dissipative structures in reactiondiffusion systems[END_REF] as a model for the phase turbulence in reaction diffusion systems, and by Sivashinsky [START_REF] Sivashinsky | Nonlinear analysis of hydrodynamic instability in laminar flames I: Derivation of basic equations[END_REF], as a model the physical phenomena of plane flame propagation, were the combined influence of diffusion and thermal conduction of a gas is described.

The K-S equation with non-constant coefficients describing the diffusion σ = σ(x), and the anti-diffusion γ = γ(x), is given as

               yt + (σ(x)yxx)xx + γ(x)yxx + yyx = g, ∀(t, x) ∈ Q, y(t, 0) = h1(t), y(t, 1) = h2(t), ∀t ∈ (0, T ), yx(t, 0) = h3(t), yx(t, 1) = h4(t), ∀t ∈ (0, T ), y(0, x) = y0(x), ∀x ∈ (0, 1), (1)
where Q := (0, T )×(0, 1), σ : [0, 1] → R * + , and the functions y0, g, hj are the initial condition, the source term and the boundary data respectively. All these terms are assumed to be known and compatible.

In this nonlinear pde, the fourth-order term models the diffusion, and the second-order term models the incipient instabilities. We consider the inverse problem of retrieving the anti-diffusion coefficient γ from boundary measurements of the solution. This corresponds for instance to getting information on the instability of a reaction-diffusion media by measuring a single solution, which could represent a flame propagating on the domain. Concerning the boundary measurements we will make, it is worth to mention that in a fourth-order parabolic problem like KS, boundary data uxx and uxxx are referred to as Neumann data, which in fact represent heat flux [START_REF] Liu | Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation[END_REF] in this kind of models.

To the knowledge of the authors there are no results in the literature concerning the determination of coefficients for this nonlinear equation. However, a Carleman estimate has been used to obtain the null-controllability of the K-S equation in reference [START_REF] Cerpa | Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation[END_REF] for the constant coefficient case. Other results on the control of the KS equation can be found in [START_REF] Hu | Robust control of the Kuramoto-Sivashinsky equation[END_REF][START_REF] Armaou | Feedback control of the Kuramoto-Sivashinsky equation[END_REF][START_REF] Christofides | Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control[END_REF][START_REF] Liu | Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation[END_REF][START_REF] Cerpa | Null controllability and stabilization of a linear Kuramoto-Sivashinsky equation[END_REF].

Since the linearized equation is parabolic, we know that boundary measurements will not be sufficient to prove stability and we must consider an additional measurement of the full solution for a given time T0 (as in [START_REF] Benabdallah | Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation[END_REF][START_REF] Yu | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF] among others).

Our first result involves the local well-posedness of the nonlinear equation [START_REF] Armaou | Feedback control of the Kuramoto-Sivashinsky equation[END_REF]. A less regular framework can be used for this equation but the method applied in this paper requires the solution and its time-derivative to be at least in L 2 (0, T ; H 4 (0, 1)). Therefore, let us introduce the following notations for the functional spaces appearing in this paper:

Y k := C([0, T ]; H k (0, 1)) ∩ L 2 (0, T ; H k+2 (0, 1)), for k ∈ N; F := {f ∈ L 2 (0, T ; H 4 (0, 1)) ft ∈ L 2 (0, T ; L 2 (0, 1))}; Z := {z ∈ Y6 zt ∈ Y2}. (2)
Theorem 1.1 Let γ ∈ H 4 (0, 1) and σ ∈ H 4 (0, 1) be such that σ(x) ≥ σ0 > 0, ∀x ∈ (0, 1).

(

) 3
There exists ε > 0 such that if y0 ∈ H 6 (0, 1), g ∈ F, and hj ∈ H 2 (0, T ) for j = 1, . . . , 4

satisfy the compatibility conditions

y0(0) = h1(0), y0,x(0) = h3(0), y0(1) = h2(0), y0,x(1) = h4(0), (4)
and

y0 H 6 (0,1) ≤ ε, g F ≤ ε, hj H 2 (0,T ) ≤ ε for j = 1, . . . , 4, (5)
then the K-S equation (1) has a unique solution y ∈ Z .

Once the existence of solutions to the K-S equation has been established (see Section 2), the following inverse problem is addressed:

Is it possible to retrieve the anti-diffusion coefficient γ = γ(x) from the measurement of yxx(t, 0) and yxxx(t, 0) on (0, T ) and from the measurement of y(T0, x) on (0, 1), where y is the solution to Equation (1) and T0 ∈ (0, T )?

A local answer for this nonlinear inverse problem is given (see section 4). To be more specific, let γ be given and fixed. We denote by ỹ the solution to Equation (1) with γ replaced by γ.

This paper focuses on the following question concerning the unknown γ and y.

Stability: Is it possible to estimate γ -γ L 2 (0,1) by suitable norms ỹ(T0, x) -y(T0, x)

in space and ỹxx(t, 0) -yxx(t, 0) , ỹxxx(t, 0) -yxxx(t, 0) in time?

Of course, a positive answer implies the usual uniqueness result.

Uniqueness: Do the equalities of the measurements ỹxx(t, 0) = yxx(t, 0) and ỹxxx(t, 0) = yxxx(t, 0) for t ∈ (0, T ) and ỹ(T0, x) = y(T0, x) for x ∈ (0, 1) imply γ = γ on (0, 1)?

In order to answer these questions, we use the Bukhgeȋm-Klibanov method. First, a global Carleman estimate for the linearized K-S equation with non-constant coefficients is obtained. It is then used to prove the main result which can be stated as follows.

To precisely state the results we prove in this article, we introduce, for m > 0, the set

L ∞ ≤m (0, 1) = γ ∈ L ∞ (0, 1)s.t. γ L ∞ (0,1) ≤ m .
Theorem 1.2 Let us consider σ ∈ H 4 (0, 1) satisfying (3), γ ∈ H 4 (0, 1), g ∈ F and the data y0 ∈ H 6 (0, 1) and hj ∈ H 2 (0, T ) for j = 1, . . . , 4 under the compatibility conditions (4). Let y ∈ Z be the solution of (1), and ỹ ∈ Z the solution corresponding to a given γ ∈ H 4 (0, 1) instead of γ. We assume that there exists η > 0 and T0 ∈ (0, T ) such that

inf {|ỹxx(T0, x)| , x ∈ (0, 1)} ≥ η, (6)
Then, given M > 0, there exists a positive constant C depending on the parameters (T, m, M, η), such that for every γ ∈ L ∞ ≤m (0, 1),

γ -γ 2 L 2 (0,1) ≤ C yxx(•, 0) -ỹxx(•, 0) 2 H 1 (0,T ) + C yxxx(•, 0) -ỹxxx(•, 0) 2 H 1 (0,T ) + C y(T0, •) -ỹ(T0, •) 2 H 4 (0,1) + C y(T0, •) -ỹ(T0, •) 4 H 1 (0,1) (7)
for all y satisfying

y Z ≤ M.
This inequality states the stability of the inverse problem. Before giving the outline of our paper and the proofs of the different steps, we want to give several comments on this result.

Remark 1.3 For numerical purposes it would be interesting to know explicitly how the constant C in (7) depends on the diffusion σ or on the time T . This kind of question has been addressed in [START_REF] Coron | Singular optimal control: A linear 1-D parabolic-hyperbolic example[END_REF][START_REF] Fernández-Cara | Global carleman inequalities for parabolic systems and applications to controllability[END_REF] for observability constant in the framework of second-order parabolic equations. In those papers the authors got an exponential dependence on both the constant diffusion and the time.

Remark 1.4 One can show that there exist solutions satisfying assumption [START_REF] Boulakia | Some inverse stability results for the bistable reaction-diffusion equation using Carleman inequalities[END_REF]. We present two different arguments:

1. We take ε > 0 given by Theorem 1.1, and some y 0 ∈ H 6 (0, 1) such that inf x∈(0,1)

y 0 xx ≥ ε/2.
For arbitrary boundary data and source term belonging to the corresponding spaces, by Theorem 1.1 there exists a solution ỹ ∈ C([0, 1]; H 6 (0, 1)) with ỹ(0, •) = y 0 . Using Sobolev injection and continuity, we obtain the existence of a time T0 > 0 such that (6) is fulfilled with η = ε/4.

2.

We can also prove that there exist solutions satisfying (6) without asking T0 to be small, but instead, constraining the source term and boundary data as follows: Let y0 be the initial data and T0 belong to (0, T ). Let us pick up a state y1 = y1(x) strictly convex.

We consider the trajectory ỹ(t, x) = T 0 -t T 0 y0(x) + t T 0 y1(x), which is the solution of equation (1) with source term given by g = ỹt + (σ(x)ỹxx)xx + γ(x)ỹxx + ỹ ỹx and the boundary data given by the traces of ỹ. Thus, ỹ(T0, x) = y1(x) and hence the trajectory

ỹ satisfies (6).
Therefore, the set of data and solutions where our stability result is valid is not empty.

Remark 1.5 We obtain the same result if ỹ has a different initial condition than y. See in Section 4, that the term v(x, 0) of system (46) does no play any role in the result.

Remark 1.6 We can complete inequality (7) by the following:

yxx(•, 0) -ỹxx(•, 0) 2 H 1 (0,T ) + yxxx(•, 0) -ỹxxx(•, 0) 2 H 1 (0,T ) + y(T0, •) -ỹ(T0, •) 2 H 4 (0,1) + y(T0, •) -ỹ(T0, •) 4 H 1 (0,1) ≤ C y -ỹ 2 H 1 (0,T ;H 4 (0,1)) + y -ỹ 4 C([0,T ];H 1 (0,1)) .
This inequality follows directly from standard Sobolev injections. It indicates that the required measurements are finite if y and ỹ belong to the space H 1 (0, T ; H 4 (0, 1)) and this is true if y and ỹ are solutions in Z provided by Theorem 1.1.

Remark 1.7

As stated in the introduction, an internal measurement at t = T0 is required if this method, using Carleman estimates, is used to prove the stability for this type of inverse problem for parabolic equations. Nevertheless, this is probably a technical point since there is no counter-example that demonstrates whether this assumption is required for stability. In [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF], uniqueness (but not stability) is proven using a very different technique in an inverse problem for a parabolic equation and without any internal measurements in the whole space domain. One can also mention a method in [START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF] that can deliver uniqueness from hyperbolic equations to parabolic ones.

Remark 1.8 In this paper, the boundary measurements are located at x = 0, but the result would be the same if we measure at x = 1 instead. Indeed, the choice of a suitable weight function in the proof of the Carleman estimate in Section 3 is critical to impose the side of measurement.

This article is organized as follows. The well-posedness result stated in Theorem 1.1 is proved in Section 2. A global Carleman estimate for a general K-S equation is given and proved in Section 3. Finally, Section 4 contains the use of the Bukhgeim-Klibanov method to prove the Lipschitz stability of the inverse problem stated in Theorem 1.2.

On the Cauchy problem for KS equation

This section presents a proof of Theorem 1.1 in a more general case including time dependent lower-order coefficients. We consider the following K-S system

               yt + (σ(x)yxx)xx + γ(x)yxx + G1yx + G2y + yyx = g, ∀(t, x) ∈ Q, y(t, 0) = h1(t), y(t, 1) = h2(t), ∀t ∈ (0, T ), yx(t, 0) = h3(t), yx(t, 1) = h4(t), ∀t ∈ (0, T ), y(0, x) = y0(x), ∀x ∈ (0, 1), (8)
where G1, G2 belong to H 1 (0, T ; H 4 (0, 1)), g ∈ F and y0 ∈ H 6 (0, 1) is compatible with

hj ∈ H 2 (0, T ) for j = 1, . . . , 4. Recall that the coefficients satisfy γ ∈ H 4 (0, 1), σ ∈ H 4 (0, 1)

and hypothesis [START_REF] Beilina | Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems[END_REF].

First, we only consider the main part of the linear differential operator in the next proposition.

Proposition 2.1 Let z0 ∈ H 6 ∩ H 2 0 (0, 1) and f ∈ F. Then, the following equation

               zt + (σ(x)zxx)xx = f, ∀(t, x) ∈ Q, z(t, 0) = 0, z(t, 1) = 0, ∀t ∈ (0, T ), zx(t, 0) = 0, zx(t, 1) = 0, ∀t ∈ (0, T ), z(0, x) = z0(x), ∀x ∈ (0, 1), (9)
has a unique solution z ∈ Z and there exists C > 0 such that

z Z ≤ C ( f F + z0 H 6 ) .
Proof. The operator

H 4 ∩ H 2 0 (0, 1) ⊂ L 2 (0, 1) -→ L 2 (0, 1) z -→ (σ(x)z ′′ (x)) ′′ ,
is simultaneously positive, coercive and self-adjoint. By the Hille-Yosida-Phillips Theorem (see [START_REF] Cazenave | An Introduction to Semilinear Evolution Equations[END_REF]), it generates a strongly continuous semigroup in L 2 (0, 1). Therefore, for

each z0 ∈ H 4 ∩ H 2 0 (0, 1) and f ∈ C 1 ([0, T ]; L 2 (0, 1)), Equation (9) has a unique solution z ∈ C([0, T ]; H 4 ∩ H 2 0 (0, 1)) ∩ C 1 ([0, T ]; L 2 (0, 1)
). We will demonstrate that the solutions z ∈ Z (refer to the notation introduced in (2)), can be obtained by taking z0 and f sufficiently regular.

We now search for some energy estimates that indicate the space where the solutions lie on depending on the regularity of the data. Suppose that there are solutions sufficiently regular to perform the following computations. Equation ( 9) is multiplied by z and integrated over (0, 1) in space. Some integrations by parts give [START_REF] Cerpa | Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation[END_REF] Throughout this paper, C denotes a positive constant that may vary from line to line. To make the reading easier, we denote for any function u of x and t,

d dt 1 0 |z(t, x)| 2 dx + 1 0 |zxx(t, x)| 2 dx ≤ C 1 0 |f (t, x)| 2 dx + 1 0 |z(t, x)| 2 dx .
Q u = T 0 1 0 u(t, x) dxdt
Using Gronwall's lemma, we first obtain that for all t > 0,

1 0 |z(t, x)| 2 dx ≤ C Q |f | 2 + 1 0 |z0| 2 dx . (11)
Then, [START_REF] Cerpa | Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation[END_REF] is integrated over [0, T ] and ( 11) is used to get

Q |zxx| 2 ≤ C Q |f | 2 + 1 0 |z0| 2 dx . (12)
Inequalities ( 11) and ( 12) finally imply that

z 2 Y 0 ≤ C Q |f | 2 + C 1 0 |z0| 2 dx. (13)
Now, equation ( 9) is multiplied by (σzxx)xx and integrated over (0, 1) in space. Some integrations by parts give also 1 2

d dt 1 0 σ|zxx(t, x)| 2 dx + 1 0 |(σzxx(t, x))xx| 2 dx = 1 0 f (t, x)(σzxx(t, x))xx dx.
Using the inequality ab

≤ 1 2 a 2 + 1 2 b 2 , we get d dt 1 0 σ|zxx(t, x)| 2 dx + 1 0 |(σzxx(t, x))xx| 2 dx ≤ 1 0 |f (t, x)| 2 dx. (14)
Using Gronwall's lemma, from( 14) and ( 3) we obtain that for all t > 0,

1 0 |zxx(t, x)| 2 dx ≤ C Q |f | 2 + 1 0 |z ′′ 0 | 2 dx . (15)
Then, ( 14) is integrated over [0, T ] and ( 15) is used to get

Q |(σzxx)xx| 2 ≤ C Q |f | 2 + 1 0 |z ′′ 0 | 2 dx , (16)
and then, taking into account that σ ∈ H 4 , we get

Q |zxxxx| 2 ≤ C Q |f | 2 + 1 0 |z ′′ 0 | 2 dx + C z L 2 (0,T ;H 3 (0,1)) . (17)
For any ε > 0, from Ehrling's Lemma (see Theorem 7.30 in [START_REF] Renardy | An Introduction to Partial Differential Equations[END_REF]) and ( 11), we have that

z L 2 (0,T ;H 3 (0,1)) ≤ ε z L 2 (0,T ;H 4 (0,1)) + C z L 2 (0,T ;L 2 (0,1)) ≤ ε z L 2 (0,T ;H 4 (0,1)) + C Q |f | 2 + 1 0 |z0| 2 dx . (18)
Taking ε > 0 small enough, inequalities (15), ( 17) and [START_REF] Yu | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF] imply that

z 2 Y 2 ≤ C Q |f | 2 + C z0 2 L 2 (0,T ;H 2 (0,1)) . (19)
On the other hand, Equation ( 9) is derived with respect to time. Thus q := zt satisfies

               qt + (σ(x)qxx)xx = ft, ∀(t, x) ∈ Q, q(t, 0) = 0, q(t, 1) = 0, ∀t ∈ (0, T ), qx(t, 0) = 0, qx(t, 1) = 0, ∀t ∈ (0, T ), q(0, x) = f (0, x) -(σz ′′ 0 (x)) ′′ , ∀x ∈ (0, 1). ( 20
)
Using estimate [START_REF] Ignat | Inverse Problem for the heat equation and the Schrodinger equation on a tree[END_REF], we obtain q ∈ Y2 if (f (0, x) -(σz ′′ 0 (x)) ′′ ) ∈ H 2 (0, 1) and ft ∈ L 2 (0, T ; L 2 (0, 1)). These hypotheses are fulfilled if z0 ∈ H 6 ∩ H 2 0 (0, 1) and f ∈ F. Note that F ⊂ C([0, T ]; H 2 (0, 1)). From the equation satisfied by z and the fact that f ∈ F and zt ∈ Y2, we determine that z ∈ Y6, which concludes the proof of Proposition 2.1.

Then, we focus on the linear problem with non-homogenous boundary conditions and low-order coefficients that depend on time.

Proposition 2.2 Let z0 ∈ H 6 (0, 1), f ∈ F, G1, G2 ∈ H 1 (0, T ; H 4 (0, 1)) and hj ∈ H 2 (0, T ) for j = 1, . . . , 4 satisfying the compatibility conditions with z0. Then, the equation

               zt + (σ(x)zxx)xx + γ(x)zxx + G1zx + G2z = f , ∀(t, x) ∈ Q, z(t, 0) = h1(t), z(t, 1) = h2(t), ∀t ∈ (0, T ), zx(t, 0) = h3(t), zx(t, 1) = h4(t), ∀t ∈ (0, T ), z(0, x) = z0(x), ∀x ∈ (0, 1), (21)
has a unique solution z ∈ Z and there exists C > 0 such that

z Z ≤ C f F + z0 H 6 + 4 j=1
hj H 2 .

Proof. We first prove this result for null boundary data (i.e. for hj = 0 for j = 1, . . . , 4 and therefore z0 ∈ H 6 ∩ H 2 0 (0, 1)). For any ŵ ∈ Z, Π( ŵ) is defined as the solution of ( 9) with f = ( f -γ(x) ŵxx -G1 ŵx -G2 ŵ).

Note that f ∈ F and therefore Π( ŵ) ∈ Z is well defined.

If T is small enough, then Π is a contraction. Indeed, for any w, ŵ ∈ Z, we have

Π( ŵ) -Π(w) Z ≤ C γ(x)(wxx -ŵxx) + G1(wx -ŵx) + G2(w -ŵ) F ≤ C w -ŵ L 2 (H 6 ) + C wt -ŵt L 2 (H 2 ) (22)
≤ CT

1 4 w -ŵ L 4 (H 6 ) + CT 1 4 wt -ŵt L 4 (H 2 ) ≤ CT 1 4 w -ŵ Y 6 + CT 1 4 wt -ŵt Y 2 ≤ CT 1 4 w -ŵ Z , (23)
where the space L m (0, T ;

H n (0, 1)) is denoted as L m (H n ).
Hence, the operator Π has a unique fixed point in Z, which is the solution of ( 21) with

hj = 0 for j = 1, . . . , 4. Using standard arguments and the linearity of this equation, the solution can be extended to a larger time interval.

In order to prove the general case, take hj ∈ H 2 (0, T ), j = 1, . . . , 4 compatible with z0. It is not difficult to find a function ψ ∈ H 2 (0, T ; C ∞ ([0, 1])) satisfying the boundary conditions of [START_REF] Klibanov | Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data[END_REF]. For instance take ψ(x, t) = 4 j=1 pj(x)hj(t) where p1(x) = 2x 3 -3x 2 + 1, p2(x) = -2x 3 + 3x 2 , p3(x) = x 3 -2x 2 + x and p4(x) = x 3 -x 2 . In particular we have Lψ := ψt + (σ(x)ψxx)xx + γ(x)ψxx + G1ψx + G2ψ ∈ F. Then, if w is the solution of equation [START_REF] Klibanov | Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data[END_REF] with null boundary data, initial condition w0 -ψ(•, 0), and right-hand side equal to f -Lψ, let us define z = w + ψ. It is not difficult to see that z is the required solution.

where

Lv = vt + (σvxx)xx + q2vxx + q1vx + q0v with qj ∈ L ∞ (Ω) for j = 0, 1, 2.
Consider β ∈ C 4 ([0, 1]) such that for some r > 0 we have, for all x ∈ (0, 1):

0 < r ≤ β(x), 0 < r ≤ β ′ (x), β ′′ (x) ≤ -r < 0, |σ ′ (x)β ′ (x)| ≤ r 4 min z∈[0,1] {σ(z)}. (26)
For instance, if σ is constant, we can consider β(x) = √ 1 + x.

On the other hand, given T0 ∈ (0, T ) we can choose φ0 ∈ C 1 ([0, T ]) such that φ0(0) = φ0(T ) = 0, and 0 < φ0(t) ≤ φ0(T0) for each t ∈ (0, T ). ( 27)

For example, if T0 = T /2, we can use φ0(t) = t(T -t).

We finally define the function

φ(t, x) = β(x) φ0(t) , (28)
for (t, x) ∈ (0, T ) × [0, 1], which is the weight function of the Carleman estimate. From [START_REF] Renardy | An Introduction to Partial Differential Equations[END_REF] and ( 27) it is not difficult to see that φ satisfies the following properties:

∃C > 0 such that φ ≤ Cφx and φ n ≤ Cφ m for each positive integers n < m.

(29) Theorem 3.1 Let φ be a function defined by [START_REF] Sivashinsky | Nonlinear analysis of hydrodynamic instability in laminar flames I: Derivation of basic equations[END_REF] and m > 0. Then there exists λ0 > 0 and a constant C = C(T, λ0, r, m) > 0 such that if qi L ∞ ((0,T )×(0,1)) ≤ m for i = 0, 1, 2 then we have

T 0 1 0 e -2λφ |vt| 2 + |(σvxx)xx| 2 λφ + λ 7 φ 7 |v| 2 + λ 5 φ 5 |vx| 2 + λ 3 φ 3 |vxx| 2 + λφ|vxxx| 2 dxdt ≤ C T 0 1 0 e -2λφ |Lv| 2 dxdt + C T 0 e -2λφ(t,0) λ 3 φ 3 x (t, 0)σ(0) 2 |vxx(t, 0)| 2 + λφx(t, 0)σ 2 (0)|vxxx(t, 0)| 2 dt (30)
for all v ∈ V, for all λ ≥ λ0.

As we pointed out in the Introduction, a Carleman estimate for the K-S equation with constant coefficients σ and γ was previously obteined in [START_REF] Cerpa | Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation[END_REF]. The final goal in that work was to prove null-controllability with boundary controls. Thus, (30) is a generalization to the case of non-constant coefficients.

Proof. Consider the following operator P defined in W λ := {e -λφ v : v ∈ V} by P w = e -λφ L(e λφ w).

We then obtain the decomposition P w = P1w + P2w + Rw, where

P1w = 6λ 2 φ 2 x σwxx + λ 4 φ 4 x σw + (σwxx)xx + 6λ 2 (φ 2 x σ)xwx (31) P2w = wt + 4λ 3 φ 3 x σwx + 4λφxσwxxx + 4λ 3 φx(φ 2 x σ)xw (32)
Rw = λφtw + 2λφxσxxwx + λ 2 φ 2 x σxxw + λφxxσxxw + 6λφxσxwxx + 6λ 2 φxφxxσxw + 6λφxxσxw + 2λφxxxσxw + 4λ 2 φxφxxxσw + 6λφxxσwxx + 3λ 2 φ 2 xx σw + 4λφxxxσwx + λφxxxxσw + q0w + q1wx + q1λφxw + q2wxx + 2λq2φxwx + λ 2 q2φ 2 x w + λφxxq2w -2λ 3 φ 2 x φxxσw -2λ 3 φ 3 x σxw. ( 33
)
Thus,

P w -Rw 2 L 2 (Q) = P1w 2 L 2 (Q) + 2 P1w, P2w + P2w 2 L 2 (Q)
where

•, • is the L 2 (Q) scalar product.
For any v ∈ V we obtain vt ∈ L 2 (0, T ; L 2 (0, 1)) and then v ∈ C([0, T ]; L 2 (0, 1)). From the construction of φ (see ( 27)), we obtain w ∈ C([0, T ]; L 2 (0, 1)) and w(x, 0) = w(x, T ) = 0 for any w ∈ W λ .

Let us define the notations

I(w) = -6λ 7 T 0 1 0 φ 6 x φxxσ 2 |w| 2 dxdt, I(wx) = -λ 5 T 0 1 0 φ 4 x σ(30φxxσ + 12φxσx)|wx| 2 dxdt, I(w2x) = -λ 3 T 0 1 0 φ 2 x σ(58φxxσ + 40φxσx)|wxx| 2 dxdt, I(w3x) = -λ T 0 1 0 σ(2φxxσ -4φxσx)|wxxx| 2 dxdt, and
Ix = T 0 (10λ 3 φ 3 x σ 2 |wxx| 2 + 2λφxσσxx|wxx| 2 + 2λφxσ 2 |wxxx| 2 ) 1 x=0 dt
The following weighted norm is defined, for any w ∈ W λ , as

w 2 λ,φ = T 0 1 0 λ 7 φ 7 |w| 2 + λ 5 φ 5 |wx| 2 + λ 3 φ 3 |wxx| 2 + λφ|wxxx| 2 dxdt.
We first require the following Lemma 3.2 Under the hypothesis of Theorem 3.1, there exists δ > 0 such that

P1w, P2w L 2 (Q) ≥ δ w 2 λ,φ + Ix (34)
for λ large enough and for all w ∈ W λ .

Proof. It is sufficient to prove that

P1w, P2w L 2 = 3 k=0 I(w kx ) + R0(w) + Ix (35)
for a large enough λ, for all w ∈ W λ , where |R0(w)| ≤ λ -1 w 2 λ,φ . Indeed, let us first assume that we have (35). From the hypotheses in [START_REF] Renardy | An Introduction to Partial Differential Equations[END_REF] we easily check that there exists ε > 0 such that φ satisfies for all x ∈ (0, 1),

φxx(x) ≤ -εφ < 0, 30φxx(x)σ(x) + 12φx(x)σx(x) ≤ -εφ < 0, 58φxx(x)σ(x) + 40φx(x)σx(x) ≤ -εφ < 0, and 2φxx(x)σ(x) -4φx(x)σx(x) ≤ -εφ < 0. ( 36
)
Then from (29) and assuming (35) we obtain, for λ large enough,

P1w, P2w L 2 = 3 k=0 I(w kx ) + R0(w) + Ix ≥ 2δ w 2 λ,φ -|R0(w)| + Ix ≥ δ w 2 λ,φ + Ix. (37)
Let us now prove (35

): we write P1w, P2w L 2 (Q) = 4 i,j=1
Ii,j where Ii,j denotes the L 2product between the i-th term of P1w in (31) and the j-th term of P2w in (32).

Integrations by parts in time or space are performed on each expression Ii,j. Each resulting expression will be included in one of the terms of the right-hand side of (35). The results are listed below, and we indicate for each term where it will be included.

• I1,1 = -I4,1 + 3λ 2 Q (φ 2 x σ)t|wx| 2 R 0 (w) • I1,2 = -12λ 5 Q (φ 5 x σ 2 )x|wx| 2 I(wx ) . • I1,3 = -12λ 3 Q (φ 3 x σ 2 )x|wxx| 2 I(w 2x ) + 12λ 3 T 0 φ 3 x σ 2 |wxx| 2 1 0 dt Ix • I1,4 = 12λ 5 Q [φ 3 x σ(φ 2 x σ)x]xx|w| 2 R 0 (w) -24λ 5 Q φ 3 x σ(φ 2 x σ)x|wx| 2 I(wx ) . • I2,1 = - λ 4 2 Q (φ 4 x σ)t|w| 2 R 0 (w) . • I2,2 = -2λ 7 Q (φ 7 x σ 2 )x|w| 2 I(w) . • I2,3 = -2λ 5 Q (φ 5 x σ 2 )xxx|w| 2 R 0 (w) + 6λ 5 Q (φ 5 x σ 2 )x|wx| 2 I(wx ) . • I2,4 = 4λ 7 Q φ 5 x σ(φ 2 x σ)x|w| 2 I(w) . • I3,1 = 1 2 1 0 σ|wxx| 2 T 0 dx = 0. • I3,2 = -2λ 3 Q [(φ 3 x σ)xxσ]x|wx| 2 R 0 (w) + 4λ 3 Q (φ 3 x σ)xσ|wxx| 2 I(w 2x ) + 2λ 3 Q (φ 3 x )xσ 2 |wxx| 2 I(w 2x ) -2λ 3 T 0 φ 3 x σ 2 |wxx| 2 1 0 dt Ix . • I3,3 = 2λ T 0 φxσσxx|wxx| 2 1 0 dt Ix -2λ Q (φxσσxx)x|wxx| 2 R 0 (w) + 8λ Q φxσσx|w3x| 2
I(wxxx ) + 2λ T 0 φxσ 2 |wxxx| 2 1 0 dt Ix -2λ Q (φxσ 2 )x|wxxx| 2 I(wxxx ) . • I3,4 = 4λ 3 Q (φx(φ 2 x σ)x)xxσwwxx R 0 (w) -4λ 3 Q (φx(φ 2 x σ)x)xσ|wx| 2 R 0 (w) + 4λ 3 Q φx(φ 2 x σ)xσ|w2x| 2 I(w 2x ) . • I4,1 = 6λ 2 Q (φ 2 x σ
)xwxwt, which is canceled when adding with I1,1.

• I4,2 = 24λ 5 Q (φ 2 x σ)xφ 3 x σ|wx| 2 I(wx)
.

• I4,3 = 12λ 3 Q [(φ 2 x σ)xφxσ]xx|wx| 2 R 0 (w) -24λ 3 Q (φ 2 x σ)xφxσ|wxx| 2 I(w 2x ) . • I4,4 = -12λ 5 Q (φ 2 x σ)x(φ 3 x σ)x|w| 2 R 0 (w)
.

Summing up all the terms, we obtain (35).

Then, we will prove a Carleman inequality for the conjugated operator P .

Lemma 3.3 There exists λ0 > 0 such that for all λ ≥ λ0 we have, for all w ∈ W λ ,

T 0 1 0 λ 7 φ 7 |w| 2 + λ 5 φ 5 |wx| 2 + λ 3 φ 3 |wxx| 2 + λφ|wxxx| 2 dxdt + P1w 2 L 2 (Q) + P2w 2 L 2 (Q) ≤ C P w 2 L 2 (Q) -Ix.
Proof.

From hypothesis [START_REF] Renardy | An Introduction to Partial Differential Equations[END_REF] and the inequalities listed in (36), we know that there exists δ > 0 such that

3 k=0 I(w kx ) ≥ δ w 2 λ,φ (38)
for a parameter λ large enough.

Besides, from the definition (33), the fact qi L ∞ ((0,T )×(0,1)) ≤ m for i = 0, 1, 2, and

(29), it is trivial to check that Rw 2

L 2 ((0,T )×(0,1))

≤ C λ 6 Q φ 6 |w| 2 + λ 2 Q φ 2 |wx| 2 + λ 2 Q φ 2 |wxx| 2 ≤ Cλ -1 w 2 λ,φ . (39)
Thus, for λ large enough, we have

P1w 2 L 2 + 2 P1w, P2w + P2w 2 L 2 = P w -Rw 2 L 2 ≤ 2 P w 2 L 2 + 2 Rw 2 L 2 ≤ 2 P w 2 L 2 + Cλ -1 w 2 λ,φ . (40)
From Lemma 3.2 and estimates (40) and (38), we conclude the proof of Lemma 3.3.

To complete the proof of Theorem 3.1, we have to deal with the norms fo P1w and P2w

appearing in Lemma 3.3. From the definition of P2w, and because ( 26) holds, we have

1 λφ |wt| 2 ≤ 2 λφ |P2w| 2 + C λ 5 φ 5 |w| 2 + λ 5 φ 5 |wx| 2 + λφ|wxxx| 2 and Q 1 λφ |wt| 2 ≤ C Q |P2w| 2 + C w 2 λ,φ
for λ large enough. A similar result is proven for (σwxx)xx and P1w, and we then have

Q 1 λφ |wt| 2 + |(σwxx)xx| 2 ≤ C Q |P1w| 2 + |P2w| 2 + C w 2 λ,φ . (41)
From (41) and Lemma 3.3 we obtain

Q 1 λφ (|wt| 2 + |(σwxx)xx| 2 ) + λ 7 φ 7 |w| 2 + λ 5 φ 5 |wx| 2 + λ 3 φ 3 |wxx| 2 + λφ|wxxx| 2 ≤ C Q |P w| 2 -CIx. ( 42
)
To handle the terms in Ix, we note that for any x ∈ (0, 1) and λ large enough,

-Cλ T 0 φx(x, t)σ(x)σxx(x)|wxx(x, t)| 2 dt ≤ Cλ 3 T 0 φx(x, t) 3 σ(x) 2 |wxx(x, t)| 2 dt.
Then

-CIx ≤ Cλ 3 T 0 φx(0, t) 3 σ(0) 2 |wxx(0, t)| 2 dt + Cλ T 0 φx(0, t)σ(0) 2 |wxxx(0, t)| 2 dt (43)
and from (42) and (43) we obtain

Q 1 λφ |wt| 2 + |(σwxx)xx| 2 + w 2 λ,φ ≤ C Q |P w| 2 + Cλ 3 T 0 φx(0, t) 3 σ(0) 2 |wxx(0, t)| 2 dt + Cλ T 0 φx(0, t)σ(0) 2 |wxxx(0, t)| 2 dt. ( 44
)
Computing the derivatives of e λφ w it is trivial to prove that

∂ k x v 2 = ∂ k x (e λφ w) 2 ≤ C k j=0 λ k-j φ k-j ∂ j x w 2
for each k = 0, . . . , 3. Therefore

Inverse Problem

In this section, the local stability of the nonlinear inverse problem stated in Theorem 1.2 will be proved following the ideas of [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] and [START_REF] Klibanov | Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data[END_REF]. The proof is splited in several steps.

Step 1. Local study of the inverse problem Let γ, γ, y and ỹ be defined as in Theorem 1.2. If we set u = y -ỹ and f = γ -γ, then u solves the following K-S equation:

               ut + (σ(x)uxx)xx + γuxx + ỹux + ỹxu + uux = f (x)ỹxx(x, t), ∀(t, x) ∈ Q, u(t, 0) = u(t, 1) = 0, ∀t ∈ (0, T ), ux(t, 0) = ux(t, 1) = 0, ∀t ∈ (0, T ), u(0, x) = 0, ∀x ∈ (0, 1). (45)
Then, in order to prove the stability of the inverse problem mentioned in the introduction, it is sufficient to obtain an estimate of f in terms of uxx(•, 0), uxxx(•, 0) and u(T0, •), where γ and ỹ are given, γ ∈ H 4 (0, 1) and u is the solution of Equation (45).

We begin by deriving Equation (45) with respect to time. Thus, v = ut satisfies the following equation:

               vt + (σvxx)xx + γvxx + ỹvx + ỹxv = f ỹxxt -g, ∀(t, x) ∈ Q, v(t, 0) = v(t, 1) = 0, ∀t ∈ (0, T ), vx(t, 0) = vx(t, 1) = 0, ∀t ∈ (0, T ), v(0, x) = f R(x, 0), ∀x ∈ (0, 1), (46)
where g(x, t) = u(x, t)yxt(x, t) + ux(x, t)yt(x, t).

The proof of Theorem 1.2 relies on the use of the Carleman estimate given in Theorem 3.1. This result will be used twice. First, Equation (46) allows to estimate v in terms of f , ỹxx and g. Then, Equation (45) will be used to handle the terms u and ux, which appear in the expression of the source term g. The details are given in the next step below.

Step 2. First use of the Carleman estimate

Similarly to the proof of the Carleman estimate, we set w = e -λφ v. Then, we work on the term

I = 2 1 0 T 0 0 w(t, x)wt(t, x) dtdx.
On the one hand, we can calculate I and bound it from below. Indeed, using w(0, x) = e -λφ(0,x) v(0, x) = 0 for all x ∈ (0, 1) and Equation (45), we can easily obtain

I = 1 0 |w (T0, x)| 2 dx = 1 0 e -2λφ(T 0 ,x) |(f ỹxx -(σuxx)xx -γuxx -ỹux -ỹxu -uux) (T0, x)| 2 dx ≥ 1 0 e -2λφ(T 0 ,x) |f (x)| 2 |ỹxx (T0, x)| 2 dx -C u (T0) 2 H 4 (0,1) -C u (T0) 4 H 1 (0,1)
where C depends on γ L ∞ (0,1) , ỹ(T0) W 1,∞ (0,1) and σ W 2,∞ (0,1) .

On the other hand, in order to estimate I from above we apply the Carleman estimate (44)

to Equation (46) using q0 = ỹx and q1 = ỹ, which are uniformly bounded in L ∞ ((0, T ) × (0, 1)) by the hypothesis in Theorem 1.2. We obtain that I = 2 T 0 e -2λφ(0,t) (λ 3 φ 3 x (0, t)σ 2 (0)|vxx(0, t)| 2 + λφx(0, t)σ 2 (0)|vxxx(0, t)| 2 ) dt.

Step 3. Second use of the Carleman estimate

Considering that g = uyxt + uxyt, we will now use a Carleman estimate for the solution of Equation (45) in order to manage the term in g of the previous inequality. The unknown trajectory y is nevertheless such that yxt and yt belong to L ∞ (0, T ; L ∞ (0, 1)) since y ∈ Z. T 0 e -2λφ(0,t) (λ 3 φ 3 x (0, t)σ 2 (0)|uxx(0, t)| 2 + λφx(0, t)σ 2 (0)|uxxx(0, t)| 2 ) dt.

Thus, we have

Gathering all the estimates of I and g that were obtained above, we have T 0 e -2λφ(0,t) (λ 3 φ 3 x (0, t)σ 2 (0)|uxx(0, t)| 2 + λφx(0, t)σ 2 (0)|uxxx(0, t)| 2 ) dt

+ Cλ -3
T 0 e -2λφ(0,t) (λ 3 φ 3 x (0, t)σ 2 (0)|vxx(0, t)| 2 + λφx(0, t)σ 2 (0)|vxxx(0, t)| 2 ) dt.

From the hypothesis of the theorem, we have ỹ ∈ C([0, T ]; H 6 (0, 1)), ỹt ∈ C([0, T ]; H 2 (0, 1)), and |ỹxx(T0, •)| > η > 0 in (0, 1). Using also that the Carleman weight function satisfies [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] thus e -2λφ(t,x) ≤ e -2λφ(T 0 ,x) in (0, T ) × (0, 1), we obtain T 0 e -2λφ(0,t) (λ 3 φ 3 x (0, t)σ 2 (0)|uxx(0, t)| 2 + λφx(0, t)σ 2 (0)|uxxx(0, t)| 2 ) dt +λ -3

T 0 e -2λφ(0,t) (λ 3 φ 3 x (0, t)σ 2 (0)|vxx(0, t)| 2 + λφx(0, t)σ 2 (0)|vxxx(0, t)| 2 ) dt .

Therefore, the regularity of φ (that come from the assumptions on β and φ0) allows to prove that choosing λ0 large enough, we obtain the existence of a constant C that depends on r, K, T, λ0, m such that ∀λ > λ0, f (x) 2 L 2 (0,1) ≤ C u(T0, •) 2 H 4 (0,1) + u (T0, •) 4

H 1 (0,1)

+ uxx(•, 0) 2 H 1 (0,T ) + uxxx(•, 0) 2 H 1 (0,T ) .

This estimate leads to the local stability of the initial inverse problem since f = γ -γ and u = y -ỹ and we have proved Theorem 1.2.

e -2λφ λ 7 φ 7 Remark 3 . 4

 734 |e λφ w| 2 + λ 5 φ 5 |(e λφ w)x| 2 + λ 3 φ 3 |(e λφ w)xx| 2 + λφ|(e λφ w)xxx| 2 dxdt ≤ C w λ,φ .Considering finally that P w = e -λϕ Lv, we obtain Carleman estimate (30). We considered the function β to be increasing. This allows the Carleman inequality to be obtained with boundary terms at x = 0. If a decreasing function β was used instead, then an inequality with boundary terms at x = 1 would have been obtained. As discussed in the following section, the boundary terms in the Carleman inequality are related to the location of the observations in the inverse problem.

e 0 T 0 e

 00 -2λφ |f (x)ỹxxt(x, t)| 2 dxdt + Cλ -3 1 -2λφ |g(x, t)| 2 dxdt +Cλ -3

Q e -2λφ |g| 2 ≤ 2 Qe -2λφ |u| 2 |yxt| 2 + 2 Qe -2λφ |ux| 2 |yt| 2 ≤ C Q e -2λφ |u| 2 + |ux| 2 . 5 Qe -2λφ λ 7 |u| 2 + λ 5 |ux| 2 ≤- 5 Qe -2λφ |f ỹxx| 2 +

 2222252252 Then we can apply Carleman estimate (44) to equation (45), using the identity ỹxu + uux = uyx, and taking q0 = yx and q1 = ỹ, which are bounded in L ∞ ((0, T ) × (0, 1)). We can choose λ0 as large as possible in Theorem 3.1: we then obtain Q e -2λφ |g| 2 ≤ Cλ -Cλ Cλ-5

1 0e 4 H 1 (0, 1 )≤ Cλ - 3 Qe -2λφ |f ỹxxt| 2 +- 8 Qe -2λφ |f ỹxx| 2 +

 14113282 -2λφ(T 0 ,x) |f (x)| 2 |ỹxx (T0, x)| 2 dx -C u (T0) 2 H 4 (0,1) -C u (T0) Cλ Cλ -8

1 0e 1 0e 4 H 1 (

 1141 -2λφ(T 0 ,x) |f (x)| 2 dx ≤ C λ -3 -2λφ(T 0 ,x) |f (x)| 2 dx + u (T0) 2 H 4 (0,1) + u (T0)

Acknowledgments: This work began while L. Baudouin and E. Crépeau were visiting the Universidad Técnica Federico Santa María on the framework of the MathAmsud project CIP-PDE. This study was partially supported by Fondecyt #11080130, Fondecyt #11090161, ANR C-QUID and CISIFS, and CMM-Basal grants.

Remark 2.3 The third-order term zxxx can be added to Equation [START_REF] Klibanov | Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data[END_REF]. Indeed, in that case [START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF] becomes C w -ŵ L 2 (H 7 ) + C wt -ŵt L 2 (H 3 ) , which is bounded by

This last expression is bounded by [START_REF] Kuramoto | On the formation of dissipative structures in reactiondiffusion systems[END_REF]. The remainder of the proof is the same.

Again, by using a fixed point theorem, we can prove Theorem 1.1 for equation [START_REF] Cazenave | An Introduction to Semilinear Evolution Equations[END_REF].

Let y0 ∈ H 6 (0, 1), hj ∈ H 2 (0, 1) compatible with y0, and g ∈ F. For any v ∈ Z, we define Λ(v) as the solution of ( 21) with f = (g -vvx) and z0 = y0. Note that f ∈ F and therefore Λ(v) ∈ Z is well defined. Indeed, if v ∈ Y3 and vt ∈ Y0, then we have

Furthermore, we can prove

hj H 2 .

(

Let ε > 0 and suppose that y0, hj and g satisfy [START_REF] Benabdallah | Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation[END_REF]. Consider v such that v Z ≤ R with R > 0 satisfying C(6ε + R 2 ) < R. From [START_REF] Liu | Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation[END_REF], we obtain Λ(v) Z < R. Thus, the application Λ maps the ball BR := {v ∈ Z v Z ≤ R} into itself.

We will now prove that Λ : BR → BR is a contraction. For any z, v ∈ BR,

is the solution of ( 21) with z0 = 0, hj = 0 for j = 1, . . . , 4 and f = vvx -zzx. We obtain the estimate

Using the definition (2) of the space F, v, z ∈ C([0, 1]; H 6 (0, 1)) ֒→ L ∞ (0, T ; W 5,∞ (0, 1))

and vt, zt ∈ C([0, 1]; H 6 (0, 1)) ֒→ L ∞ (0, T ; W 1,∞ (0, 1)), we obtain

which implies that Λ is a contraction if R is chosen small enough. More precisely, we can choose R, ε such that 2CR < 1 and C(6ε + R 2 ) < R. Hence, the map Λ has a unique fixed point y ∈ Z, which is the unique solution of (8). Thus, we have proven Theorem 1.1.

Global Carleman inequality

In this section, a global Carleman inequality will be proved for the linearized K-S equation.

We define the space V = {v ∈ L 2 (0, T ; H 4 ∩ H 2 0 (0, 1)) Lv ∈ L 2 ((0, T ) × (0, 1))} [START_REF] Puel | On a global estimate in a linear inverse hyperbolic problem[END_REF]