Michel Grabisch
email: michel.grabisch@lip6.fr

Eric Raufaste
email: raufaste@univ-tlse2.fr

An empirical study of statistical properties of the Choquet and Sugeno integrals

This paper investigates the statistical properties of the Choquet and Sugeno integrals, used as multiattribute models. The investigation is done on an empirical basis, and focuses on two topics: the distribution of the output of these integrals when the input is corrupted with noise, and the robustness of these models, when they are identified using some set of learning data through some learning procedure.

I. INTRODUCTION

The Choquet integral [START_REF] Choquet | Theory of capacities[END_REF] and the Sugeno integral [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF], also known under the generic name of fuzzy integral, have become widely used aggregation functions, especially in multicriteria decision making [START_REF] Grabisch | The application of fuzzy integrals in multicriteria decision making[END_REF], [START_REF] Grabisch | Fuzzy measures and integrals in MCDA[END_REF], subjective evaluation [START_REF] Büyüközkan | Providing elucidations of web site evaluation based on a multi-criteria aggregation with the Choquet integral[END_REF], [START_REF] Grabisch | Subjective evaluation of discomfort in sitting position[END_REF], [START_REF] Kwon | A hierarchical subjective evaluation model using non-monotonic fuzzy measures and the Choquet integral[END_REF], [START_REF] Verkeyn | Sugeno integrals for the modelling of noise annoyance aggregation[END_REF], pattern classification [START_REF] Grabisch | Fuzzy integral for classification and feature extraction[END_REF], [START_REF] Keller | Fuzzy integrals in image processing and recognition[END_REF], image processing [START_REF] Grabisch | Texture recognition by Choquet integral filters[END_REF], [START_REF] Grabisch | Mathematical morphology, order filters and fuzzy logic[END_REF], [START_REF] Keller | Fuzzy integrals in image processing and recognition[END_REF], information fusion [START_REF] Auephanwiriyakul | Generalized Choquet fuzzy integral fusion[END_REF], [START_REF] Gader | Rényi entropy with respect to choquet capacities[END_REF], [START_REF] Wang | Multiple neural networks fusion model based on choquet fuzzy integral[END_REF], regression analysis [START_REF] Wang | Nonlinear nonnegative multiregressions based on Choquet integrals[END_REF], etc. (see also [START_REF] Grabisch | Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference[END_REF] for a detailed study and many references).

Their mathematical properties as aggregation functions have been studied extensively [START_REF] Dubois | The use of the discrete Sugeno integral in decision making: a survey[END_REF], [START_REF] Grabisch | Fuzzy integral in multicriteria decision making[END_REF], [START_REF] Marichal | On Sugeno integral as an aggregation function[END_REF], [START_REF] Murofushi | Some quantities represented by the Choquet integral[END_REF], and it is known that many classical aggregation functions are particular cases of these so-called fuzzy integrals, e.g., the weighted arithmetic mean, ordered weigthed averages (OWA), weighted minimum and maximum, etc.

It is surprising that almost no study (at least to the knowledge of the authors) has been done concerning the statistical properties of the Choquet and Sugeno integrals, since this question is of primary importance in any application, where the robustness of models against noise has to be evaluated. The answer may lie in the mathematical difficulty to analyze the statistical behavior of these integrals, due to their nonlinear character. However, a very recent theoretical work has been done in this direction by Marichal, who obtained the mathematical expression of the distribution of the Sugeno integral [START_REF] Marichal | Cumulative distribution functions and moments of weighted lattice polynomials[END_REF]. We will present this result in Section III.

This paper aims to fill this gap, by providing an empirical analysis of statistical properties of the Choquet and Sugeno integrals, based on synthetic and real data. Two questions are addressed: for an input vector corrupted with Gaussian noise, what is the distribution, mean and variance of the output of Choquet and Sugeno integrals? Second, what is the impact of noise corrupting learning data on the fuzzy measure, i.e., the parameters of the model, for a given learning procedure? This second question in fact addresses the problem of robustness of such models.

The paper is organized as follows. Section II recalls basic definitions on fuzzy integrals, while Section III states precisely the kinds of problem we study. Sections IV and V give the result of the empirical studies, concerning respectively the analysis of output when the input vector is corrupted with noise, and the analysis of the fuzzy measure when the learning data is corrupted with noise. Section VI concludes the paper.

All tables and figures showing results of experiments are put in appendix. We give also in appendix an improved version of the HLMS algorithm, whose original version [START_REF] Grabisch | A new algorithm for identifying fuzzy measures and its application to pattern recognition[END_REF] is well known by practitioners of the Choquet integral.

II. BASIC DEFINITIONS

Throughout the paper, we assume that input vectors are ndimensional nonnegative vectors, and N := {1, . . . , n} is the index set. We call for commodity attributes the dimensions of x.

Definition 1: A fuzzy measure [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF] or capacity [START_REF] Choquet | Theory of capacities[END_REF] is a function µ : 2 N → R + such that µ(∅) = 0, and µ(A) ≤ µ(B) whenever A ⊆ B (monotonicity). A fuzzy measure is normalized if µ(N ) = 1. We assume in this paper that fuzzy measures are normalized. A fuzzy measure is additive if µ(A ∪ B) = µ(A) + µ(B) whenever A ∩ B = ∅. The uniform additive measure is the additive measure defined by µ({i}) = 1/n, for i = 1, . . . , n.

Definition 2: Let µ be a fuzzy measure on N , and x := (x 1 , . . . , x n ) ∈ R n + . The Choquet integral of x w.r.t. µ is defined by:

C µ (x 1 , . . . , x n ) := n i=1 (x π(i) -x π(i-1) )µ(A π(i) ) (1)
with π a permutation on N such that

x π(1) ≤ x π(2) ≤ • • • ≤ x π(n)
, with the convention x π(0) := 0, and A π(i) := {π(i), . . . , π(n)}.

An equivalent expression is

C µ (x 1 , . . . , x n ) := n i=1 x π(i) [µ(A π(i) ) -µ(A π(i+1) )] (2)
with the convention A π(n+1) := ∅. From this formula, we deduce that the Choquet integral reduces to a classical weighted arithmetic mean n i=1 w i x i when the fuzzy measure is additive, with w i := µ({i}).

Definition 3: Let µ be a fuzzy measure on N , and x ∈ [0, 1] n . The Sugeno integral of x w.r.t. µ is defined by:

S µ (x 1 , . . . , x n ) := n i=1 x π(i) ∧ µ(A π(i) )
with same notations as above.

When these integrals are used to model the relationship between an input vector x and some output y, the parameters of the model are the 2 n -2 values of the fuzzy measure µ for all subsets of N , except for ∅ and N whose values are fixed. The exponential complexity of these models obliges to look for either simpler models or for interpretative tools. The notion of k-additive fuzzy measures [START_REF] Grabisch | k-order additive discrete fuzzy measures and their representation[END_REF] provides simpler models, ranging from the purely additive one (k = 1), to the general case (k = n), and the general notion of interaction [START_REF] Grabisch | k-order additive discrete fuzzy measures and their representation[END_REF], closely linked to k-additive fuzzy measures, provides a way to interpret the Choquet integral model. We recall here very briefly the essential notions, since they will be used in the sequel. The interested reader can find more details in e.g., [START_REF] Grabisch | Alternative representations of discrete fuzzy measures for decision making[END_REF], [START_REF] Grabisch | The interaction and Möbius representations of fuzzy measures on finite spaces, k-additive measures: a survey[END_REF], [START_REF] Grabisch | Set function over finite sets: transformations and integrals[END_REF], [START_REF] Grabisch | Equivalent representations of set functions[END_REF].

Definition 4: Let µ be a fuzzy measure on N . The Möbius transform of µ is a function m : 2 N → R defined by:

m(A) := B⊆A (-1) |A\B| µ(B).
The Möbius transform is invertible since one can recover µ from m by: µ

(A) = B⊆A m(B).
If the fuzzy measure is additive, then m is non null only for singletons, and m({i}) = µ({i}). Definition 5: A fuzzy measure µ is said to be k-additive for some integer k ∈ {1, . . . , n} if m(A) = 0 whenever |A| > k, and there exists some A with |A| = k such that m(A) = 0. Clearly, 1-additive fuzzy measures are additive measures, and a k-additive fuzzy measure needs only k i=1 n i -1 values to be defined.

Definition 6: Let µ be a fuzzy measure on N . The Shapley value of i ∈ N is defined by:

φ(i) := K⊆N \i (n -k -1)!k! n! [µ(K ∪ i) -µ(K)] with k := |K|.
This notion has been introduced by Shapley [START_REF] Shapley | A value for n-person games[END_REF] in cooperative game theory. It represents the overall importance of attribute i in the model, and it has the property that n i=1 φ(i) = 1. For an additive measure, the Shapley value, the Möbius transform and µ coincide in the sense that µ({i}) = m({i}) = φ(i), i = 1, . . . , n. It could be said that if φ(i) = 1/n, attribute i is neither important nor unimportant, since the Shapley value of the uniform additive measure satisfies this property for every i ∈ N .

Another useful notion is the notion of interaction between two attributes i, j (originally introduced by Murofushi and Soneda [START_REF] Murofushi | Techniques for reading fuzzy measures (III): interaction index[END_REF] and later generalized to more attributes by Grabisch [START_REF] Grabisch | Alternative representations of discrete fuzzy measures for decision making[END_REF]).

Definition 7: Let µ be a fuzzy measure on N , and i, j ∈ N . The interaction index between i and j is defined by:

I(i, j) := K⊆N \{i,j} (n -k -2)!k! (n -1)! [µ(K ∪ {i, j}) -µ(K ∪ i) -µ(K ∪ j) + µ(K)].
If µ is additive, then I(i, j) = 0 for all i, j ∈ N . Positive (resp. negative) interaction values represent a kind of positive synergy or complementarity (resp. negative synergy, redundancy) between attributes. Together with the Shapley value, the interaction index is a valuable tool to interpret the model (see for example [START_REF] Grabisch | Subjective evaluation of discomfort in sitting position[END_REF]). It can be proved that if µ is 2-additive, the values φ(i) and I(i, j) for i ∈ N and j = i uniquely determine µ.

III. DEFINITION OF THE EMPIRICAL STUDY

The aim of the paper is to study the statistical properties of the Choquet and Sugeno integrals. Two important questions arise in an applied context:

• What is the statistical behavior of y = C µ (x) and y = S µ (x) when input vector x is corrupted with noise whose distribution is known? • For a given set of learning data supposed to be corrupted with noise whose distribution is known, what are the statistical properties of the parameters of the model, namely µ, for a given learning method? It is very difficult to give a mathematical answer to these questions, especially the second one; hence our position is to undertake an empirical study. Section IV addresses the first question, while Section V addresses the second one.

We give below some insights to the first question. This will show why a mathematical analysis is difficult to undertake. Let us take the case of the Choquet integral, and consider first an additive fuzzy measure. In this case, the Choquet integral writes:

y = n i=1 µ({i})x i .
Considering x 1 , . . . , x n as independent random variables with expected values m 1 , . . . , m n , we know that the distribution of y is the convolution product of the distributions of the x i 's, up to multiplicative constants µ({i}). Hence, in the normal case, the result is particularly simple since we know that if X 1 , . . . , X n are independent Gaussian random variables, with means m 1 , . . . , m n and variances σ 2 1 , . . . , σ 2 n , then the distribution of the sum α 1 X 1 + • • • + α n X n is again a Gaussian distribution, with mean and variance given by:

m =α 1 m 1 + • • • + α n m n (3) σ 2 =α 2 1 σ 2 1 + • • • + α 2 n σ 2 n . (4)
The additive case is thus solved when the input variables are independent Gaussian. Let us consider the general case where the fuzzy measure is not additive. Rearranging terms in Def. 2, the Choquet integral can be written as in [START_REF] Büyüközkan | Providing elucidations of web site evaluation based on a multi-criteria aggregation with the Choquet integral[END_REF]. It is known that if X 1 , . . . , X n are identically distributed and independent, with cumulative distribution function F (t) := P (X ≤ t), then the distribution function of X π(k) (called the kth order statistic) is given by:

F X π(k) (t) = P (X π(k) ≤ t) = n j=k n j F j (t)(1 -F (t)) n-j .
Supposing that this result permits to compute the distributions of X π(k) , k = 1, . . . , n, it does not seem obvious to compute y, because the X π(k) 's are no more statistically independent, and thus the classical results on the sum of independent random variables (e.g., the sum of Gaussian r.v. is still Gaussian, or the p.d.f. of the sum is the convolution product of the p.d.f.'s) do not apply. We briefly cite the recent result of Marichal concerning the Sugeno integral [START_REF] Marichal | Cumulative distribution functions and moments of weighted lattice polynomials[END_REF]. Consider n independent random variables X 1 , . . . , X n , with cumulative distribution functions F 1 , . . . , F n , and a fuzzy measure µ on N . Let H be the Heaviside step function defined by H(x) := 1 if x ≥ 0, and 0 otherwise. For any c ∈ R, we also introduce H c (x) := H(xc). Then the cumulative distribution function of Y := S µ (X 1 , . . . , X n ) is given by

F (y) = 1 - A⊆N 1 -H µ(A) (y) i∈N \S F i (y) i∈S [1 -F i (y)]. (5)

IV. STATISTICAL STUDY OF THE OUTPUT VALUE

In this section, we study the statistical properties of y = C µ (x) and y = S µ (x) when the vector x is corrupted with noise, that is, x = x 0 + ν, with ν a Gaussian white noise with zero mean. A first experiment is done where the vector x 0 has components which are scattered on the range [0, 1], and a second one where on the contrary components of x 0 are all equal.

In both cases, we generate 1000 6-dimensional samples. We consider the three following fuzzy measures:

• An additive measure µ add defined by: µ add ({1}) µ add ({2}) µ add ({3}) 0.0931979 0.0203709 0.276359 µ add ({4}) µ add ({5}) µ add ({6}) 0.0824232 0.0631751 0.464474 • A 2-additive measure µ 2-add defined by:

µ 2-add ({1}) µ 2-add ({2}) µ 2-add ({3}) 0.130094 0.061056 0.246516 µ 2-add ({4}) µ 2-add ({5}) µ 2-add ({6}) 0.214887 0.400155 0.508465 µ 2-add ({1, 2}) µ 2-add ({1, 3}) µ 2-add ({1, 4}) 0.19115 0.376611 0.344981 µ 2-add ({1, 5}) µ 2-add ({1, 6}) µ 2-add ({2, 3}) 0.530249 0.508466 0.307572 µ 2-add ({2, 4}) µ 2-add ({2, 5}) µ 2-add ({2, 6}) 0.275943 0.400156 0.569521 µ 2-add ({3, 4}) µ 2-add ({3, 5}) µ 2-add ({3, 6}) 0.246517 0.727001 0.826465 µ 2-add ({4, 5}) µ 2-add ({4, 6}) µ 2-add ({5, 6})
0.615042 0.7555 0.569521 • A fuzzy measure coming from identification on some real data set, denoted µ 0 . We do not display the 64 values of µ 0 since this would take up a lot of room, and be little informative, however, fuzzy measures µ add and µ 2-add come from the same real data set, and so can be considered respectively as additive and 2-additive approximations of µ 0 . This real data set comes from experiments done on subjective evaluation of mental workload [START_REF] Grabisch | Application of the Choquet integral to the evaluation of subjective mental workload[END_REF]. We consider both the Choquet and Sugeno integrals in these two experiments.

A. Experiment 1

We choose x 0 = (0, 0.2, 0.4, 0.6, 0.8, 1), with standard deviation of the noise being successively 0.01, 0.05 and 0.1. Table I in appendix gives the mean and standard deviation of the output y = C µ (x) and y = S µ (x) for the different µ and x defined above. Note that if σ is low, then x has a high probability to satisfy x 1 ≤ x 2 ≤ x 3 ≤ x 4 ≤ x 5 ≤ x 6 , i.e., the same permutation applies for x 0 and x. From (2), it is clear that the Choquet integral reduces to a weighted sum whose weights are: µ 0 ({1, 2, 3, 4, 5, 6})µ 0 ({2, 3, 4, 5, 6}) = 0 µ 0 ({2, 3, 4, 5, 6})µ 0 ({3, 4, 5, 6}) = 0 µ 0 ({3, 4, 5, 6})µ 0 ({4, 5, 6}) = 0.2818

µ 0 ({4, 5, 6}) -µ 0 ({5, 6}) = 0 µ 0 ({5, 6}) -µ 0 ({6}) = 0.2588 µ 0 ({6}) = 0.4594.

B. Experiment 2

We choose x 0 = (0.5, 0.5, 0.5, 0.5, 0.5, 0.5), with standard deviation of the noise being successively 0.01, 0.05 and 0.1. Table II in appendix gives the mean and standard deviation of the output y = C µ (x) and y = S µ (x) for the different µ and x defined above.

C. Interpretation and comments

1) Choquet integral: For the additive case, it is possible to compute the theoretical mean and variance of the Choquet integral thanks to (3) and (4). Thus the mean is nothing else that the value of the Choquet integral of x 0 without noise (theoretical value in the tables). For both Experiments 1 and 2, these theoretical values are recovered with great precision (error less than 1%).

The variance writes in our case:

σ 2 y = σ 2 (0.0931979 2 +0.0203709 2 +0.276359 2 +0.0824232 2 + 0.0631751 2 + 0.464474 2 ) = 0.3119959σ 2 .
We obtain the following theoretical values for the standard deviation:

standard deviation of x 0.01 0.05 0.1 standard deviation of y 0.0055857 0.027928 0.055857 Again, we remark that the theoretical values are recovered with great precision in both experiments in the case of the Choquet integral w.r.t. an additive measure.

Strangely enough, even if the fuzzy measure is no more additive, the values for mean and standard deviation remain very similar to the theoretical values of the additive case. This curious result may come from the fact that fuzzy measures µ 2-add and µ add are in a sense approximations of µ 0 , as explained above.

Fig. 1 and2 show some histograms of y for Experiments 1 and 2. We observe that the histograms are close to a Gaussian distribution, especially for low values of σ and for Experiment 1. This is natural, since in this case the Choquet integral reduces to a weighted sum whose weights are given above (Section IV-A), so that the variance of y is

σ 2 y = σ 2 (0.2818 2 + 0.2588 2 + 0.4594 2 ) = 0.3574σ 2 ,
and the standard deviation is σ y = 0.5979σ. This result is very close to the one obtained in Table I. For Experiment 2, the shape of the distribution is flatter than a Gaussian distribution.

2) Sugeno integral: Here results are more difficult to interpret. The mean value of y is stable, however less than for the Choquet integral, and the standard deviation is approximately proportional to the standard deviation of the noise in case of Experiment 2, but not for Experiment 1.

Fig. 3 and4 show some histograms of y for Experiments 1 and 2. The Sugeno integral gives rise to very various forms for the histograms. While Experiment 2 shows histograms similar to a Gaussian density, Experiment 1 shows rather curious shapes, but which can be explained for low values of variance. Indeed, in this case the resulting random variable is approximated by, using Def. 2 and values of µ 0 :

Y = (X 1 ∧ 1) ∨ (X 2 ∧ 1) ∨ (X 3 ∧ 1) ∨ (X 4 ∧ 0.7181) ∨ (X 5 ∧ 0.7181) ∨ (X 6 ∧ 0.4594) ≈ X 1 ∨ X 2 ∨ X 3 ∨ [(X 4 ∨ X 5 ) ∧ 0.7181] ∨ 0.4594
since the X 0 's are centered on 0, 0.2, 0.4, 0.6, 0.8 and 1 respectively, and they have a low variance. Simplifying further, we get: y ≈ X 5 ∧ 0.7181 which well corresponds to the histogram (see Figure 3 left. For σ = 0.01, we have seen that the histogram has only one slot in 0.7181).

V. STATISTICAL STUDY OF THE PARAMETERS OF THE MODEL

A. Description of the experiment

We begin by general considerations. Let X := {x l 1 , . . . , x l n ; y l } l∈L be a set of learning data we have at disposal for the identification of the model (in this section we restrict to the case of the Choquet integral), and some given learning procedure M . We obtain after learning a fuzzy measure µ X ,M , which represents the set of all parameters of the model. An important question in practice is the robustness of the model, specifically:

If a Gaussian noise with zero mean is added to the learning data, what is the influence on the parameters of the model?

The answer depends on the kind of learning procedure which is used. The most commonly used learning procedures for the Choquet integral are the following:

1) Heuristic Least Mean Squares (HLMS) [START_REF] Grabisch | A new algorithm for identifying fuzzy measures and its application to pattern recognition[END_REF]. This algorithm uses as basic idea the gradient algorithm to minimize the sum of squared errors, under the constraint of monotonicity of the fuzzy measure. The algorithm is not optimal, but is very fast and uses few memory. It has also the property of giving a fuzzy measure as close as possible to the uniform additive fuzzy measure µ({i}) = 1/n, i = 1, . . . , n. We use here an improved version of HLMS (described in appendix). Values of parameters for this method are α = 0.01 (coefficient of the gradient) and 300 iterations. 2) Quadratic programming (QUAD). The error criterion is again the sum of squared errors, under the constraint of monotonicity of the fuzzy measure. This leads to a quadratic program with linear constraints, and gives a (non unique) optimal solution. 3) k-additive quadratic programming (k-ADD). It is the same as above, but the model is supposed to be a Choquet integral w.r.t. a k-additive fuzzy measure. In this experiment, we use k = 2 (2-ADD).

For a careful study of the methods based on quadratic learning, the reader is referred to [START_REF] Miranda | Optimization issues for fuzzy measures[END_REF]. We describe our experimental setting and define more precisely our aims. We consider a theoretical model (the Choquet integral) with 4 attributes, built with several different fuzzy measures, and the identification of these models will be done with the above mentionned learning procedures, and also several data sets, each of them illustrating a different practical situation. In order to master the whole set of data of the study, we use only synthetic data.

We consider the following three fuzzy measures:

• an additive measure µ add defined by µ add ({1}) µ add ({2}) µ add ({3}) µ add ({4}) 0. 0.5 0.2 1 The number of free parameters of these three models is respectively 3, 9, and 14.

m 0 ({1}) m 0 ({2}) m 0 ({3}) 0.4 0.1 0.1 m 0 ({4}) m 0 ({1, 2}) m 0 ({1, 3}) 0 0.1 0 m 0 ({1, 4}) m 0 ({2, 3}) m 0 ({2, 4}) 0 -0.1 0.1 m 0 ({3, 4}) m 0 ({1, 2, 3}) m 0 ({1, 2, 4}) 0 0 0 m 0 ({1, 3, 4}) m 0 ({2, 3, 4}) m 0 ({1,
We consider 9 learning data sets, which we describe below, illustrating three typical situations in practice:

• 100 data (x l ; y l ), with vectors x l uniformly distributed on [0, 1] 4 , and y l = C µ (x l ), with µ being one of the three fuzzy measures above defined. We denote by X 100 µ add , X 100 µ2-add and X 100 µ0 these 3 learning data sets. Since data are in sufficient number regarding the number of parameters and uniformly distributed in the whole space, this is an ideal situation for learning, which is not always encountered in practice. We call it Situation S1. • same as above, but the number of learning data is only 10.

We denote these learning data sets by X 10 µ add , X 10 µ2-add and X 10 µ0 . The aim is to study the behavior of learning procedures with very few learning data, but still uniformly distributed in the space. We call it Situation S2.

• we select in X 100 µ add , X 100 µ2-add and X 100 µ0 the data such that x l 1 ≤ x l 4 , which gives about 50 data. We denote by X 1≤4 µ add , X 1≤4 µ2-add and X 1≤4 µ0 these learning data sets. This data set illustrates the situation where there are apparently enough data regarding the number of parameters, but there is no data in some part of the space, which makes difficult the learning of some parameters. We call this situation S3. In our case, imposing x l 1 ≤ x l 4 amounts to cancel all data which would allow the identification of the values of µ(A) such that A ∋ 1 but A ∋ 4, i.e., these are µ(1), µ(1, 2), µ(1, 3) and µ(1, 2, 3). Remark that these values are the highest in the three fuzzy measures defined above. This means that we have learning data with a considerable lack of information. As said above, we use the following learning procedures: HLMS, QUAD and 2-ADD. We have verified beforehand that the above defined learning data sets (without noise) permit a perfect identification of the 3 fuzzy measures, up to a precision of 10 -6 for QUAD, and 10 -2 for HLMS.

For each learning data set and each learning procedure, we add to the data a Gaussian noise with zero mean and standard deviation being successively 0.01, 0.02, and 0.05, and we have done 100 realizations of each data set. We try to answer the following questions for each situation S1, S2 or S3:

What is the behavior of the estimation of each parameter, described in terms of bias and standard deviation? What is the best learning procedure, i.e., with minimum bias and standard deviation?

The results shown are:

• The average bias and average standard deviation of the values of the fuzzy measure, denoted by b µ , σ µ , of the Shapley value, denoted by b φ , σ φ , and of the interaction, denoted by b I , σ I , obtained on 100 realizations. All these results are given in Tables III to XI. • In the case of incomplete data sets X 1≤4 µ add and X 1≤4 µ0 the values of the fuzzy measure found by the different learning procedures, together with their Shapley values and interactions (Tables XII andXIII). For this case, we have taken σ = 0.01. Caution: Shapley values are multiplied by n (hence by 4) in tables, so that a value of 1 indicates a neutral value for overall importance (see Section II). Consequently, for comparing standard deviations of the Shapley value with those of interactions, it is necessary to divide the standard deviation of the Shapley value by 4.

B. Results and comments

We address first Tables III to XI.

• The computation of the overall bias on µ, φ, I being done by an arithmetic mean of the biases on each value of µ, φ, I, the bias on φ is always close to 0, since the sum of the Shapley values is equal to 1. This figure being not significant, we discard it from our analysis.

• Generally speaking, the bias on µ remains very low, and can be considered as being 0, even with a low number of data (X 10 ). However, in the case of learning data which do not cover the whole space [0, 1] 4 (Tables IX to XI), a non negligible negative bias may occur, in particular for QUAD.

There is no clear linearity relation between the bias and σ: although the bias is generally increasing with the level of noise, in many cases, it is observed that the bias is smaller for higher levels of noise. More or less the same conclusions hold for the bias of the interaction.

• Results are easier to interpret for the standard deviation of µ. For Situations S1 and S2 (that is, with uniform distribution of the data), Tables III to V and VI to VIII show that σ µ is approximately proportional to σ (although more experiments should be done to confirm this), and is relatively independent of µ, especially for X 10 . A linear regression σ µ = ασ done for each learning procedure and each situation, and using results for all three fuzzy measures gives the following result: clearly shows that in the case of a large number of uniformly distributed data (X 100 ), 2-ADD, and QUAD in second position give the best results, always better than HLMS. By contrast, when the number of data is insufficient (X 10 ), HLMS gives always the best results, and QUAD the worst ones. This tendency is even stronger with X 1≤4 (see Tables IX to XI), where QUAD and 2-ADD give similar results, far worse than HLMS (for these cases, since σ µ is no more independent of µ, we did not performed any linear regression). Lastly, remark that the linear hypothesis is questionable for Situation S2 and QUAD, 2-ADD.

α
• The same phenomena can be observed on the standard deviation on the Shapley value, even in a more noticeable way. For the interaction, the behavior of its standard deviation is the same as the one of µ. Lastly, we comment on Tables XII and XIII, illustrating Situation S3. The effect of a lack of learning data in a half-space can be well observed there. As explained above, this lack forbids a correct estimation of µ({1}), µ({1, 2}), µ({1, 3}), and µ({1, 2, 3}). This is exactly what can be observed: these values are very badly estimated, and only these ones.

The consequence is that the Shapley values are more or less heavily perturbed. Indeed, in the computation of φ(i), all terms µ(A ∪ i)µ(A), A ⊆ N \ i, are used. It is easy to see that in our case, for φ(1) and φ(4), there are 4 terms among 8 which are spoilt, while there are only 2 among 8 for φ(2) and φ(3). This can be well observed on Table XIII (remark that Table XII is not significant for this case since the measure is additive, so that φ(i) = µ({i})). For the computation of I, the perturbation is even stronger and tends to spread over all values.

Let us come back to the estimation of µ({1}), µ({1, 2}), µ({1, 3}) et µ({1, 2, 3}). The results that are closest to the true value are given by HLMS and 2-ADD, while QUAD gives erratic results.

The good performance of HLMS in Situation S3 (and also S2) is explained by the fact that, in the absence of information for values µ(A) for some subsets A ⊆ N , HLMS assign values for those µ(A)'s which are as close as possible (under monotonicity constraints) to the uniform additive measure, i.e., µ(A) = |A| n . This can be checked in Tables XII andXIII.

VI. CONCLUSION

The following fundamental conclusions can be drawn from the different experimentations:

• The exact distribution of y = C µ (x) and y = S µ (x) seem to be very complicated to obtain, even if for the latter case, an analytical expression exists. Empirically, it is observed that if x follows a Gaussian distribution, then in general the distribution of y is similar to a Gaussian one (unimodal and symmetric). For the Sugeno integral, the results are more unpredictable, a peak or a Gaussian distribution can be obtained. • The quality of the learning data set is of considerable influence on the quality of the identification. If the number of data is sufficient with respect to the number of parameters of the model, and if they are uniformly distributed in [0, 1] n (Situation S1), then the optimal methods QUAD, 2-ADD give the best results in term of robustness. The standard deviation on µ is roughly proportional to the standard deviation σ of the noise. By contrast, the value of the bias is not clearly related to σ, but remains very low.

If few learning data are available, but still uniformly distributed (Situation S2), HLMS becomes slightly better than 2-ADD, and significantly better than QUAD. Conclusions on the relation between the standard deviation on µ, the bias and σ remain the same than in Situation S1.

If some parts of [0, 1] n are not covered (Situation S3), then HLMS gives the most reliable results, then 2-ADD in second position since this method uses a reduced number of parameters. In this case, which is in practice the most frequent one, it seems better to avoid the use of QUAD. We mention that the above learning procedures, as well as tools to compute the Möbius transform, interaction indices, Shapley values, etc., are all available in the free Kappalab package [START_REF] Grabisch | Using the Kappalab R package for Choquet integral based multi-attribute utility theory[END_REF], a package running under the R environment for statistics (see http://www.polytech.univ-nantes.fr/kappalab).

As a final remark, we would like to stress the fact that this work is a first step towards a more complete and theoretical analysis of the statistical properties of these methods, now more and more used in practical applications. In particular, the least square estimation of fuzzy measures can be seen as an extension of classical multiple linear regression estimation, where classical results could be applied here with benefit.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. Histogram of y = Cµ(x) for Experiment 1, µ 0 and σ = 0.01 (left), σ = 0.1 (right)

  bµ -0.0567233 -0.0571832 -0.0562092 -0.1188823 -0.1042598 -0.0871266 -0.0588714 -0.0720761 -0.0593036 b φ -0.0000025 0.000005 -0.0000175 0.0000075 -0.0000125 -0.0000225

VII. ACKNOWLEDGMENTS

We address all our thanks to INRS (Institut National de Recherche et de Sécurité) for supporting this study (contract N • 5033704). We thank one of the anonymous referees for providing the theoretical result about the Sugeno integral.

APPENDIX

An improvement of HLMS

The HLMS algorithm for the identification of a fuzzy measure when the model is a Choquet integral has been proposed by Grabisch in [START_REF] Grabisch | A new algorithm for identifying fuzzy measures and its application to pattern recognition[END_REF]. We propose here a simplification of it, which leads to better performance. We first recall the basic ideas of HLMS. Consider a learning datum (x 1 , . . . , x n ; y). We compute the model error e = C µ (x)y. Let us denote by u(0), u(1), . . . , u(n) the values of the fuzzy measure µ used in the computation of C µ (x). The new values are computed as follows (gradient algorithm):

where α ∈ [0, 1] is a constant, and e max is the maximum value of the error. e max = 1 if y takes its values in [0, 1], and π is a permutation on N as in Def. 2. The problem is then to ensure the monotonicity of µ, while modification of its values are done. In the original HLMS algorithm, this was done in two steps. In a first step, for each datum, after application of Eq. ( 6), verification of monotonicity was done only for values of µ already modified (for previous data). Then in a second step, when all data have ben used, unmodified values of µ are checked to see if they satisfy monotonicity, and they are modified accordingly. Then a kind of uniformization is performed on the whole set of unmodified values of µ.

It has been observed in experiments that is some rare cases, monotonicity may be violated when HLMS is used. This is due to the intricate way of modifying the values of µ. The new version of the algorithm avoids this drawback, and is much simpler (particularly in step 2). We describe it below.

• step 0 : the fuzzy measure is initialized at the uniform additive measure. • step 1.1 : consider a learning datum (x 1 , . . . , x n ; y), inducing a permutation π. Compute e = C µ (x)y, and denote by u(0) := µ(∅) = 0, . . . , u(i) := µ(A π(n-i+1) , . . . , u(n) := µ(N ) = 1 the values of the fuzzy measure µ used in the computation of C µ (x). • step 1.2 : compute the new value u new (i) by Eq. ( 6).

• step 1.3 : verify the monotonicity relations. If e > 0, the verification is done for all lower neighbors, i.e., all µ(A π(n-i+1) \ j) for all j ∈ A, and if e < 0, for all upper neighbors, i.e., all µ(A π(n-i+1) ∪ j) for all j ∈ N \ A.

If a monotonicity relation is violated, say with µ(K), then

Repeat steps 1.2 and 1.3 for i = 1, . . . , n -1, in the following order:

-if e > 0, we begin by u(1), u(2), . . . , u(n -1) -if e < 0, we begin by u(n -1), u(n -2), . . . , u(1) Repeat steps 1.1 to 1.3 for all learning data.

• step 2: for every µ(K) left unmodified in step 1 (scanning begins by singletons, then pairs, etc.), adjust its value considering the values of its upper and lower neighbors, in order to homogenize the whole. This is done by computing the minimum distance between µ(K) and its upper (resp. lower) neighbors, denoted d min , (resp. d min ). Then

Step 1 and 2 form one iteration, and they can be repeated in the same order. [START_REF] Büyüközkan | Providing elucidations of web site evaluation based on a multi-criteria aggregation with the Choquet integral[END_REF][START_REF] Dubois | The use of the discrete Sugeno integral in decision making: a survey[END_REF] 0 -0.0018243 0.0032101 0.0004247 I add [START_REF] Choquet | Theory of capacities[END_REF][START_REF] Dubois | The use of the discrete Sugeno integral in decision making: a survey[END_REF] 0 -0.0420457 0.0075858 -0.001587

APPENDIX

Tables and Figures


Pierre Gilles Lemarié-Rieusset
email: pierregilles.lemarierieusset@univ-evry.fr

Highly singular (frequentially sparse) steady solutions for the 2D Navier-Stokes equations on the torus

Keywords: Navier-Stokes equations, steady solutions, lacunary Fourier series, nonuniqueness, Koch and Tataru theorem AMS classification : 35K55, 35Q30, 76D05

We construct non-trivial steady solutions in H -1 for the 2D Navier-Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to redefine the notion of solutions.

Introduction

In this paper, we are looking for steady solutions u of the 2D Navier-Stokes equations on the torus T d = R 2 /2πZ 2 , i.e. for solutions of the equations

∆ u -P( u • ∇ u) = 0 div u = 0 ( 1
)
where u is a periodical distribution vector field, with mean value 0:

T 2 u(x) dx = 0.
Such a periodical distribution vector field u can be written as a Fourier series u(x) =

k∈Z 2 + cos(k • x) v k + sin(k • x) w k
where k ∈ Z 2 + if and only if k = (k 1 , k 2 ) ∈ Z 2 \ {(0, 0)} and arg(k 1 + ik 2 ) ∈ (-π/2, π/2]. The convergence in D is given by a slow growth of the coefficients:

| v k | + | w k | ≤ C|k| N
for some constants C and N ; in particular, u ∈ H s (T 2 ) for s < -N -1.

If k ∈ Z 2 , k = (0, 0) and k / ∈ Z 2 + , then we may replace k with -k (with -k ∈ Z 2 + ) and write cos(k

•x) v k +sin(k •x) w k = cos((-k)•x) v k +sin((-k)•x)(-w k ), hence the condition k ∈ Z 2
+ is not essential. In our equations, P is the Leray projection operator on solenoidal vector fields, defined by

P( k∈Z 2 + cos(k • x) v k + sin(k • x) w k ) = k∈Z 2 + ρ k cos(k • x + θ k )k ⊥ (2)
with

ρ k cos(k • x + θ k ) = cos(k • x) v k • k ⊥ |k| 2 + sin(k • x) w k • k ⊥ |k| 2 where (k 1 , k 2 ) ⊥ = (-k 2 , k 1 ), |(k 1 , k 2 )| 2 = k 2 1 + k 2 2 and ρ k = ( v k • k ⊥ ) 2 + ( w k • k ⊥ ) 2 |k| 2 .
It is easy to check that, if the solution u satisfies u ∈ H 1 (T 2 ), then u = 0. Thus, we shall not require (weak) diffentiability for u. Usually, it is customary to rewrite u • ∇ u in the equations as ∂ 1 (u 1 u) + ∂ 2 (u 2 u) (since div u = 0), where the derivatives are taken in the sense of distributions. In order to define u i u, one then usually requires that u be square integrable.

As we whall see it, it is easy to check that, if the solution u satisfies u ∈ L p (T 2 ) for some p > 2, then u = 0. This is even true when u belongs to the Lorentz space L 2,1 (T 2 ). This is still probably the case for u ∈ L 2 (T 2 ), thus we will search for some solution u which is not square integrable. We need however to be able to define P( u • ∇ u) when u is no longer square integrable.

Definition 1 (Admissible vector fields).

A divergence free periodical distribution vector field is admissible for the steady problem on T 2 if it is of the form u =

k∈Z 2 + ρ k cos(k • x + θ k )k ⊥ (3)
with

k 1 ∈Z 2 + k 2 ∈Z 2 + ρ k 1 ρ k 2 P cos(k 1 •x + θ k 1 ) sin(k 2 •x + θ k 2 )(k ⊥ 1 •k 2 )k ⊥ 2 H -N < +∞ (4)
for some N .

If u = k∈Z 2 + ρ k cos(k •x+θ k )k ⊥ = k∈Z 2 +
u k is an admissible vector field, we will then define P( u • ∇ u) as

P( u • ∇ u) = k 1 ∈Z 2 + k 2 ∈Z 2 + P( u k 1 • ∇ u k 2 ). ( 5
)
The main result in this paper is then the following one:

Theorem 1. There exists non-trivial solutions to the equations

∆ u -P( u • ∇ u) = 0 div u = 0 ( 6
)
where u is an admissible vector field (with mean value 0)

with u ∈ H -1 (T 2 ) ∩ BM O -1 .
The paper is organized in the following manner: in Section 1, we recall classical results on steady solutions on T d ; in section 2, we describe some examples of admissible vector fields; in section 3, we prove Theorem 1; in section 4, we comment on the Koch and Tataru theorem.

1 Steady solutions for the Navier-Stokes problem on T d : known results.

In this section, we recall known results on steady solutions for the Navier-

Stokes problem in L 2 (T d ), for d ≥ 2. Case u ∈ H 1 (T d ) ∩ L 4 (T d ). If u ∈ H 1 (T d ) ∩ L 4 (T d ) (recall that, when d ≤ 4, H 1 (T d ) ⊂ L 4 (T d )), we can compute - T d | ∇⊗ u| 2 dx = T d u•P( u•∇ u) dx = T d u•( u•∇ u) dx = T d div ( | u| 2 2 u) dx = 0.
Thus u = 0 (we are interested in vector fields with null mean value).

Case u ∈ L p (T d ), p > d.

As p is subcritical when p > d, the initial value problem

     ∂ t v = ∆ v -P( v • ∇ v) div v = 0 v(0, x) = u 0 (x) (7)
with u 0 ∈ L p has a unique solution in C([0, T ), L p ) for some time T . This solution v is smooth on (0, T ) × T d . If u is a steady solution of (1), then v(t, x) = u(x) defines a solution of the Cauchy problem [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF] with initial value u 0 = u. Hence, if u ∈ L p , we find that u ∈ H 1 ∩ L ∞ and finally u = 0.

Case u ∈ L d (T d ), d ≥ 3.
The value p = d is critical for the Cauchy problem [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF]. When u 0 ∈ L d , the problem has a solution v in C([0, T ), L d ) for some time T and this solution v is smooth on (0, T ) × T d . But uniqueness of solutions in C([0, T ), L d ) is known only for d ≥ 3. Thus, if u is a steady solution of [START_REF] Buckmaster | Nonuniqueness of weak solutions to the Navier-Stokes equation[END_REF]

with u ∈ L d (T d ) and if d ≥ 3, then u = 0. Case u ∈ L 2 (T d ), d ≥ 4.
Recently, Luo [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF] constructed non trivial steady solutions in L 2 (T d ), d ≥ 4 (this solution belongs to L p (T d ) for some p ∈ (2, d)). His proof was following the scheme of convex integration developed by De Lellis and Székelyhidi [START_REF] De Lellis | The Euler equations as a differential inclusion[END_REF] in the case of non-steady solutions for the Euler equations and by Buckmaster and Vicol [START_REF] Buckmaster | Nonuniqueness of weak solutions to the Navier-Stokes equation[END_REF] in the case of non-steady solutions for the Navier-Stokes equations. However, his proof requires the spatial dimension d to be no lesser than 4.

Case u ∈ L 2 (T 2 ).

Uniqueness of solutions of the Cauchy problem [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF]

in C([0, T ), L 2 (T 2 )) is not known. Proofs of uniqueness in C([0, T ), L 3 (T 3
)) are based on maximal regularity properties which are no longer true in the 2D case [START_REF] Furioli | Sur l'unicité dans L 3 (R 3 ) des solutions "mild" de l'équation de Navier-Stokes[END_REF][START_REF] Lions | Unicité des solutions faibles de Navier-Stokes dans L N (Ω)[END_REF][START_REF] Meyer | Wavelets, paraproducts and Navier-Stokes equations[END_REF][START_REF] Monniaux | Uniqueness of mild solutions of the Navier-Stokes equation and maximal L p -regularity[END_REF][START_REF] Furioli | Unicité dans L 3 (R 3 ) et d'autres espaces limites pour Navier-Stokes[END_REF][START_REF] May | Extension d'une classe d'unicité pour les équations de Navier-Stokes[END_REF]. On the other hand, the proof of non-uniqueness in C([0, T ), L 2 (T 3 )) is based on convex integration methods which cannot be applied in the 2D case [START_REF] Buckmaster | Nonuniqueness of weak solutions to the Navier-Stokes equation[END_REF].

However, we have uniqueness in C([0, T ), L 2,1 (T 2 )), where L 2,1 is a Lorentz space:

Proposition 1. If v 1 and v 2 are two solutions of the Cauchy problem (7) with v 1 , v 2 ∈ C([0, T ), L 2,1 (T 2 )) and v 1 (0, .) = v 2 (0, .) = u 0 , then v 1 = v 2 .
Proof. We follow the lines of [START_REF] Furioli | Unicité dans L 3 (R 3 ) et d'autres espaces limites pour Navier-Stokes[END_REF] and [START_REF] Meyer | Wavelets, paraproducts and Navier-Stokes equations[END_REF]. If

T * = sup{S ≥ 0 / v 1 = v 2 on [0, S)} and if T * < T , then v 1 (T * , .) = v 2 (T * , .): it is obvious if T * = 0,
and is a consequence of continuity if 0 < T * < T . Moreover, we can write the integral formulation of the Navier-Stokes equations with initial time T * : for

T * ≤ t < T and j = 1, 2 v j (t, .) = e (t-T * )∆ v j (T * , .) + t T * R 2 K(t -s, . -y)( v j (s, y) ⊗ v j (s, y) dy ds with |K(t, x)| ≤ C 1 ( √ t + |x|) 3 ≤ C(1 |x|<π 1 t 3/2 + 1 |x|≥π 1 |x| 3 ). Let K per (t, x) = k∈Z 2 K(t, x -2πk). Then K per (t, .) ∈ L 1 (T 2 ) ∩ L ∞ (T 2 ), with K per (t, .) 1 ≤ C 1 √ t and K per (t, .) ∞ ≤ C(1 + 1 t 3/2 ). We write v j = w 0 + w j , with w 0 = e (t-T * )∆ v j (T * , .). By density of L ∞ (T 2 ) in L 2,1 (T 2 ), we have lim δ→0 + sup T * <t<T * +δ √ t -T * w 0 (t, .) ∞ = 0,
while, by continuity of v j and w 0 , we have, for j = 1, 2,

lim δ→0 + sup T * <t<T * +δ w j (t, .) L 2,1 (T 2 ) = 0.
We now write, for w = v 1 -v 2 and T * ≤ t < T ,

w(t, x) = t T * T 2
K per (t -s, x -y)( w(s, y) ⊗ w 0 (s, y)) dy ds

+ t T * T 2 K per (t -s, x -y)( w(s, y) ⊗ w 1 (s, y)) dy ds + t T * T 2
K per (t -s, x -y)( w 0 (s, y) ⊗ w(s, y)) dy ds

+ t T * T 2 K per (t -s, x -y)( w 2 (s, y) ⊗ w(s, y)) dy ds = z 1 (t, x) + z 2 (t, x) + z 3 (t, x) + z 4 (t, x)
and we estimate w(t, .) L 2,∞ (T 2 ) . We first write, for w = v 1 -v 2 and T * ≤ t < T ,

z 1 (t, .) L 2,∞ + z 3 (t, .) L 2,∞ ≤C t T * K per (t -s, .) 1 w 0 (s, .) ∞ w(s, .) L 2,∞ ds ≤ C t T * 1 √ t -s √ s -T * ds sup T * <s<t √ s -T * w 0 (s, .) ∞ sup T * <s<t w(s, .) L 2,∞ =πC sup T * <s<t √ s -T * w 0 (s, .) ∞ sup T * <s<t w(s, .) L 2,∞ .
For A > 0, we write

z 2 (t, x) + z 4 (t, x) = t sup(t-A,T * ) T 2 K per (t -s, x -y)( w(s, y) ⊗ w 1 (s, y)) dy ds + sup(t-A,T * ) T * T 2 K per (t -s, x -y)( w(s, y) ⊗ w 1 (s, y)) dy ds + t sup(t-A,T * ) T 2 K per (t -s, x -y)( w 2 (s, y) ⊗ w(s, y)) dy ds + sup(t-A,T * ) T * T 2 K per (t -s, x -y)( w 2 (s, y) ⊗ w(s, y)) dy ds = z 5,A (t, x) + z 6,A (t, x) + z 7,A (t, x) + z 8,A (t, x).
Since the pointwise product is bounded from L 2,1 × L 2,∞ to L 1 , we have

z 5,A (t, .) 1 ≤C t sup(t-A,T * ) K per (t -s, .) 1 w 1 (s, .) L 2,1 w(s, .) L 2,∞ ds ≤C t sup(t-A,T * ) 1 √ t -s ds sup T * <s<t w 1 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ ≤2C √ A sup T * <s<t w 1 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ .
Similarly

z 7,A (t, .) 1 ≤ C √ A sup T * <s<t w 2 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ .
On the other hand, we have (for T * ≤ t < min(T, T * + 1))

z 6,A (t, .) ∞ ≤C sup(t-A,T * ) T * K per (t -s, .) ∞ w 1 (s, .) L 2,1 w(s, .) L 2,∞ ds ≤C sup(t-A,T * ) T * 1 (t -s) 3/2 ds sup T * <s<t w 1 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ ≤2C 1 √ A sup T * <s<t w 1 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ .
Similarly

z 8,A (t, .) ∞ ≤ C 1 √ A sup T * <s<t w 2 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ . As L 2,∞ = [L 1 , L ∞ ]1 2 ,∞ , we find that, for T * ≤ t < min(T, T * + 1), z 2 (t, .) L 2,∞ ≤ C sup T * <s<t w 1 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ and z 4 (t, .) L 2,∞ ≤ C sup T * <s<t w 2 (s, .) L 2,1 sup T * <s<t w(s, .) L 2,∞ .
Putting together those estimates, we get that, for 0

< δ < min(1, T -T * ), sup T * ≤t≤T * +δ w(t, .) L 2,∞ ≤ CA(δ) sup T * ≤t≤T * +δ w(t, .) L 2,∞ with
A(δ) = sup T * <t<T * +δ √ t -T * w 0 (t, .) ∞ + w 1 (t, .) L 2,1 (T 2 ) + w 2 (t, .) L 2,1 (T 2 ) .
As lim δ→0 + A(δ) = 0, we get that w = v 1 -v 2 is equal to 0 on [0, T * + δ] for δ small enough, in contradiction with the definition of T * . Thus T * = T , and v 1 = v 2 .

Corollary 1.

If u is a steady solution of (1) with u ∈ L 2,1 (T 2 ), then u = 0.

Proof. We consider the Cauchy problem [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF] where the initial value u 0 is equal to our steady solution u. We can construct a mild solution v 1 on a small time interval [0, T ] such that v 1 ∈ C([0, T ], L 2,1 ), sup 0<t<T √ t v 1 (t, .) ∞ < +∞ and lim t→0 + √ t v 1 (t, .) ∞ = 0. We have another solution in C([0, T ], L 2,1 ), namely v 2 (t, .) = u. By uniqueness, we find that u = v 1 ( T 2 , .) ∈ L ∞ , and thus u = 0.

2 Admissible vector fields.

In this section, we describe some examples of admissible divergence free periodical distribution vector fields

u = k∈Z 2 + ρ k cos(k • x + θ k )k ⊥ = k∈Z 2 + u k . ( 8
)

Square integrable vector fields

The most obvious example is the case u ∈ L 2 , i.e.

k∈Z 2 + u k 2 2 < +∞. We have u k 1 ⊗ u k 2 ∈ L 1 while the frequencies appearing in u k 1 ⊗ u k 2 are k 1 + k 2 and k 1 -k 2 (if k 1 = k 2 , since u k • ∇ u k = 0). Thus, P( u k 1 • ∇ u k 2 ) H -N ≤ C N u k 1 2 u k 2 2 (|k 1 + k 2 | -N -1 + |k 1 -k 2 | -N -1 ). If N > 1, we have j∈Z 2 \{(0,0)} |j| -N -1 < +∞, hence j∈Z 2 \{(0,0)} |j| -N -1 u k 2 +j 2 ∈ l 2
and thus

k 1 ∈Z 2 + k 2 ∈Z 2 + P( u k 1 • ∇ u k 2 ) H -N < +∞.

Lacunary Fourier series

Let us consider a lacunary Fourier series

u = +∞ j=0 ρ k j cos(k j • x + θ k j )k ⊥ j = +∞ j=0 u k j with |k j+1 | > 8|k j | and +∞ j=0 ρ 2 k j |k j | -2N < +∞
where N ≥ 0 (so that u ∈ H -N -1 ). We have.

P( u k j • ∇ u kp ) H -2N -3 ≤Cρ k j ρ kp |k j ||k p |(|k j | + |k p |) -2N -2 ≤C |k j | -N ρ k j |k p | -N ρ kp min(|k j |, |k p |) max(|k j |, |k p |) N +1
.

Noticing that

+∞ j=1 0≤p≤j-1 ρ k j ρ kp |k j ||k p | (|k j | + |k p |) 2N +2 ≤ C +∞ j=1 0≤p≤j-1 ρ k j |k j | N ρ kp |k p | N 8 -(j-p)(N +1) , we find that +∞ p=0 +∞ j=0 P( u kp • ∇ u k j ) H -2N -3 < +∞.
Remark: We could have proved that P( u • ∇ u) ∈ H -2N -3 in another way:

we have u ∈ H -N -1 ∩ B -N -1
∞,∞ ; using paradifferential calculus and decomposing the product u • ∇ u in two paraproducts and a remainder, we see that the paraproducts are controlled in

H -2N -3 by u H -N -1 u B -N -1
∞,∞ , while the remainder is equal to 0.

Lacunary resonant Fourier modes

u = +∞ j=0 ρ k j (cos(k j • x + θ k j )k ⊥ j + cos((k j + ω j ) • x + η k j )(k j + ω j ) ⊥ ) = +∞ j=0 u k j with |k j+1 | > 8|k j |, |k j | > 8|ω j |, ω j • k j = 0 and +∞ j=0 ρ 2 k j |k j | |ω j | < +∞.
We write

v k j = ρ k j cos(k j • x + θ k j )k ⊥ j and w k j = ρ k j cos((k j + ω j ) • x + η k j )(k j + ω j ) ⊥ . In particular, we have that +∞ j=0 ρ 2 k j < +∞, so that u ∈ H -1 ∩ B -1 ∞,∞
. Following the computations of the case of lacunary solutions, we find that +∞ p=0 0≤j,j =p

P( v kp • ∇ v k j ) H -3 < +∞, +∞ p=0 0≤j,j =p P( v kp • ∇ w k j ) H -3 < +∞, +∞ p=0 0≤j,j =p P( w kp • ∇ v k j ) H -3 < +∞, +∞ p=0 0≤j,j =p P( w kp • ∇ w k j ) H -3 < +∞.
We now estimate the diagonal terms P( u k j • ∇ u k j ). We have

v k j • ∇ v k j = w k j • ∇ w k j = 0, while v k j • ∇ w k j =ρ 2 k j (k ⊥ j • ω j ) cos(k j • x + θ k j ) cos((k j + ω j ) • x + η k j + π 2 )(k j + ω j ) ⊥ = 1 2 ρ 2 k j (k ⊥ j • ω j ) cos((2k j + ω j ) • x + θ k j + η k j + π 2 )(k j + ω j ) ⊥ + 1 2 ρ 2 k j (k ⊥ j • ω j ) cos(ω j • x -θ k j + η k j - π 2 )(k j + ω j ) ⊥ and w k j • ∇ v k j =ρ 2 k j (k j • ω ⊥ j ) cos(k j • x + θ k j + π 2 ) cos((k j + ω j ) • x + η k j )k ⊥ j = 1 2 ρ 2 k j (k j • ω ⊥ j ) cos((2k j + ω j ) • x + θ k j + η k j + π 2 )k ⊥ j + 1 2 ρ 2 k j (k j • ω ⊥ j ) cos(ω j • x -θ k j + η k j - π 2 )k ⊥ j .
We have

P(ρ 2 k j (k ⊥ j • ω j ) cos((2k j + ω j ) • x + θ k j + η k j + π 2 )(k j + ω j ) ⊥ ) H -3 ≤ Cρ 2 k j |ω j | |k j | and P(ρ 2 k j (k j • ω ⊥ j ) cos((2k j + ω j ) • x + θ k j + η k j - π 2 )k ⊥ j ) H -3 ≤ Cρ 2 k j |ω j | |k j | .
On the other hand, we have

P(ρ 2 k j (k j • ω ⊥ j ) cos(ω j • x -θ k j + η k j + π 2 )k ⊥ j ) = 0 and P(ρ 2 k j (k ⊥ j •ω j ) cos(ω j •x-θ k j +η k j - π 2 )(ω j +k j ) ⊥ ) = ρ 2 k j (k ⊥ j •ω j ) cos(ω j •x-θ k j +η k j - π 2 )ω ⊥ j so that P(ρ 2 k j (k ⊥ j • ω j ) cos(ω j • x -θ k j + η k j - π 2 )(ω j + k j ) ⊥ ) H -3 ≤ Cρ 2 k j |k j | |ω j | .
Thus, we get

+∞ j=0 P( v k j • ∇ w k j ) H -3 < +∞, +∞ j=0 P( w k j • ∇ v k j ) H -3 < +∞,
and

P( u • ∇ u) is well defined in H -3 .
3 2D steady solutions.

We are going to prove Theorem 1 following the lines of [START_REF] Buckmaster | Nonuniqueness of weak solutions to the Navier-Stokes equation[END_REF] and [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF], i.e. applying the convex integration scheme by using intermittencies in the Fourier spectrum of the solution. In our case, however, computations will be much more simple than in the ones in [START_REF] Buckmaster | Nonuniqueness of weak solutions to the Navier-Stokes equation[END_REF] and [START_REF] Luo | Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier-Stokes Equations in High Dimensions Arch[END_REF], as we don't bother on convergence in L 2 . We shall look for a solution

u = +∞ j=0 u j = u 0 + +∞ j=1 v j + w j
where

• u 0 = ρ 0 cos(k 0 • x)k ⊥ 0 with 0 < ρ 0 < 1 and k 0 ∈ Z 2 \ {(0, 0)}, • v j = ρ j cos(k j • x)k ⊥ j and w j = ρ j cos((k j + ω j ) • x + η j )(k j + ω j ) ⊥ with ρ j > 0 and k j , ω j ∈ Z 2 \ {(0, 0)}, • for j ≥ 1, |k j | > 8|k j-1 |, |k j | > 8|ω j |, ω j • k j = 0.
k j , ω j and η j will be constructed by induction and we'll check that

+∞ j=1 ρ 2 j |k j | |ω j | < +∞,
so that u is an admissible vector field such that u ∈ H -1 (hence ∆ u ∈ H -3 ) and P( u • ∇ u) ∈ H -3 . Defining U n = n j=0 u j , we have the convergence of ∆ U n -P(

U n • ∇ U n ) to ∆ u -P( u • ∇ u) in H -3 . We write, for n ≥ 1, ∆ U n -P( U n • ∇ U n ) = V 0 + n j=1 V j + W j • V 0 = ∆ u 0 (= ∆ u 0 -P( u 0 • ∇ u 0 )) = -ρ 0 |k 0 | 2 cos(k 0 • x)k ⊥ 0 • for n ≥ 1, V n = ∆ U n -P( u n • ∇ U n-1 ) -P( U n-1 • ∇ u n ) - 1 2 (k ⊥ n • ω n )ρ 2 n P(cos((2k n + ω n ) • x + η n + π 2 )(k n + ω n ) ⊥ ) - 1 2 (k n • ω ⊥ n )ρ 2 n P(cos((2k n + ω n ) • x + η n - π 2 )k ⊥ n ) • for n ≥ 1, W n = -1 2 ρ 2 n (k ⊥ n • ω n ) cos(ω n • x + η n -π 2 )ω ⊥ n
Let us write A n for the set of frequencies involved in the expansion of V n :

V n = k∈An cos(k • x + α n,k ) v n,k = k∈An λ n,k cos(k • x + α n,k )k ⊥ , with λ n,k = v n,k •k ⊥ |k| 2 .
Using the formula

P(cos(α. • x + θ)α ⊥ • ∇(cos(β • x + η)β ⊥ ) + cos(β • x + η)β ⊥ • ∇(cos(α. • x + θ)α ⊥ )) = -P((α ⊥ • β) cos(α. • x + θ) sin(β • x + η)β ⊥ ) -P((β ⊥ • α) sin(α. • x + θ) cos(β • x + η)α ⊥ ) = - 1 2 P((cos((α + β) • x + θ + η - π 2 )((α ⊥ • β)β ⊥ + (β ⊥ • α)α ⊥ )) - 1 2 P((cos((α -β) • x + θ -η - π 2 )(-(α ⊥ • β)β ⊥ + (β ⊥ • α)α ⊥ )) = - 1 2 cos((α + β) • x + θ + η + π 2 )(β ⊥ • α) |β| 2 -| α| 2 |α + β| 2 (α + β) ⊥ - 1 2 cos((α -β) • x + θ -η + π 2 )(α ⊥ • β) |β| 2 -| α| 2 |α + β| 2 (α -β) ⊥
we see that we have more precisely 8n -1 frequencies in A n for n ≥ 1:

• k = k n with λ n,k = -ρ n |k n | 2 and η n,k = 0 • k = k n + ω n with λ n,k = -ρ n |k n + ω n | 2 and η n,k = η n • k = 2k n + ω n with λ n,k = -1 2 (k ⊥ n •ωn) 2 |2kn+ωn| 2 ρ 2 n and η n,k = η n -π 2 • for j = 0, . . . , n -1, k = k n + k j with λ n,k = 1 2 ρ n ρ j (k ⊥ j • k n ) |k j | 2 -|kn| 2 |kn+k j | 2 and η n,k = π 2 • for j = 0, . . . , n -1, k = k n -k j with λ n,k = 1 2 ρ n ρ j (k ⊥ n • k j ) |k j | 2 -| kn| 2 |kn-k j | 2 and η n,k = π 2 • for j = 1, . . . , n -1, k = k n + k j + ω j with η n,k = η j + π 2 and λ n,k = 1 2 ρ n ρ j ((k j + ω j ) ⊥ • k n ) |k j +ω j | 2 -|kn| 2 |kn+k j +ω j | 2
For k ∈ A n , we find that and the frequencies occuring in A n+1 are greater than those occuring in A n . We then write A 0 = {γ 1 }, A 1 = {γ 2 , . . . , γ 8 }, . . . , A n = {γ 4n 2 -5n+3 , . . . , γ 4n 2 +3n+1 }, . . . We write, for j ≥ 0,

V j = γp∈A j λ p cos(γ p • x + α p )γ ⊥ p = γp∈A j |λ p | cos(γ p • x + α p + p π)γ ⊥ p with p ∈ {0, 1}. Thus we have ∆ U n -P( U n • ∇ U n ) = n j=0 γp∈A j |λ p | cos(γ p • x + α p + p π)γ ⊥ p - n j=1 1 2 ρ 2 j (k ⊥ j • ω j ) cos(ω j • x + η j - π 2 )ω ⊥ j .
We know the values of ρ 0 , k 0 , hence of γ 1 = k 0 , |λ 1 | = ρ 0 |k 0 | 2 and α 0 + 0 π = π. We shall define by induction ω n , k n , ρ n and η n for n ≥ 1: we remark that γ n ∈ A j(n) for some j(n) < n (as n < 4n 2 -5n + 3). Thus, if we already know ω j , k j , ρ j and η j for 0 ≤ j ≤ n -1, we already know γ n , |λ n | and α n + n π.

The main idea is then to require that

|λ n | cos(γ n • x + α n + n π)γ ⊥ n = 1 2 ρ 2 n (k ⊥ n • ω n ) cos(ω n • x + η n - π 2 )ω ⊥ n .
We thus make the following choices:

• We take ω n = γ n .

• We take k n = N n ω ⊥ n , where the integer N n will fulfill some requirements. Our first requirement will be that N n ∈ N is large enough to grant that N n > 8 and |k n | > 8|k n-1 |.

• We then have

1 2 ρ 2 n (k ⊥ n •ω n ) cos(ω n •x+η n - π 2 )ω ⊥ n . = 1 2 ρ 2 n N n |ω n | 2 cos(ω n •x+η n + π 2 )ω ⊥ n .
Thus, we take

ρ n = 2|λ n | N n |ω n | 2 and η n = α n + n π - π 2 .
• We shall add another requirement on N n in order to grant that (if we take N n ≥ 4C 6 0 ρ -4 0 ). We have

ρ 2 n |k n | |ω n | = N n ρ 2 n = 2|λ n | |ω n | 2 ≤ 2C 3 0 ρ 0 N -1/4 j(n)

+∞ j=1 ρ 2 j 1 4nN

 21 |k j | |ω j | < +∞, Recall that 0 < ρ 0 < 1. Take N 0 = 1. We first check by induction that ρ n ≤ ρ 0 N -(≤ 1). Indeed, there is a constant C 0 such that |ω n | ≥ 1 C 0 |k j(n) | and |λ n | ≤ C 0 |k j(n) | 2 ρ j(n) sup(1, ρ 0 , . . . , ρ j(n) ) so that, by induction, |λ n | ≤ C 0 |k j(n) | 2 ρ 0

Thus,

Hence, our last requirement on N n will be that N n ≥ (8n -1) 12 .

Theorem 1 is proved.

4 A remark on the Koch-Tataru theorem.

In our construction, we have

By the Koch-Tataru theorem [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF], for ρ 0 small enough, the evolutionary problem

will have a smooth solution on (0, +∞) × T 2 such that

The steady solution u is another solution of the evolutionary problem (9), with u ∈ C([0, +∞), BM O -1 ). Of course, v = u as lim t→+∞ v(t, .) H -1 = 0.