Dietrich Kuske

IS RAMSEY'S THEOREM ω-AUTOMATIC?

Keywords: 1998 ACM Subject Classification: F.4.1 Logic in computer science, Automata, Ramsey theory. These results

We study the existence of infinite cliques in ω-automatic (hyper-)graphs. It turns out that the situation is much nicer than in general uncountable graphs, but not as nice as for automatic graphs.

More specifically, we show that every uncountable ω-automatic graph contains an uncountable co-context-free clique or anticlique, but not necessarily a context-free (let alone regular) clique or anticlique. We also show that uncountable ω-automatic ternary hypergraphs need not have uncountable cliques or anticliques at all.

Introduction

Every infinite graph has an infinite clique or an infinite anticlique -this is the paradigmatic formulation of Ramsey's theorem [START_REF] Ramsey | On a problem of formal logic[END_REF]. But this theorem is highly non-constructive since there are recursive infinite graphs whose infinite cliques and anticliques are all non-recursive (not even in Σ 0 2 , [START_REF] Jockusch | Ramsey's theorem and recursion theory[END_REF], cf. [Gas98, Thm. 4.6]). Recall that a graph is recursive if both its set of nodes and its set of edges can be decided by a Turing machine. Replacing these Turing machines by finite automata, one obtains the more restrictive notion of an automatic graph: the set of nodes is a regular set and whether a pair of nodes forms an edge can be decided by a synchronous two-tape automaton (this concept is known since the beginning of automata theory, a systematic study started with [START_REF] Khoussainov | Automatic presentations of structures[END_REF][START_REF] Blumensath | Finite presentations of infinite structures: Automata and interpretations[END_REF], see [START_REF] Rubin | Automata presenting structures: A survey of the finite string case[END_REF] for a recent overview). In this context, the situation is much more favourable: every infinite automatic graph contains an infinite regular clique or an infinite regular anticlique (cf. [START_REF] Rubin | Automata presenting structures: A survey of the finite string case[END_REF]).

Soon after Ramsey's paper from 1930, authors got interested in a quantitative analysis. For finite graphs, one can ask for the minimal number of nodes that guarantee the existence of a clique or anticlique of some prescribed size. This also makes sense in the infinite: how many nodes are necessary and sufficient to obtain a clique or anticlique of size ℵ 0 (Ramsey's theorem tells us: ℵ 0 ) or ℵ 1 (here one needs more than 2 ℵ 0 nodes [START_REF] Sierpiński | Sur un problème de la thèorie des relations[END_REF][START_REF] Erdős | A partition calculus in set theory[END_REF]).

Since automatic graphs contain at most ℵ 0 nodes, we need a more general notion for a recursion-theoretic analysis of this situation. For this, we use Blumensath & Grädel's [START_REF] Blumensath | Finite presentations of infinite structures: Automata and interpretations[END_REF] ω-automatic graphs: the names of nodes form a regular ω-language and the edge relation (on names) as well as the relation "these two names denote the same node" can be decided by a synchronous 2-tape Büchi-automaton. In this paper, we answer the question whether these ω-automatic graphs are more like automatic graphs (i.e., large cliques or anticliques with nice properties exist) or like general graphs (large cliques need not exist).

Our answer to this question is a clear "somewhere in between": We show that every ω-automatic graph of size 2 ℵ 0 contains a clique or anticlique of size 2 ℵ 0 (Theorem 3.1) -this is in contrast to the case of arbitrary graphs where such a subgraph need not exist [START_REF] Sierpiński | Sur un problème de la thèorie des relations[END_REF]. But in general, there is no regular clique or anticlique (Theorem 3.13) -this is in contrast with the case of automatic graphs where we always find a large regular clique or anticlique. Finally, we also provide an ω-automatic "ternary hypergraph" of size 2 ℵ 0 without any clique or anticlique of size ℵ 1 , let alone 2 ℵ 0 (Theorem 3.11).

For Theorem 3.1, we re-use the proof from [START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF] that was originally constructed to deal with infinity quantifiers in ω-automatic structures. The proof of Theorem 3.13 makes use of the "ultimately equal" relation. This relation was also crucial in the separation of injectively from general ω-automatic structures [START_REF] Hjorth | From automatic structures to borel structures[END_REF] as well as in the handling of infinity quantifiers in [START_REF] Kuske | First-order and counting theories of ω-automatic structures[END_REF] and [START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF]. In the ternary hypergraph from Theorem 3.11, a 3-set {x, y, z} of infinite words with x < lex y < lex z forms an undirected hyperedge iff the longest common prefix of x and y is shorter than the longest common prefix of y and z.

¿From Theorem 3.1 (i.e., the existence of large cliques or anticliques in ω-automatic graphs), we derive that any ω-automatic partial order of size 2 ℵ 0 contains an antichain of size 2 ℵ 0 or a copy of the real line.

Preliminaries

Ramsey-theory

For a set V and a natural number

k ≥ 1, let [V ] k denote the set of k-element subsets of V . A (k, ℓ)-partition is a pair G = (V, E 1 , . . . , E ℓ ) where V is a set and (E 1 , . . . , E ℓ ) is a partition of [V ] k into (possibly empty) sets. For 1 ≤ i ≤ ℓ, a set W ⊆ V is E i - homogeneous if [W ] k ⊆ E i ; it is homogeneous if it is E i -homogeneous for some 1 ≤ i ≤ ℓ. The case k = ℓ = 2 is special: any (2, 2)-partition G = (V, E 1 , E 2 )
can be considered as an (undirected loop-free) graph (V, E 1 ). Homogeneous sets in G are then complete or discrete induced subgraphs of (V, E 1 ).

Ramsey theory is concerned with the following question: Does every (k, ℓ)-partition G = (V, E 1 , . . . , E ℓ ) with |V | = κ have a homogeneous set of size λ (where κ and λ are cardinal numbers and k, ℓ ≥ 2 are natural numbers). If this is the case, one writes κ → (λ) k ℓ (a notation due to Erdős and Rado [START_REF] Erdős | A partition calculus in set theory[END_REF]). This allows to formulate Ramsey's theorem concisely:

Theorem 1.1 (Ramsey [Ram30]). If k, ℓ ≥ 2, then ℵ 0 → (ℵ 0 ) k ℓ .
In particular, every graph with ℵ 0 nodes contains a complete or discrete induced subgraph of the same size. If one wants to find homogeneous sets of size ℵ 1 , the base set has to be much larger:

Theorem 1.2 (Sierpiński [Sie33]). If k, ℓ ≥ 2, then 2 ℵ 0 → (ℵ 1 ) k ℓ and therefore in particular 2 ℵ 0 → (2 ℵ 0 ) k ℓ .
Erdős and Rado [START_REF] Erdős | A partition calculus in set theory[END_REF] proved that partitions of size properly larger than 2 ℵ 0 have homogeneous sets of size ℵ 1 . For more details on infinite Ramsey theory, see [Jec02, Chapter 9].

ω-languages

Let Γ be a finite alphabet. With Γ * we denote the set of all finite words over the alphabet Γ. The set of all nonempty finite words is Γ + . An ω-word over Γ is an infinite ω-sequence x = a 0 a 1 a 2 • • • with a i ∈ Γ, we set x[i, j) = a i a i+1 . . . a j-1 for natural numbers i ≤ j. In the same spirit, x[i, ω) denotes the ω-word a i a i+1 . . . . The set of all ω-words over Γ is denoted by Γ ω and Γ ∞ = Γ * ∪ Γ ω . For a set V ⊆ Γ + of finite words let V ω ⊆ Γ ω be the set of all ω-words of the form v 0 v 1 v 2 • • • with v i ∈ V . Two infinite words x, y ∈ Γ ω are ultimately equal, briefly x ∼ e y, if there exists i ∈ N with x[i, ω) = y[i, ω). By ≤ lex , we denote the lexicographic order on the set Σ ω (with some, implicitly assumed linear order on the letters from Σ) and ≤ pref the prefix order on Σ ∞ .

For Σ = {0, 1}, the support supp(x) ⊆ N is the set of positions of the letter 1 in the word

x ∈ Σ ω . A (nondeterministic) Büchi-automaton M is a tuple M = (Q, Γ, δ, ι, F ) where Q is a finite set of states, ι ∈ Q is the initial state, F ⊆ Q is the set of final states, and δ ⊆ Q × Γ × Q is the transition relation. If Γ = Σ n for some alphabet Σ, then we speak of an n-dimensional Büchi-automaton over Σ. A run of M on an ω-word x = a 0 a 1 a 2 • • • is an ω-word r = p 0 p 1 p 2 • • • over the set of states Q such that (p i , a i , p i+1 ) ∈ δ for all i ≥ 0.
The run r is successful if p 0 = ι and there exists a final state from F that occurs infinitely often in r. The ω-language L(M ) ⊆ Γ ω defined by M is the set of all ω-words that admit a successful run. An ω-language L ⊆ Γ ω is regular if there exists a Büchi-automaton M with L(M ) = L.

Alternatively, regular ω-languages can be represented algebraically. To this end, one defines ω-semigroups to be two-sorted algebras S = (S + , S ω ; •, * , π) where • : S + × S + → S + and * : S + × S ω → S ω are binary operations and π : (S + ) ω → S ω is an ω-ary operation such that the following hold:

• (S + , •) is a semigroup, • s * (t * u) = (s • t) * u, • s 0 • π((s i ) i≥1 ) = π((s i ) i≥0 ), • π((s 1 i • s 2 i • • • s k i i ) i≥0 ) = π((t j ) j≥0
) whenever (t j ) j≥0 = (s 1 0 , s 2 0 , . . . , s k 0 0 , s 1 1 , . . . , s k 1 1 , . . . ) . The ω-semigroup S is finite if both, S + and S ω are finite. The free ω-semigroup generated by Γ is Γ ∞ = (Γ + , Γ ω ; •, * , π) where u • v and u * x are the natural operations of prefixing a word by the finite word u, and π((u i ) i≥0 ) is the omega-word u 0 u 1 u 2 . . . . A homomorphism h : Γ ∞ → S of ω-semigroups maps finite words to elements of S + and ω-words to elements of S ω and commutes with the operations •, * , and π. The algebraic characterisation of regular ω-languages then reads as follows.

Proposition 1.3. An ω-language L ⊆ Γ ω is regular if and only if there exists a finite ω-semigroup S, a set T ⊆ S ω , and a homomorphism η : Γ ∞ → S such that L = η -1 (T ).

Hence, every Büchi-automaton is "equivalent" to a homomorphism into some finite ω-semigroup together with a distinguished set T (and vice versa).

For ω-words

x i = a 0 i a 1 i a 2 i • • • ∈ Γ ω , the convolution x 1 ⊗ x 2 ⊗ • • • ⊗ x n ∈ (Γ n ) ω is defined by (x 1 , . . . , x n ) ⊗ = (a 0 1 , . . . , x 0 n ) (a 1 1 , . . . , a 1 n ) (a 2 1 , . . . , a 2 n ) • • • . An n-ary relation R ⊆ (Γ ω ) n is called ω-automatic if the ω-language {(x 1 , . . . , x n ) ⊗ | (x 1 , . . . , x n ) ∈ R} is regular.
To describe the complexity of ω-languages, we will use language-theoretic terms. Let LANG denote the class of all languages (i.e., sets of finite words over some finite set of symbols) and ωLANG the class of all ω-languages. By REG and ωREG, we denote the regular languages and ω-languages, resp. An ω-language is context-free if it can be accepted by a pushdown-automaton with Büchi-acceptance (on states), it is co-context-free if its complement is context-free. We denote by ωCF the set of context-free ω-languages and by co-ωCF their complements. An ω-language belongs to LANG * if it is of the form 1≤i≤n U i V ω i with U i , V i ∈ LANG. Then ωREG ⊆ LANG * and ωCF ⊆ LANG * where the sets U i and V i are regular and context-free, resp [START_REF] Staiger | ω-languages[END_REF]. In between these two classes, we define the class ωerCF of eventually regular context-free ω-languages that comprises all sets of the form 1≤i≤n U i V ω i with U i ∈ LANG context-free and V i ∈ LANG regular. Alternatively, eventually regular context-free ω-languages are the finite unions of ω-languages of the form C • L where C is a context free-language and L a regular ω-language. Let co-ωerCF denote the set of complements of eventually regular context-free ω-languages.

A final, rather peculiar class of ω-languages is Λ: it is the class of ω-languages L such that (R, ≤) embeds into (L, ≤ lex ) (the name derives from the notation λ for the order type of (R, ≤)).

ω-automatic (k, ℓ)-partitions

An ω-automatic presentation of a (k, ℓ)-partition (V, E 1 , . . . , E ℓ ) is a pair (L, h) consisting of a regular ω-language L and a surjection h :

L → V such that {(x 1 , x 2 , . . . , x k ) ∈ L k | {h(x 1 ), h(x 2 ), . . . , h(x k )} ∈ E i } for 1 ≤ i ≤ k and R ≈ = {(x 1 , x 2 ) ∈ L 2 | h(x 1 ) = h(x 2 )} are ω-automatic. An ω-automatic presentation is injective if h is a bijection. A (k, ℓ)-partition is (injectively) ω-automatic if it has an (injective) ω-automatic presentation. From [BKR08],
it follows that an uncountable ω-automatic (k, ℓ)-partition has 2 ℵ 0 elements. This paper is concerned with the question whether every (injective) ω-automatic presentation (L, h) of a (k, ℓ)-partition admits a "simple" set H ⊆ L such that h(H) has λ elements and is homogeneous. More precisely, let C be a class of ω-languages, k, ℓ ≥ 2 natural numbers, and κ and λ cardinal numbers. Then we write (κ, ωA) → (λ, C) k ℓ if the following partition property holds: for every ω-automatic presentation (L, h) of a (k, ℓ)-partition G of size κ, there exists H ⊆ L in C such that h(H) is homogeneous in G and of size λ.

(κ, ωiA) → (λ, C) k ℓ is to be understood similarly where we only consider injective ω-automatic presentations.

Remark 1.4. Let G = (V, E 1 , . . . , E ℓ ) be some (k, ℓ)-partition with ω-automatic presentation (L, h). Then the partition property above requires that there is a "large" homogeneous set X ⊆ V and an ω-language H ∈ C such that h(H) = X, in particular, every element of X has at least one representative in H. Alternatively, one could require that h -1 (X) ⊆ L is an ω-language from C. In this paper, we only encounter classes C of ω-languages such that the following closure property holds: if H ∈ C and R is an ω-automatic relation, then also

R(H) = {y | ∃x ∈ H : (x, y) ∈ R} ∈ C. Since h -1 h(H) = R ≈ (H)
, all our results also hold for this alternative requirement h -1 (X) ∈ C.

This paper shows

(0) if k, ℓ ≥ 2, then (ℵ 0 , ωA) → (ℵ 0 , ωREG) k ℓ , but (2 ℵ 0 , ωA) → (ℵ 0 , ωREG) k ℓ , see Theo- rem 2.1. (1) if ℓ ≥ 2, then (2 ℵ 0 , ωA) → (2 ℵ 0 , co-ωerCF) 2 ℓ , see Theorem 3.1. (2) if k ≥ 3, ℓ ≥ 2, and λ > ℵ 0 , then (2 ℵ 0 , ωiA) → (λ, ωLANG) k ℓ , see Theorem 3.11. (3) if k, ℓ ≥ 2 and λ > ℵ 0 , then (2 ℵ 0 , ωiA) → (λ, ωCF) k
ℓ , see Theorem 3.13. Here, the first part of (0) is a strengthening of Ramsey's theorem since the infinite homogeneous set is regular. The second part might look surprising since larger (k, ℓ)-partitions should have larger homogeneous sets -but not necessarily regular ones! In contrast to Sierpiński's result, (1) shows that ω-automatic (2, ℓ)-partitions have a larger degree of homogeneity than arbitrary (2, ℓ)-partitions. Even more, the complexity of the homogeneous set can be bound in language-theoretic terms (there is always a homogeneous set that is the complement of an eventually regular context-free ω-language). Statement (2) is an analogue of Sierpiński's Theorem 1.2 showing that (injective) ω-automatic (k, ℓ)-partitions are as in-homogeneous as arbitrary (k, ℓ)-partitions provided k ≥ 3. The complexity bound from (1) is shown to be optimal by (3) proving that one cannot always find context-free homogeneous sets. Hence, despite the existence of large homogeneous sets for k = 2, for some ω-automatic presentations, they are bound to have a certain (low) level of complexity that is higher than the regular ω-languages.

Countably infinite homogeneous sets

Let k, ℓ ≥ 2 be arbitrary. Then, from Ramsey's theorem, we obtain immediately (ℵ 0 , ωA) → (ℵ 0 , ωLANG) k ℓ and (2 ℵ 0 , ωA) → (ℵ 0 , ωLANG) k ℓ , i.e., all infinite ω-automatic (k, ℓ)-partitions have homogeneous sets of size ℵ 0 . In this section, we ask whether such homogeneous sets can always be chosen regular:

Theorem 2.1. Let k, ℓ ≥ 2. Then (a) (ℵ 0 , ωA) → (ℵ 0 , ωREG) k ℓ . (b) (2 ℵ 0 , ωiA) → (ℵ 0 , ωREG) k ℓ . (c) (2 ℵ 0 , ωA) → (ℵ 0 , LANG * ) k
ℓ , and therefore in particular [START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF], there exists L ′ ⊆ L regular such that (L ′ , h) is an injective ω-automatic presentation of G. From a Büchi-automaton for L ′ , one can compute a finite automaton accepting some language K such that (K, h ′ ) is an injective automatic presentation of G [START_REF] Blumensath | Automatic structures[END_REF]. Hence, by [START_REF] Rubin | Automata presenting structures: A survey of the finite string case[END_REF], there exists a regular set H ′ ⊆ K such that h ′ (H ′ ) is homogeneous in G and countably infinite. From this set, one obtains a regular ω-language H ⊆ L ′ ⊆ L with h(H) = h ′ (H ′ ), i.e., h(H) is a homogeneous set of size ℵ 0 . This proves (a).

(2 ℵ 0 , ωA) → (ℵ 0 , ωCF) k ℓ and (2 ℵ 0 , ωA) → (ℵ 0 , ωREG) k ℓ . Proof. Let (L, h) be an ω-automatic presentation of some (k, ℓ)-partition G = (V, E 1 , . . . , E ℓ ) with |V | = ℵ 0 . By
To prove (b), let (L, h) be an injective ω-automatic presentation of some (k, ℓ)-partition G = (V, E 1 , . . . , E ℓ ) of size 2 ℵ 0 . Then there exists a regular ω-language L ′ ⊆ L with

|L ′ | = ℵ 0 . Consider the sub-partition G ′ = (h(L ′ ), E ′ 1 , . . . , E ′ ℓ ) with E ′ i = E i ∩ [h(L ′ )] k
. This (k, ℓ)-partition has as ω-automatic presentation the pair (L ′ , h). Then, by (a), there exists L ′′ ⊆ L ′ regular and infinite such that h(L ′′ ) is homogeneous in G ′ and therefore in G. Since

h is injective, this implies |h(L ′ )| = |L ′ | = ℵ 0 .
Finally, we show (c) by a counterexample. Let L = {0, 1} ω , V = L/∼ e , and h : L → V the canonical mapping. Furthermore, set

E 1 = [L] k . Then G = (V, E 1 , ∅, . . . , ∅) is a (k, ℓ)- partition with ω-automatic presentation (L, h).
Now let H = 1≤i≤n U i V ω i ⊆ L for some non-empty languages U i , V i ⊆ {0, 1} + such that h(H) is homogeneous and infinite.

If

|V ω i | = 1, then U i V ω i /∼ e is finite. Since h(H) is infinite, there exists 1 ≤ i ≤ n with |V ω i | > 1 implying the existence of words v, w ∈ V + i such that |v| = |w| and v = w. For u ∈ U i , the set u{v, w} ω ⊆ H has 2 ℵ 0 equivalence classes wrt. ∼ e . Hence |h(H)| = 2 ℵ 0 .

Uncountable homogeneous sets

A Ramsey theorem for ω-automatic (2, ℓ)-partitions

The main result of this section is the following theorem that follows immediately from Prop. 3.7 and Lemma 3.5. Theorem 3.1. For all ℓ ≥ 2, we have

(2 ℵ 0 , ωA) → (2 ℵ 0 , co-ωerCF ∩ Λ) 2 ℓ .
3.1.1. The proof. The proof of this theorem will construct a language from co-ωerCF that describes a homogeneous set. This language is closely related to the following language

N = 1{0, 1} ω ∩ n≥0 {0, 1} n (0{0, 1} n 00 ∪ 10 n {01, 10}){0, 1} ω ,
i.e., an ω-word x belongs to N iff it starts with 1 and, for every n ≥ 0, we have x[n, 2n+3) ∈ 0{0, 1} * 00 ∪ 10 * 01 ∪ 10 * 10. We first list some useful properties of this language N : Lemma 3.2. The ω-language N is contained in (1 + 0 + ) ω , belongs to co-ωerCF ∩ Λ, and supp(x) ∩ supp(y) is finite for any x, y ∈ N distinct.

Proof. Let b i ∈ {0, 1} for all i ≥ 0 and suppose the word x = b 0 b 1 . . . belongs to N . Then b 0 = 1, hence the word x contains at least one occurrence of 1. Note that, whenever b n = 1, then {b 2n+1 , b 2n+2 } = {0, 1}, hence x contains infinitely many occurrences of 1 and therefore infinitely many occurrences of 0, i.e., N ⊆ (1

+ 0 + ) ω . Note that the complement of N equals 0{0, 1} ω ∪ n≥0 {0, 1} n (0{0, 1} n {01, 10, 11} ∪ 1{0, 1} n {00, 11}){0, 1} ω =   0 ∪ n≥0 {0, 1} n (0{0, 1} n {01, 10, 11} ∪ 1{0, 1} n {00, 11})   {0, 1} ω .
Since the expression in square brackets denotes a context-free language, {0, 1} ω \ N is an eventually regular context-free ω-language.

By Lemma 3.3, there are words (x α ) α<2 ℵ 0 in L such that [x α ] ∼e ∩ [x β ] ≈ = ∅ for all α < β.

In the following, we only need the words x 0 , x 1 , . . . , x C with C = |S| • |T |. Then [BKR08, Sections 3.1-3.3]1 first constructs two ω-words y 1 and y 2 and an infinite sequence 1 ≤ g 1 < g 2 < . . . of natural numbers such that in particular y 1 [g 1 , g 2 ) < lex y 2 [g 1 , g 2 ). Set u = y 2 [0, g 1 ), v = y 1 [g 1 , g 2 ), and w = y 2 [g 1 , g 2 ). In the following, let h : {0, 1} ∞ → Σ ∞ be the homomorphism from Def. 3.4 and set χ(x) = uh(x) for x ∈ {0, 1} * . As in [START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF], one can then show that all the words from H u,v,w belong to the ω-language L. In the following, set x •• = χ((01) ω ) and x •• = χ((10) ω ). Then obvious alterations in the proofs by Bárány et al. show:

(1) [BKR08, Lemma 3.4] 2 If x, y ∈ {0, 1} ω with supp(x) \ supp(y) and supp(y) \ supp(x) infinite, then

{δ(χ(x), χ(y)), δ(χ(y), χ(x))} = {δ(x •• , x •• ), δ(x •• , x •• )} . (2) [BKR08, Lemma 3.5] x •• ≈ x •• . There exists 0 ≤ i ≤ ℓ with {x •• , x •• } ∈ E i . Then (2) implies i > 0.
Let x, y ∈ N be distinct. Then supp(x) ∩ supp(y) is finite by Lemma 3.2. Since, on the other hand, supp(x) and supp(y) are both infinite, the two differences supp(x) \ supp(y) and supp(y)\supp(x) are infinite. Hence we obtain δ(χ(x), χ(y

)) ∈ {δ(x •• , x •• ), δ(x •• , x •• )} from (1). Hence (b) implies {χ(x), χ(y)} ∈ E i , i.e., H u,v,w is E i -homogeneous.
Since H u,v,w ∈ co-ωerCF ∩ Λ by Lemma 3.5, the result follows.

Proposition 3.7. Let G = (V, E ′ 1 , . . . , E ′ ℓ ) be some (2, ℓ)-partition with automatic presentation (L, h). Then there exist u, v, w ∈ Σ + with v and w distinct of equal length such that h(H u,v,w ) is homogeneous and of size 2 ℵ 0 . Proof. To apply Prop. 3.6, consider the following (2, 1 + ℓ)-partition G = (L, E 0 , . . . , E ℓ ):

• The underlying set is the ω-language L, • E 0 comprises all sets {x, y} with h(x) = h(y) and x = y, and • E i (for 1 ≤ i ≤ ℓ) comprises all sets {x, y} with {h(x), h(y)} ∈ E ′ i . Then (L, id) is an injective ω-automatic presentation of the (2, 1 + ℓ)-partition G. By Prop. 3.6, there exists 1 ≤ i ≤ ℓ and words u, v and w such that H u,v,w is i-homogeneous in G. Since (E 0 , . . . , E ℓ ) is a partition of [L] 2 , we have {x, y} / ∈ E 0 (and therefore h(x) = h(y)) for all x, y ∈ H u,v,w distinct. Hence h is injective on

H u,v,w . Furthermore [H u,v,w ] 2 ⊆ E i implies [h(H u,v,w )] 2 ⊆ E ′ i . Hence h(H u,v,w
) is an i-homogeneous set in G ′ of size 2 ℵ 0 . This finishes the proof of Theorem 3.1.

3.1.2. Effectiveness. Note that the proof above is non-constructive at several points: Lemma 3.3 is not constructive and the proof proper uses Ramsey's theorem [BKR08, page 390] and makes a Ramseyan factorisation coarser [BKR08, begin of section 3.2]. We now show that nevertheless the words u, v, and w can be computed. By Prop. 3.7, it suffices to decide for a given triple (u, v, w) whether h(H u,v,w ) is i-homogeneous for some fixed 1 ≤ i ≤ ℓ.

To be more precise, let (V, E 1 , . . . , E ℓ ) be some (2, ℓ)-partition with ω-automatic presentation (L, h). Furthermore, let u, v, w ∈ Σ + with v = w of the same length and write H

The authors of [START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF] only require one of the two differences to be infinite, but the proof uses that they both are infinite.

for H u,v,w . We have to decide whether H ⊆ L and

H ⊗ H ⊆ L i ∪ L = . Note that H ⊆ L iff L ∩ Σ ω \ H = ∅. But Σ ω \ H is context-free,
so the intersection is context-free. Hence the emptiness of the intersection can be decided.

Towards a decision of the second requirement, note that

(Σ × Σ) ω \ (H ⊗ H) = (Σ ω \ H ⊗ Σ ω ) ∪ (Σ ω ∪ Σ ω \ H)
is the union of two context-free ω-languages and therefore context-free itself. Since

L i ∪ L = is regular, the intersection (L i ∪ L = ) ∩ (Σ × Σ) ω \ (H ⊗ H) is context-free implying that its emptiness is decidable. But this emptiness is equivalent to H ⊗ H ⊆ L 1 ∪ L = .
3.1.3. ω-automatic partial orders. ¿From Theorem 3.1, we now derive a necessary condition for a partial order of size 2 ℵ 0 to be ω-automatic. A partial order (V, ⊑) is ω-automatic iff there exists a regular ω-language L and a surjection h : L → V such that the relations

R = = {(x, y) ∈ L 2 | h(x) = h(y)} and R ⊑ = {(x, y) ∈ L 2 | h(x) ⊑ h(y)} are ω-automatic. Corollary 3.8 ([BKR08] 3 ). If (V, ⊑) is an ω-automatic partial order with |V | ≥ ℵ 1 , then (R, ≤)
or an antichain of size 2 ℵ 0 embeds into (V, ⊑).

Proof. Let (V, ⊑) be a partial order, L ⊆ Σ ω a regular ω-language and h : L → V a surjection such that R = and R ⊑ are ω-automatic. Define an injective ω-automatic

(2, 4)- partition G = (L, E 0 , E 1 , E 2 , E 3 ): • E 0 comprises all pairs {x, y} ∈ [L] 2 with h(x) = h(y), • E 1 comprises all pairs {x, y} ∈ [L] 2 with h(x) ⊏ h(y) and x < lex y,
• E 2 comprises all pairs {x, y} ∈ [L] 2 with h(x) ⊐ h(y) and x < lex y, and

• E 3 = [L] 2 \ (E 0 ∪ E 1 ∪ E 2 ) comprises all pairs {x, y} ∈ [L]
2 such that h(x) and h(y) are incomparable. From |L| ≥ |V | > ℵ 0 , we obtain |L| = 2 ℵ 0 . Hence, by Prop. 3.6, there exists H ⊆ L 1-, 2or 3-homogeneous with (R, ≤) ֒→ (H, ≤ lex ). Since [H] 2 ⊆ E 1 ∪ E 2 ∪ E 3 and since G is a partition of L, the mapping h acts injectively on H.

If [H] 2 ⊆ E 1 (the case [H] 2 ⊆ E 2 is symmetrical) then (R, ≤) ֒→ (H, ≤ lex ) ∼ = (h(H), ⊑). If [H] 2 ⊆ E 3 , then h(H) is an antichain of size 2 ℵ 0 .
A linear order (L, ⊑) is scattered if (Q, ≤) cannot be embedded into (L, ⊑). Automatic partial orders are defined similarly to ω-automatic partial orders with the help of finite automata instead of Büchi-automata.

Corollary 3.9 ([BKR08]3 ). Any scattered ω-automatic linear order (V, ⊑) is countable. Hence,

• a scattered linear order is ω-automatic if and only if it is automatic, and

• an ordinal α is ω-automatic if and only if α < ω ω .
Proof. If (V, ⊑) is not countable, then it embeds (R, ≤) by the previous corollary and therefore in particular (Q, ≤). The remaining two claims follow immediately from [START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF] ("countable ω-automatic structures are automatic") and [START_REF] Ch | Automaticité des ordinaux et des graphes homogènes[END_REF] ("an ordinal is automatic iff it is properly smaller than ω ω "), resp.

Contrast Theorem 3.1 with Theorem 1.2: any uncountable ω-automatic (k, ℓ)-partition contains an uncountable homogeneous set of size 2 ℵ 0 . But we were able to prove this for k = 2, only. One would also wish the homogeneous set to be regular and not just from co-ωerCF. We now prove that these two shortcomings are unavoidable: Theorem 3.1 does not hold for k = 3 nor is there always an ω-regular homogeneous set. These negative results hold even for injective presentations. We first concentrate on the question whether some form of Theorem 3.1 holds for k ≥ 3. The following lemma gives the central counterexample for k = 3 and ℓ = 2, the below theorem then derives the general result.

Lemma 3.10. (2 ℵ 0 , ωiA) → (ℵ 1 , ωLANG) 3 2 . Proof. Let Σ = {0, 1}, V = L = {0, 1} ω . Furthermore, for H ⊆ L, we write H ∈ Σ ∞ for the longest common prefix of all ω-words in H, {x, y} is also written x ∧ y. Then let E 1 consist of all 3-sets {x, y, z} ∈ [L] 3 with x < lex y < lex z and x ∧ y < pref y ∧ z; E 2 is the complement of E 1 . This finishes the construction of the (3, 2)-partition (V, E 1 , E 2 ) of size 2 ℵ 0 with injective ω-automatic presentation (L, id).

Note that 1 * 0 ω is a countable E 1 -homogeneous set and that 0 * 1 ω is a countable E 2homogeneous set. But there is no uncountable homogeneous set: First suppose H ⊆ L is infinite and x ∧ y < pref y ∧ z for all x < lex y < lex z from H. Let u ∈ Σ * such that H ∩ u0Σ ω and H ∩ u1Σ ω are both nonempty and let x, y ∈ H ∩ u0Σ ω with x ≤ lex y and z ∈ H ∩ u1Σ ω . Then x ∧ y > pref u = y ∧ z and therefore x = y (for otherwise, we would have x < lex y < lex z in H with x ∧ y > pref y ∧ z). Hence we showed |H ∩ u0Σ ω | = 1. Let u 0 = H and H 1 = H ∩ u 0 1Σ ω . Since H ∩ u 0 0Σ ω is finite, the set H 1 is infinite. We proceed by induction:

u n = H n and H n+1 = H n ∩ u n 1Σ ω satisfying |H n ∩ u n 0Σ ω | = 1. Then u 0 < pref u 0 1 ≤ pref u 1 < pref u 1 1 ≤ pref u 2 • • • with H = n≥0 (H ∩ u n 0Σ ω ) ∪ n≥0 (H ∩ u n 1Σ ω ) .
Then any of the sets H ∩ u n 0Σ ω = H n ∩ u n 0Σ ω and (H ∩ u n 1Σ ω ) is a singleton, proving that H is countable. Thus, there cannot be an uncountable E 1 -homogeneous set.

So let H ⊆ L be infinite with x ∧ y ≥ pref y ∧ z for all x < lex y < lex z. Since we have only two letters, we get x ∧ y > pref y ∧ z for all x < lex y < lex z which allows to argue symmetrically to the above. Thus, indeed, there is no uncountable homogeneous set in L.

Theorem 3.11. For all k ≥ 3, ℓ ≥ 2, and λ > ℵ 0 , we have (2 ℵ 0 , ωiA) → (λ, ωLANG) k ℓ . Proof. Let G be the (3, 2)-partition from Lemma 3.10 that does not have homogeneous sets of size λ and let (L, id) be an injective ω-automatic presentation of G

= (V, E 1 , E 2 ) (in particular, V = L). For a set X ∈ [L] k , let X 1 < lex X 2 < lex X 3 be the three lexicographically least elements of X. Then set G ′ = (V, E ′ 1 , E ′ 2 , . . . , E ′ ℓ ) with E ′ 1 = {X ∈ [V ] k | {X 1 , X 2 , X 3 } ∈ E 1 }, E ′ 2 = {X ∈ [V ] k | {X 1 , X 2 , X 3 } ∈ E 2 }, and
E ′ i = ∅ for 3 ≤ i ≤ ℓ .
Then (L, id) is an injective ω-automatic presentation of G ′ . Now suppose H ′ ⊆ L is homogeneous in G ′ and of size λ. Then there exists H ⊆ H ′ of size λ such that for any words x 1 < lex x 2 < lex x 3 from H, there exists X ⊆ H ′ with X i = x i for 1 ≤ i ≤ 3 (if necessary, throw away some lexicographically largest elements of H ′ ). Hence H is homogeneous in G, contradicting Lemma 3.10.

Complexity of homogeneous sets in ω-automatic (2, ℓ)-partitions

Having shown that k = 2 is a central assumption in Theorem 3.1, we now turn to the question whether homogeneous sets of lower complexity can be found.

Construction. Let V = L denote the regular ω-language (1 + 0 + ) ω . Furthermore, E 1 ⊆ [L] 2 comprises all 2-sets {x, y} ⊆ L such that supp(x) ∩ supp(y) is finite or x ∼ e y. The set E 2 is the complement of E 1 in [L] 2 . This completes the construction of the (2, 2)-partition G = (L, E 1 , E 2 ). Note that (L, id L ) is an injective ω-automatic presentation of G.
By Theorem 3.1, G has an E 1 -or an E 2 -homogeneous set of size 2 ℵ 0 . We convince ourselves that G has large homogeneous sets of both types. By Lemma 3.2, there is an ωlanguage N ⊆ (1 + 0 + ) ω of size 2 ℵ 0 such that the supports of any two words from N have finite intersection. Hence [N ] 2 ⊆ E 1 and N has size 2 ℵ 0 . But there is also an E 2 -homogeneous set L 2 of size 2 ℵ 0 : Note that the words from N are mutually non-∼ e -equivalent and let L 2 denote the set of all words 1a 1 1a 2 1a 3 . . . for a 1 a 2 a 3 • • • ∈ N . Then for any x, y ∈ L 2 distinct, we have 2N ⊆ supp(x) ∩ supp(y) and x ∼ e y, i.e., {x, y} ∈ E 2 . Lemma 3.12. Let H ∈ LANG * have size λ > ℵ 0 . Then H is not homogeneous in G.

Proof. By definition of LANG * , there are languages U i , V i ∈ LANG with H = 1≤i≤n U i V ω i . Since H is infinite, there are 1 ≤ i ≤ n and x, y ∈ U i V ω i distinct with x ∼ e y and therefore {x, y} ∈ E 1 .

Since |H| > ℵ 0 , there is 1 ≤ i ≤ n with |U i V ω i | > ℵ 0 ; we set U = U i and V = V i . From |U | ≤ ℵ 0 , we obtain |V ω | > ℵ 0 . Hence there are v 1 , v 2 ∈ V + distinct with |v 1 | = |v 2 |. Since uv ω 1 ∈ H and each element of H contains infinitely many occurrences of 1, the word v 1 belongs to {0, 1} * 10 * . Let u ∈ U be arbitrary (such a word exists since U V ω = ∅) and consider the ω-words x ′ = u(v 1 v 2 ) ω and y ′ = u(v 1 v 1 ) ω from U V ω ⊆ H. Then x ′ ∼ e y ′ since v 1 = v 2 and |v 1 | = |v 2 |. At the same time, supp(x ′ ) ∩ supp(y ′ ) is infinite since v 1 contains an occurrence of 1. Hence {x ′ , y ′ } ∈ E 2 .

Thus, we found ω-words x, y, x ′ , y ′ ∈ H with {x, y} ∈ E 1 and {x ′ , y ′ } / ∈ E 1 proving that H is not homogeneous. Thus, we found a (2, 2)-partition G = (V, E 1 , E 2 ) with 2 ℵ 0 elements and an injective ω-automatic presentation (L, h) such that

(1) G has sets L 1 and L 2 in co-ωerCF of size 2 ℵ 0 with [L i ] 2 ⊆ E i for 1 ≤ i ≤ 2.

(2) There is no ω-language H ∈ LANG * with H ⊆ L such that h(H) is homogeneous of size 2 ℵ 0 . Since all context-free ω-languages belong to LANG * , the following theorem follows the same way that Lemma 3.10 implied Theorem 3.11. Theorem 3.13. For all k, ℓ ≥ 2 and λ > ℵ 0 , we have (2 ℵ 0 , ωiA) → (λ, ωCF) k ℓ and (2 ℵ 0 , ωiA) → (λ, ωREG) k ℓ .

This result can be understood as another Sierpiński theorem for ω-automatic (k, ℓ)partitions. This time, it holds for all k ≥ 2 (not only for k ≥ 3 as Theorem 3.11). The price to be paid for this is the restriction of homogeneous sets to "simple" ones. In particular the non-existence f regular homogeneous sets provides a Sierpiński theorem in the spirit of automatic structures.

Open questions

Our positive result Theorem 3.1 guarantees the existence of some clique or anticlique of size 2 ℵ 0 (and such a clique or anticlique can even be constructed). But the following situation is conceivable: the ω-automatic graph contains large cliques without containing large cliques that can be described by a language from co-ωerCF. In particular, it is not clear whether the existence of a large clique is decidable.

A related question concerns Ramsey quantifiers. Rubin [START_REF] Rubin | Automata presenting structures: A survey of the finite string case[END_REF] has shown that the set of nodes of an automatic graph whose neighbors contain an infinite anticlique is regular (his result is much more general, but this formulation suffices for our purpose). It is not clear whether this also holds for ω-automatic graphs. A positive answer to this second question (assuming that it is effective) would entail an affirmative answer to the decidability question above.

  3.2. A Sierpiński theorem for ω-automatic (k, ℓ)-partitions with k ≥ 3

The authors of[START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF] require [xi]∼ e ∩ [xj]≈ = ∅ for all 0 ≤ i, j ≤ C distinct, but they use it only for i < j. Hence we can apply their result here.

As pointed out by two referees, the paragraph before Sect.

4.1 in[START_REF] Bárány | Cardinality and counting quantifiers on omega-automatic structures[END_REF] already hints at this result, although in a rather implicit way.

This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/3.0/.

Note that a word 10 n 0 10 n 1 10 n 2 . . . belongs to N iff, for all k ≥ 0, we have 0 ≤ n k -|10 n 0 10 n 1 . . . 10 n k-1 | ≤ 1. Hence, when building a word from N , we have two choices for any n k , say n 0 k and n 1 k with n 0 k < n 1 k . But then a 0 a 1 a 2 . . . → 10 n a 0 0 10 n a 1 1 10 n a 2 2 . . . defines an order embedding ({0, 1} ω , ≤ lex ) ֒→ (N, ≤ lex ). Since (R, ≤) ֒→ ({0, 1} ω , ≤ lex ), we get N ∈ Λ. Now let x, y ∈ N with supp(x) ∩ supp(y) infinite. Then there are arbitrarily long finite words u and v of equal length such that u1 and v1 are prefixes of x and y, resp. Since u1 is a prefix of x ∈ N , it is of the form u1 = u ′ 10 |u ′ | 1 (if |u| is even) or u1 = u ′ 10 |u ′ | 01 (if |u| is odd) and analogously for v. Inductively, one obtains u ′ = v ′ and therefore u = v. Since u and v are arbitrarily long, we showed x = y. Lemma 3.3. Let ∼ and ≈ be two equivalence relations on some set L such that any equivalence class [x] ∼ of ∼ is countable and ≈ has 2 ℵ 0 equivalence classes. Then there are elements

Proof. We construct the sequence (x α ) α<2 ℵ 0 by ordinal induction. So assume we have elements (x α ) α<κ for some ordinal κ < 2

Definition 3.4. Let u, v, and w be nonempty words with |v| = |w| and v = w. Define an ω-semigroup homomorphism h : {0, 1} ∞ → Σ ∞ by h(0) = v and h(1) = w and set

where N is the set from Lemma 3.2. Lemma 3.5. Let u, v, and w be as in the previous definition. Then H u,v,w ∈ co-ωerCF∩Λ.

Proof. Assume v < lex w. Then the mapping χ : {0, 1} ω → Σ ω : x → uh(x) (where h is the homomorphism from the above definition) embeds (N, ≤ lex ) (and hence (R, ≤)) into (H u,v,w , ≤ lex ). If w < lex v, then (R, ≤) ∼ = (R, ≥) ֒→ (N, ≥ lex ) ֒→ (H α,β,γ , ≤ lex ). This proves that H u,v,w belongs to Λ.

Since v = w, the mapping χ is injective. Hence

Since χ can be realized by a generalized sequential machine with Büchi-acceptance, χ({0, 1} ω ) is regular and χ({0, 1} ω \N ) (as the image of an eventually regular context-free ω-language) is eventually regular context-free. Hence Σ ω \ H u,v,w is eventually regular context-free.

Proposition 3.6. Let G = (L, E 0 , E 1 , . . . , E ℓ ) be some (2, 1 + ℓ)-partition with injective ω-automatic presentation (L, id) such that {(x, y) | {x, y} ∈ E 0 } ∪ {(x, x) | x ∈ L} is an equivalence relation on L (denoted ≈) with 2 ℵ 0 equivalence classes. Then there exist nonempty words u, v, and w with v and w distinct, but of the same length, such that H u,v,w is i-homogeneous for some 1 ≤ i ≤ ℓ.

Proof. There are finite ω-semigroups S and T and homomorphisms γ : Σ ∞ → S and δ : (Σ × Σ) ∞ → T such that (a) x ∈ L, y ∈ Σ ω , and γ(x) = γ(y) imply y ∈ L and (b) x, x ′ , y, y ′ ∈ L, {h(x), h(x ′ )} ∈ E i , and δ(x, x ′ ) = δ(y, y ′ ) imply {h(y), h(y ′ )} ∈ E i (for all 0 ≤ i ≤ ℓ).


Catherine Choquet

Laurent Chupin

Marguerite Gisclon

Convergence to the Reynolds approximation with a double effect of roughness

Keywords: Boundary conditions, upscaling, two-scale convergence, micro-fluidic, Reynolds and Stokes equations, rough boundaries, thin films. AMS Subject Classification -35Q30, 76A20, 76D08, 78M35, 78M40

We prove that the lubrication approximation is perturbed by a non-regular roughness of the boundary. We show how the flow may be accelerated using adequate rugosity profiles on the bottom. We explicit the possible effects of some abrupt changes in the profile. The limit system is mathematically justified through a variant of the notion of two-scale convergence. Finally, we present some numerical results, illustrating the limit system in the three-dimensional case.

Introduction

We study in this paper the effect of different very small domain irregularities on a thin film flow governed by the Stokes equations. There exist already some references on the subject. Roughly speaking, they may be ordered into three categories. The first one is devoted to the study of the roughness effects on the flow in a channel: the height of the channel, denoted h and depending on the horizontal component x, is fixed and the effect of small scales of the boundary is studied. In such studies, the height of the channel is written as (see [START_REF] Achdou | Effective boundary conditions for laminar flows over periodic rough boundaries[END_REF][START_REF] Jäger | Couette flows over a rough boundary and drag reduction[END_REF])

h(x) = h 2 x, x ε .
The second category is concerned with the specific study of the thin film assumption, that is when the height of channel is assumed to be small, of the form (see [START_REF] Dridi | Comportement asymptotique des équations de Navier-Stokes dans des domaines applatis[END_REF][START_REF] Benhaboucha | Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary[END_REF][START_REF] Dyson | Hydrodynamic lubrication of rough surface -a review work[END_REF][START_REF] Elrod | A review of theories for the fluid dynamic effects of roughness on laminar lubricating films[END_REF])

h(x) = εh 1 (x).
The third category combines the mechanical point of view of the latter lubrication studies with the analysis of the roughness effects. Various limit models, in special regimes, are obtained depending on the ratio between the size of the rugosities and the mean height of the domain. In [START_REF] Bayada | New models in the theory of the hydrodynamic lubrication of rough surfaces[END_REF][START_REF] Bayada | Homogenization of the Stokes system in a thin film flow with rapidly varying thickness[END_REF][START_REF] Dyson | Hydrodynamic lubrication of rough surface -a review work[END_REF][START_REF] Elrod | A review of theories for the fluid dynamic effects of roughness on laminar lubricating films[END_REF], the ratio is assumed to be of order one, namely

h(x) = εh 2 x, x ε
and an asymptotic analysis is performed using an homogenization process. More recently, in [START_REF] Benhaboucha | Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary[END_REF], the authors study the case where the narrow gap is smaller than the roughness, namely

h(x) = ε h 2 x, x ε α .
In all previous works, the ration between the two scales, modeled by the parameter α is such that α ≤ 1. Moreover, the asymptotic model obtained in the case α ≤ 1 is always the classical Reynolds equation. In [START_REF] Bresch | Roughness-Induced Effect at Main order on the Reynolds Approximation[END_REF], the authors consider the particular case which does not enter in the previous framework: h(x) = ε h 2 x, x ε 2 . They mathematically justisfy (through a variant of the notion of two-scale convergence) that an extra term modifies the standard Reynolds equation.

There exists now a huge number of available data for rough surfaces, due to the increasing efficiency of the measurements (optic or laser technics, see e.g. [START_REF] Bushan | Atomic force microscopy of magnetic rigid disks disks and sliders and its applications to tribology[END_REF]). On the one hand, the existence of multiple scales of rough oscillations is for instance emphasized in [START_REF] He | The fractal character of processed metal surfaces[END_REF][START_REF] Zhou | Fractal geometry model for wear prediction[END_REF]. But no analytical solution is available for complex geometries. These data are also often too cumbersome for numerical simulation. This motivates an approach using asymptotic analysis. On the other hand, the tribological literature does not provide any categoric answer for the choice of characteristic roughness parameters [START_REF] Koura | The effects of surface parameters on friction[END_REF][START_REF] Wieleba | The statistical correlation of the coefficient of friction and wear rate of PTFE composites with steel counterface roughness and hardness[END_REF]. The aim of the present paper is thus to explore the effects due to abrupt changes in the rugosity profile. A direct application of this kind of study is the production of nano-scale electronic components which are nowadays formed by self assembly. In such a process, the solvent and the monomers are confined in a thin channel. The block copolymers create abrupt changes of the geometry (see for instance [START_REF] Fitzgerald | A Study on the Combined Effects of Solvent Evaporation and Polymer Flow upon Block Copolymer Self-Assembly and Alignement on Topographic Patterns[END_REF]).

The case of a change of speed of order one could be easily deduced from our previous work [START_REF] Bresch | Roughness-Induced Effect at Main order on the Reynolds Approximation[END_REF]. We thus focus on more abrupt changes in the rough profile. In particular, we aim to compare the speed of the change, assumed of order ε α , α > 0, and those due to the characteristic speed of the rough oscillations, assumed of order ε 2 , whereas ε corresponds to the caracteristic height of the channel. We restrict ourself to the case where α ≤ 2 to ensure that our limit model respects the structure of the classical Reynolds approximation [START_REF] Reynolds | On the theory of lubrication and its application to mr beauchamp tower's experiment, including an experimental determination of the viscosity of olive oil[END_REF].

We thus consider a model profile of the form

h(x) = εh ε = ε h 1 (x) + ε 1 -ψ x ε α h 2 x ε 2 , 0 ≤ α ≤ 2.
Function ψ is introduced to mimic the change in the profile (see the Figure 1).

O(ε 2 ) O(ε 2 ) O(ε) O(1) O(1) O(ε α )
Figure 1: Domain Ω ε and the different scales

The domain occupied by the fluid is then:

Ω ε = {(x, z) ∈ R d-1 × R, x ∈ ω, 0 < z < h(x)}
where ω is a domain of R d-1 . We choose d = 2 or d = 3 for the current applications. The change in the profile occurs in a subdomain w ε T of ω. The domain ω is thus decomposed in ω = ω ε -∪ w ε T ∪ ω ε + . We assume that the function ψ belongs to L ∞ (ω) and is such that ψ

(x) = 0 if x ∈ ω ε -and ψ(x) = 1 if x ∈ ω ε + .
The positive function h 1 is the main order part of the roughness, while function h 2 describes the oscillating part. For sake of simplicity, we assume that the fonction h ε 2 defined by

h ε 2 (x) = h 2 (x/ε 2 ) is an ε 2 -periodic function.
Actually, it would be sufficient for our needs to assume that h 2 is an admissible test function for the two-scale convergence (see Proposition 2.1 below). The first step of our method consists in introducing a new vertical variable. This change of variable motivates our last technical assumptions for the rough profile. We assume that ψ ∈ C 2 (R d-1 ),

h 1 ∈ H 2 (R d-1 ) and h ε 2 ∈ C 1 (T d-1 ).
Let us present our results. We prove that the change in the rough profile gives the same type of correction at the main order than the oscillating part h 2 , but only under an additional assumption linking the behavior of the roughness jumps with the one of the oscillating part of the profile. Without this additional assumption, the Reynolds approximation contains a non-explicit contribution (a kind of "strange term coming from nowhere", see [START_REF] Cioranescu | Un terme étrange venu d'ailleurs. [A strange term brought from somewhere else[END_REF]).

Theorem 0.1 Let ω be the horizontal projection of the domain of study. (i) Let us denote by

ψ 0 ∈ L 2 (ω; H 1 (T d-1 )) (respectively ψ 0 ∈ L 2 (ω) if α <
2) the two-scale limit of the jump approximation ψ ǫ (x) = ψ(x/ε α ) as ε → 0 and by U b the velocity of the lower surface. The behavior of the thin flow is approximated by the following modified Reynolds equation on the pressure p of the fluid which only depend on the horizontal variable:

div x h 3 1 12 A∇ x p 0 = div x h 1 (BU b -Q 0 ) on ω,
where the functions A and B are defined on ω by

A = 12 N ψ e N ψ /2 1 0 e -N ψ t 2 /2 dt -1 - 12 N ψ (e N ψ /2 -1) 1 0 s 0 e N ψ (s 2 -t 2 )/2 dtds 1 0 e N ψ s 2 /2 ds , B = 1 N ψ (e N ψ /2 -1) 1 1 0 e N ψ s 2 /2 ds , where N ψ (x) = T d-1 |∇ X ((1 -ψ 0 (x, X))h 2 (X))| 2 dX, x ∈ ω. The "strange" function Q 0 is non explicit.
(ii) Assuming moreover one of the following assumption, (H1) ∇(ε 2 ψ ε h ε 2 ) strongly two-scale converges to ∇ X (ψ 0 h 2 ), (H2) ψ ε strongly two-scale converges to ψ 0 , we recover a completely explicit Reynolds-type approximation:

div x h 3 1 12 A∇ x p 0 = div x h 1 BU b on ω.
Remark 0.1 It is crucial to clearly separate the oscillating profile h 2 and the jumps characteristic function ψ in such a study. Indeed, the limit behavior described in Theorem 0.1 clearly depends on the link between the behavior of the roughness jumps with the one of the oscillating part of the profile (see Assumptions (H1) and (H2)).

Remark 0.2 Let us describe a situation allowing the use of Assumptions (H1). Assume that ω ε T is a finite collection of disconnected subsets of ω with measure of order ε α . As ε → 0, ω ε T reduces to a finite discrete set of points {x i T } i=1..N . Then

∇(ε 2 ψ ε h ε 2 ) = N i=1 χ ω i,ε + ∇ X h ε 2 + χ ω i,ε T ε 2-α ∇ x ψ ε h ε 2 .
Since the Lebesgue measure does not see accidents, one easily checks with the definition of strong two-scale convergence in Proposition 2.1 that

∇(ε 2 ψ ε h ε 2 ) 2 → N i=1 χ ω i + ∇ X h 2 .
This situation is illustrated through numerical exemples in Section 4.

Remark 0.3 We recall that the usual Reynolds approximation (corresponding to h 2 = 0) reads

div x h 3 1 12 ∇ x p 0 = div x h 1 2 U b on ω.
The latter result is thus a perturbation of the classical Reynolds equation. The low order perturbations of the rough profile (oscillations h 2 and abrupt change in the roughness ψ) both give a perturbation of the Couette component of the Reynolds model. This is a highly desirable effect for the applications of such a lubrication model. We explicitly recover a perturbation which was detected with formal computations by [START_REF] Myers | Characterization of surface roughness[END_REF], or more recently by [START_REF] Plouraboué | Sliding lubricated anisotropic rough surfaces[END_REF].

Remark 0.4 The strong two-scale convergence mentioned in (ii) of Theorem 0.1 could appear as a rather technical assumption. In fact, it means that the oscillation spectrum of the sequence belongs to the integer grid (see [START_REF] Zhikov | On two-scale convergence[END_REF] for a proof through Fourier analysis). The oscillation spectrum is then due only to the periodicity of the coefficients.

The paper is organized as follows. In the first section, we give the model and the first estimates for the velocity and the pressure. In the second section we give the definition and the properties of a convenient two-scale convergence. The third section is devoted to the rigorous study of the upscaling process in view of proving Theorem 0.1. In Section 4, we present some numerical results, illustrating the limit system in the three-dimensional lubrication context.

1 Model and estimates

Stokes model in lubricant context

The vector field U ε = (u ε , w ε ) ∈ R 2 × R (which describes the fluid velocity) and the pressure (given by the scalar function p ε ) satisfy the stationary Stokes equations4 :

-∆u ε + ∇ x p ε = 0 in Ω ε , (1)
-∆w ε + ∂ z p ε = 0 in Ω ε , (2)
div x u ε + ∂ z w ε = 0 in Ω ε , (3)
U ε = U ε b on ∂Ω ε . (4)
• The viscosity of the fluid is set equal to 1 for sake of simplicity.

• In classical lubrication framework, the boundary conditions are the following:

U ε b (x, z) =    (u b , 0) x ∈ ω, z = 0, (ε u b (z), 0) x ∈ ∂ω, (0, 0) x ∈ ω, z = εh ε ,
where u b is a constant corresponding to the imposed velocity at the bottom of the mechanism, and u b is a regular fonction (for instance an affine function) connecting u b for z = 0 and 0 for z = h ε . Notice that for such a problem (that is for an asymptotic study ε → 0) the precise value for the velocity u b is not primary. In fact, the physical quantity which persists to the limit is the total flux ∂ω εh ε 0 u b (z) dz, see [START_REF] Bayada | New models in the theory of the hydrodynamic lubrication of rough surfaces[END_REF].

• Finally, the pressure is normalized by

Ω ε p ε = 0.
We introduce a new vertical variable Z defined by Z = z h to consider the same problem in the fix domain Ω = ω × (0, 1). Problem (1)-( 4) now reads5

-∆ x u ε + ∆h h - |∇h| 2 h 2 Z∂ Z u ε + 2 ∇h h • Z∇ x ∂ Z u ε - |∇h| 2 h 2 Z 2 ∂ 2 Z u ε - 1 h 2 ∂ 2 Z u ε +∇ x p ε - ∇h h Z∂ Z p ε = 0 in Ω, ( 5
) -∆ x w ε + ∆h h - |∇h| 2 h 2 Z∂ Z w ε + 2 ∇h h • Z∇ x ∂ Z w ε - |∇h| 2 h 2 Z 2 ∂ 2 Z w ε - 1 h 2 ∂ 2 Z w ε + 1 h ∂ Z p ε = 0 in Ω, (6)
div x u ε - ∇h h • Z∂ Z u ε + 1 h ∂ Z w ε = 0 in Ω, (7)
U ε = U b on ∂Ω, (8)
Observe that ∆h h -|∇h| 2 h 2 = div ∇h h .

Estimates and dependance with respect to the small parameter ε

We begin by some estimates for the velocity.

Lemma 1.1 There exists a constant C such that the velocity components satisfy the following uniform estimates:

∇ x U ε (L 2 (Ω)) d ≤ C ε , (9)
∂ Z U ε (L 2 (Ω)) d ≤ C, (10)
U ε (L 2 (Ω)) d ≤ C. (11)
Proof: These estimates are directly derived from the original problem after an adequate lifting of the non-homogenenous boundary conditions. We write the standard energy estimate for Stokes type system and then use the change of variables to control the derivatives. We conclude with the Poincaré inequality.

We now give some uniform estimates for the pressure.

Lemma 1.2 There exists a constant C such that the pressure satisfies

∇ x p ε H -1 (Ω) ≤ C ε 2 , (12)
∂ Z p ε H -1 (Ω) ≤ C ε , (13)
p ε L 2 (Ω) ≤ C ε 2 . ( 14
)
Proof: Such estimates come from Equations ( 5)-( 6) estimated in H -1 . This provides bounds on ∇ x p ε and ∂ Z p ε . Using Poincaré-Wirtinger inequality (with the normalization hypothesis) we get the L 2 -estimates for p ε .

Notational convention:

In what follows, for any function

φ ∈ H 1 (ω × T d-1 ), we denote by ∇φ(x, x/ε 2 ) = ∇φ ε = ∇ x φ ε + ε -2 ∇ X φ ε its horizontal gradient.
2 Definition and properties of a convenient two-scale convergence

The proof of the homogenization process will be carried out by using a variant of the two-scale convergence introduced by G. Nguetseng in [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] and developed by G. Allaire in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. Let us give the basic definition and properties of this concept.

Proposition 2.1 A sequence (v ε ) of functions in L 2 (Ω) two-scale converges to a limit v 0 (x, Z, X) belonging to L 2 (Ω × T d-1 ), v ε 2 ⇀ v 0 , if lim ε→0 Ω v ε (x, Z) Ψ(x, Z, x/ε 2 ) dx dZ = Ω T d-1 v 0 (x, Z, X) Ψ(x, Z, X) dx dZ dX,
for any test function Ψ(x, Z, X), X-periodic in the third variable, satisfying

lim ε→0 Ω |Ψ(x, Z, x/ε 2 )| 2 dx dZ = Ω T d-1 |Ψ(x, Z, X)| 2 dx dZ dX.
Such a function Ψ is called an admissible test function for the two-scale convergence. Note that Z is only a parameter for this definition. (i) From each bounded sequence (v ε ) in L 2 (Ω) one can extract a subsequence which two-scale converges to some limit

v 0 ∈ L 2 (Ω × T d-1 ). The weak L 2 -limit of v ε is v(x, Z) = T d-1 v 0 (x, Z, X)dX. (ii) Let (v ε ) be a bounded sequence in L 2 (0, 1; H 1 (ω)) which converges weakly to v in L 2 (0, 1; H 1 (ω)). Then v ε 2 ⇀ v and there exists a function v 2 ∈ L 2 (Ω; H 1 (T d-1
)) such that, up to a subsequence,

∇v ε 2 ⇀ ∇v(x) + ∇ X v 2 (x, y). (iii) Let (v ε ) be a bounded sequence in L 2 (Ω) such that (ε 2 ∇ x v ε ) is bounded in (L 2 (Ω)) d-1 . Then, there exists a function v 0 ∈ L 2 (Ω; H 1 (T d-1
)) such that, up to a subsequence,

v ε 2 ⇀ v 0 and ε 2 ∇v ε 2 ⇀∇ X v 0 (x, Z, X).
• Due to our assumptions for h 1 and h 2 , functions

h 1 (x), h 2 (x/ε 2 ), ∂ x i h 1 (x), ∂ x i (ε 2 h 2 (x/ε 2 )) = ∂ X i h 2 (x/ε 2
) can obviously be considered as admissible test functions for the two-scale convergence.

• We also note that the function v 2 appearing in part (ii) of the previous proposition is the rigorous counterpart of the third term of the formal anisotropic expansion associated with the present setting:

v ε = ṽ0 (x, Z, x ε 2 ) + εṽ 1 (x, Z, x ε 2 ) + ε 2 ṽ2 (x, Z, x ε 2 ) + . . .
• An consequence of the above definition of the two-scale convergence is the following.

Lemma 2.1 Let (v ε ) be a bounded sequence in L 2 (Ω) such that (ε η ∇v ε ) is bounded in L 2 , with η < 2. Then the two-scale limit v 0 ∈ L 2 (Ω; H 1 (T d-1 )) of v ε is such that ∇ X v 0 = 0. Proof: By (iii) of Proposition 2.1, ε 2 ∇v ε 2 ⇀∇ X v 0 . Since (ε η ∇v ε ) is bounded in L 2
, it two-scale converges to some limit η 0 and (ε 2 ∇v ε = ε 2-η ε η ∇v ε ) two-scale converges to 0. Thus ∇ X v 0 = 0. Note that the above lemma is a key result to treat the anisotropy of the rough profile.

3 Rigorous derivation of the limit model

Convergence results

We infer from the estimates derived in Subsection 1.2 the following result. Lemma 3.1 There exist limit functions

p 0 ∈ L 2 (Ω; L 2 (T d-1 )), u 0 ∈ L 2 (Ω; H 1 (T d-1 )) and w 0 ∈ L 2 (Ω; H 1 (T d-1 )) such that ε 2 p ε 2 ⇀ p 0 , u ε 2 ⇀ u 0 , ∇ X u 0 = 0, w ε 2 ⇀ w 0 and ∇ X w 0 = 0.
Proof: In view of Lemma 1.1, we claim the existence of limits (up to a subsequence). Moreover, using Lemma 2.1, we assert that ∇ X u 0 = 0 and ∇ X w 0 = 0.

Before passing to the limit in the equations, we state some auxiliary results. We begin by the pressure function.

Lemma 3.2 The two-scale limit pressure is such that

∇ X p 0 = 0 and ∂ Z p 0 = 0.
Proof: We multiply Equation ( 5) by an admissible test function in the form ε 4 φ(x, Z, x/ε 2 ) and we integrate by parts. Unsing in particular ∆h

h - |∇h| 2 h 2 = div ∇h h , we get Ω ∇ x u ε • (ε 4 ∇ x φ ε + ε 2 ∇ X φ ε ) dx dZ - Ω ε 4 (∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε ) • 1 h 1 + ε(1 -ψ ε )h ε 2 ∇ x u ε ∂ Z (Zφ ε ) dx dZ - Ω ε 4 (∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε ) • 1 h 1 + ε(1 -ψ ε )h ε 2 Z∂ Z u ε (∇ x φ ε + 1 ε 2 ∇ X φ ε ) dx dZ + Ω ε 4 |∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε | 2 |h 1 + ε(1 -ψ ε )h ε 2 | 2 ∂ Z u ε • ∂ Z (Z 2 φ ε ) dx dZ + Ω ε 4 1 ε 2 |h 1 + ε(1 -ψ ε )h ε 2 | 2 ∂ Z u ε • ∂ Z φ ε dx dZ - Ω ε 2 p ε (ε 2 div x φ ε + div X φ ε ) dx dZ + Ω ε 2 (∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε ) • 1 h 1 + ε(1 -ψ ε )h ε 2 ε 2 p ε ∂ Z (Zφ ε ) dx dZ = 0.
We bear in mind that the term ε 1-α is of lower order than ε -1 if 0 ≤ α < 2 and of the same order if α = 2. Passing to the limit ε → 0, we obtain

lim ε→0 Ω ε 2 p ε div X φ ε dx dZ = Ω T d-1 p 0 div X φ dx dZ dX = 0.
Thus ∇ X p 0 = 0 and the two-scale limit and the weak L 2 limit of (ε 2 p ε ) coincide. Now let φ ∈ L 2 (ω; H 1 0 (0, 1)). We write

Ω ∂ Z (ε 2 p ε )φ dx dZ = ε ε∂ Z p ε , φ H -1 ×H 1 → 0 = - Ω ε 2 p ε ∂ Z φ dx dZ → - Ω p 0 ∂ Z φ dx dZ.
It follows that ∂ Z p 0 = 0. This ends the proof of the lemma.

We now introduce auxiliary limit functions for sake of clearness in the computations. We define,

γ 0 ∈ L 2 (Ω; H 1 (T d-1 )) and ξ 0 ∈ L 2 (Ω; H 1 (T d-1 )) such that (1 -ψ ε )u ε 2 ⇀ γ 0 , ε 2 ∇ x ((1 -ψ ε )u ε ) 2 ⇀ ∇ X γ 0 , (1 -ψ ε ) 2 u ε 2 ⇀ ξ 0 , ε 2 ∇ x ((1 -ψ ε ) 2 u ε ) 2 ⇀ ∇ X ξ 0 .
Of course, if 0 ≤ α < 2, we have ∇ X γ 0 = 0 and ∇ X ξ 0 = 0. We now study the two-scale limit of the vertical velocity component.

Lemma 3.3 The vertical velocity component is such that

w ε 2 ⇀ 0.
Proof: We already mentioned that ∇ X w 0 = 0 (see lemma 3.1). We now prove that ∂ Z w 0 = 0. We multiply the divergence Equation ( 7) by εφ(x, Z) where φ ∈ H 1 0 (Ω). Integrating by parts, we obtain

- Ω εu ε • ∇ x φ + Ω ε h 1 + ε(1 -ψ ε )h ε 2 ∇ x h 1 • u ε ∂ Z (Zφ) + Ω 1 h 1 + ε(1 -ψ ε )h ε 2 (1 -ψ ε )∇ X h ε 2 • u ε ∂ Z (Zφ) - Ω ε 2-α h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ∇ x ψ ε • u ε ∂ Z (Zφ) - Ω 1 h 1 + ε(1 -ψ ε )h ε 2 w ε ∂ Z φ = 0. ( 15
)
Thanks to the relation

lim ε→0 Ω 1 h 1 + ε(1 -ψ ε )h ε 2 ∇ X h ε 2 • (1 -ψ ε )u ε ∂ Z (Zφ) = Ω 1 h 1 T d-1 ∇ X h 2 γ 0 dX ∂ Z (Zφ),
which is justified by (1 -ψ ε )u ε 2 ⇀ γ 0 and to the relation

-lim ε→0 Ω ε 2-α h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ∇ x ψ ε • u ε ∂ Z (Zφ) = -lim ε→0 Ω 1 h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ε 2 ∇ x (ψ(x/ε α )u ε )∂ Z (Zφ) + lim ε→0 Ω 1 h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ψ ε ε 2 div x u ε ∂ Z (Zφ) = lim ε→0 Ω 1 h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ε 2 ∇ x ((1 -ψ(x/ε α ))u ε )∂ Z (Zφ) = Ω 1 h 1 ∂ Z (Zφ) T d-1 h 2 ∇ X γ 0 dX, which is justified by ε 2 ∇ x ((1 -ψ ε )u ε ) 2 ⇀ ∇ X γ 0 , we compute lim ε→0 Ω 1 h 1 + ε(1 -ψ ε )h ε 2 ∇ X h ε 2 • (1 -ψ ε )u ε ∂ Z (Zφ) - Ω ε 2-α h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ∇ x ψ ε • u ε ∂ Z (Zφ) = Ω 1 h 1 ∂ Z (Zφ) ∇ X (h 2 γ 0 ) dX = 0 because of the periodicity of h 2 γ 0 .
We thus infer from (15) that lim

ε→0 Ω 1 h 1 + ε(1 -ψ ε )h ε 2 w ε ∂ Z φ = Ω 1 h 1 w 0 ∂ Z φ = 0.
It follows that ∂ Z w 0 = 0. We conclude using the boundary conditions.

Remark 3.1 We recall that the aim of this work is to get a perturbation of the Reynolds approximation. And the previous result is characteristic of such a lubrication approximation: it states that we can neglect the vertical component of the velocity.

Divergence equation

Lemma 3. [START_REF] Bayada | Homogenization of the Stokes system in a thin film flow with rapidly varying thickness[END_REF] We have the relation

h 1 div x u 0 -∇ x h 1 • Z ∂ Z u 0 + ∂ Z (w 1 + L) = 0,
where

w 1 = (1/d -1) T d-1 w1 dX, with w1 ∈ L 2 (Ω; H 1 (T d-1
)) is defined by the following two-scale convergence εw ε 2 ⇀ 0,

∇(εw ε ) 2 ⇀ ∇ X w1 , while ∂ Z L is defined by the following weak convergence    ∂ Z L = lim ε→0 ∂ Z Zε (1 -ψ ε )h ε 2 h 1 divu ε in D ′ (Ω), L |Z=0 = L |Z=1 = 0. Remark 3.2
The term w1 is the rigorous counterpart of the first order term of a formal anisotropic expansion of w ε :

w ε = w0 (x, Z, x/ε 2 ) + ε w1 (x, Z, x/ε 2 ) + ε 2 w2 (x, Z, x/ε 2 ) + • • • Proof:
We multiply the divergence Equation ( 7) by a test function φ(x, Z) ∈ D(Ω). We obtain:

-

Ω u ε • ∇ x φ + Ω 1 h 1 + ε(1 -ψ ε )h ε 2 ∇ x h 1 • u ε ∂ Z (Zφ) dx dZ + Ω 1 h 1 + ε(1 -ψ ε )h ε 2 (1 -ψ ε ) ε ∇ X h ε 2 • u ε ∂ Z (Zφ) dx dZ - Ω 1 h 1 + ε(1 -ψ ε )h ε 2 h ε 2 ε α-1 ∇ x ψ ε • u ε ∂ Z (Zφ) dx dZ - Ω 1 h 1 + ε(1 -ψ ε )h ε 2 1 ε w ε ∂ Z φ dx dZ = 0. It follows that Ω u 0 • ∇ x φ + Ω 1 h 1 ∇ x h 1 • Z∂ Z u 0 φ dx dZ + lim ε→0 Ω 1 ε (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε h 1 + ε(1 -ψ ε )h ε 2 • Z∂ Z u ε φ dx dZ + Ω 1 ε 1 h 1 + ε(1 -ψ ε )h ε 2 w ε ∂ Z φ dx dZ = 0.( 16
)
We note that 1/(h

1 + ε(1 -ψ ε )h ε
2 ) is a computable perturbation of 1/h 1 for our convergences needs. Indeed, we have

1 h 1 + ε(1 -ψ ε )h ε 2 - 1 h 1 = -ε(1 -ψ ε )h ε 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 )
.

On the one hand, using (1

-ψ ε ) 2 u ε 2 ⇀ ξ 0 and ε 2 ∇ x (1 -ψ ε ) 2 u ε 2 ⇀ ∇ X ξ 0 , we write lim ε→0 Ω 1 ε -ε(1 -ψ ε )h ε 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • Z∂ Z u ε φ = - Ω 1 h 2 1 h 2 ∇ X h 2 Z ∂ Z ξ 0 φ dX -lim ε→0 Ω h ε 2 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) ε 2 2 ∇ x (1 -ψ(x/ε α )) 2 ∂ Z u ε Z φ + lim ε→0 Ω h ε 2 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) ε 2 2 (1 -ψ(x/ε α )) 2 ∇ x (∂ Z u ε )Z φ = - Ω 1 h 2 1 h 2 ∇ X h 2 Z ∂ Z ξ 0 φ dX + lim ε→0 Ω h ε 2 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) ε 2 2 ∇ x (1 -ψ(x/ε α )) 2 u ε ∂ Z (Z φ) -lim ε→0 Ω h ε 2 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) ε 2 2 (1 -ψ(x/ε α )) 2 ∇ x (u ε )∂ Z (Z φ) = - Ω 1 h 2 1 h 2 ∇ X h 2 Z ∂ Z ξ 0 φ dX + Ω 1 2h 2 1 h 2 2 ∇ X (ξ 0 ) ∂ Z (Z φ) dX = - Ω 1 h 2 1 h 2 ∇ X h 2 ∂ Z ξ 0 Z φ dX - Ω 1 2h 2 1 h 2 2 ∇ X (∂ Z ξ 0 ) Z φ dX = - Ω 1 2h 2 1 ∇ X h 2 2 ∂ Z ξ 0 Z φ dX = 0 10
because of the periodicity of h 2 ξ 0 .

On the other hand, we define w aux by

lim ε→0 Ω (1 -ψ ε )h ε 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) w ε ∂ Z φ = Ω T d-1 w aux dX ∂ Z φ = - Ω ∂ Z T d-1
w aux dX φ, [START_REF] Myers | Characterization of surface roughness[END_REF] which means that (1 -ψ ε )h ε 2 w ε /h 2 1 ⇀ T d-1 w aux dX weakly in L 2 (Ω). Then Relation ( 16) reads:

Ω u 0 • ∇ x φ + Ω 1 h 1 ∇ x h 1 • Z∂ Z u 0 φ + Ω ∂ Z T d-1 w aux dX φ = -A -B (18)
where

A = lim ε→0 Ω 1 ε 1 h 1 (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • Z∂ Z u ε φ, B = lim ε→0 Ω 1 ε 1 h 1 w ε ∂ Z φ.
We write

A = lim ε→0 Ω 1 h 1 ∇ x (ε(1 -ψ ε )h ε 2 ) • (Z∂ Z u ε )φ = lim ε→0 Ω ε(1 -ψ ε )h ε 2 h 1 div x u ε ∂ Z (Zφ) - Ω ε(1 -ψ ε )h ε 2 h 2 1 ∇ x h 1 • u ε ∂ Z (Zφ) + Ω ε(1 -ψ ε )h ε 2 h 1 u ε • ∇ x (∂ Z (Zφ)) = lim ε→0 Ω ε(1 -ψ ε )h ε 2 h 1 div x u ε ∂ Z (Zφ) = lim ε→0 Ω ε(1 -ψ ε )h ε 2 h 1 div x u ε Z∂ Z φ + lim ε→0 Ω ε(1 -ψ ε )h ε 2 h 1 div x u ε φ = -lim ε→0 Ω ∂ Z Z ε(1 -ψ ε )h ε 2 h 1 div x u ε φ + lim ε→0 Ω ε(1 -ψ ε )h ε 2 h 1 div x u ε φ. (19)
The first term in the right hand side will give the conservative contribution ∂ Z L to the final divergence equation. Let us compute the last term. We multiply the divergence Equation ( 7) by the test function

ε(1 -ψ ε )h ε 2 φ(x, Z). We get lim ε→0 Ω (1 -ψ ε )h ε 2 h 1 + ε(1 -ψ ε )h ε 2 w ε ∂ Z φ = lim ε→0 Ω εdiv x u ε (1 -ψ ε )h ε 2 φ + lim ε→0 Ω ε(1 -ψ ε )h ε 2 h 1 + ε(1 -ψ ε )h ε 2 ∇ x h 1 • u ε ∂ Z (Zφ) + lim ε→0 Ω h ε 2 h 1 + ε(1 -ψ ε )h ε 2 ∇ X h ε 2 • (1 -ψ ε ) 2 u ε ∂ Z (Zφ) -lim ε→0 Ω ε 2-α h ε 2 2 h 1 + ε(1 -ψ ε )h ε 2 ∇ x ψ ε • (1 -ψ ε )u ε ∂ Z (Zφ) = lim ε→0 Ω ε div x u ε (1 -ψ ε ) h ε 2 φ + Ω 1 2h 1 ∇ X (h 2 2 ) • ξ 0 ∂ Z (Zφ) dX + lim ε→0 Ω h ε 2 2 h 1 + ε(1 -ψ ε )h ε 2 ε 2 ∇ x 1 2 (1 -ψ ε ) 2 u ε ∂ Z (Zφ) -lim ε→0 Ω h ε 2 2 h 1 + ε(1 -ψ ε )h ε 2 ε 2 2 (1 -ψ ε ) 2 div x u ε ∂ Z (Zφ) = lim ε→0 Ω ε div x u ε (1 -ψ ε ) h ε 2 φ + Ω 1 2h 1 ∇ X (h 2 2 ) • ξ 0 ∂ Z (Zφ) dX + Ω 1 2h 1 h 2 2 ∇ X ξ 0 ∂ Z (Zφ) dX = lim ε→0 Ω ε div x u ε (1 -ψ ε ) h ε 2 φ + 1 2 Ω 1 h 1 ∇ X (h 2 2 ξ 0 ) ∂ Z (Zφ) dX = lim ε→0 Ω ε div x u ε (1 -ψ ε ) h ε 2 φ
because of the periodicity of h 2 ξ 0 . Bearing in mind ( 17), we infer from the latter relation that:

lim ε→0 Ω (1 -ψ ε )h ε 2 h 1 (h 1 + ε(1 -ψ ε )h ε 2 ) w ε ∂ Z φ = lim ε→0 Ω h ε 2 h 1 ε (1 -ψ ε ) φ div x u ε = - Ω ∂ Z T d-1
w aux dX φ [START_REF] Reynolds | On the theory of lubrication and its application to mr beauchamp tower's experiment, including an experimental determination of the viscosity of olive oil[END_REF] because of the definition of w aux . Let us now compute B = lim

ε→0 Ω 1 ε 1 h 1 w ε ∂ Z φ. We write Ω 1 ε 1 h 1 w ε ∂ Z φ = Ω 1 ε 2 (εw ε 1 h 1 ∂ Z φ) = 1 d -1 Ω div(I(x/ε 2 ))(εw ε 1 h 1 ∂ Z φ) = - 1 d -1 Ω I(x/ε 2 ) • ∇(εw ε 1 h 1 ∂ Z φ)
where I is the 1-periodic function such that

I(x) = x if x ∈ T d-1 . Then B = - 1 d -1 Ω T d-1 X • ∇ X w1 dX 1 h 1 ∂ Z φ = 1 d -1 Ω T d-1 w1 dX 1 h 1 ∂ Z φ = Ω w 1 h 1 ∂ Z φ = - Ω ∂ Z w 1 h 1 φ (21)
where

w 1 = 1 d-1 T d-1 w1 dX, the function w1 ∈ L 2 (Ω; H 1 (T d-1
)) being defined by

εw ε 2 ⇀ 0, ∇(εw ε ) 2 ⇀ ∇ X w1 .
Using ( 18)-( 21), we obtain the announced result.

The conservative form of the limit divergence equation obtained in the previous lemma is sufficient to get an explicit Reynolds approximation of the flow, even if function L is not explicitly computed (see [START_REF] Bresch | Roughness-Induced Effect at Main order on the Reynolds Approximation[END_REF]). By the way, we end this section by computing L under an additional assumption.

Lemma 3.5 Assuming moreover one of the following assumption, (H1) ∇(ψ ε h ε

2 ) strongly two-scale converges to ∇ X (ψ 0 h 2 ), (H2) ψ ε strongly two-scale converges to ψ 0 , (H3) α < 2, then L = 0 and the limit divergence equation reads

h 1 div x u 0 -∇ x h 1 • Z∂ Z u 0 + ∂ Z w 1 = 0.
Proof: From the assumptions (H1) or (H2), it follows that

h 2 (X)γ 0 (x, X) = h 2 (X) 1 -ψ 0 (x, X) u 0 (x, Z), h 2 (X)ξ 0 (x, X) = h 2 (X) 1 -ψ 0 (x, X) 2 u 0 (x, Z).
The two scale limit of Z ε(1

-ψ ε ) h ε 2 divu ε is h 1 L = Z (1 -ψ 0 ) h 2 div X u 1 where u 1 ∈ L 2 (Ω; H 1 (T d-1
)) is the anisotropic two-scale limit defined by

∇(εu ε ) 2 ⇀ ∇ X u 1 .
Let us compute div X u 1 . To this aim, we multiply the divergence Equation ( 7) by εφ(x, Z, x/ε 2 ) where φ is an admissible test function for the two scale convergence. Integrating by parts, we obtain

Ω εdivu ε φ ε - Ω (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • Z∂ Z u ε φ ε h 1 + ε(1 -ψ ε )h ε 2 - Ω ε∇ x h 1 • Z∂ Z u ε φ ε h 1 + ε(1 -ψ ε )h ε 2 + Ω ∂ Z w ε φ ε h 1 + ε(1 -ψ ε )h ε 2 = 0.
We recall that ε∂ Z u ε 2 → 0 (a strong two-scale convergence due to the fact that ∂ Z u ε is bounded in (L 2 (Ω)) d ) and w ε 2 ⇀ 0 (see Lemma 3.3). We also have

lim ε→0 Ω (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • Z∂ Z u ε φ ε h 1 + ε(1 -ψ ε )h ε 2 = Ω T d-1 1 h 1 div X (h 2 Z∂ Z γ 0 ) φ dX, because (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε u ε = ε 2 div x (1 -ψ ε )h ε 2 u ε -ε 2 (1 -ψ ε )h ε 2 div x u ε 2 ⇀ div X h 2 γ 0 .
We thus conclude that div

X u 1 = 1 h 1 div X (h 2 • Z∂ Z γ 0 ). ( 22
)
With the additional assumption (H1) or (H2), we infer from [START_REF] Zhikov | On two-scale convergence[END_REF] and from

∇ x u 0 = 0 that div X u 1 = 1 h 1 div X (h 2 (1 -ψ 0 )Z∂ Z u 0 ) = 1 h 1 ∇ X (h 2 (1 -ψ 0 )) • Z∂ Z u 0 .
We conclude by computing

T d-1 L dX = 1 h 2 1 ∂ Z (Z 2 ∂ Z u 0 ) T d-1 h 2 (1 -ψ 0 ) ∇ X (h 2 (1 -ψ 0 )) dX = 0.
Assume now (H3). Following the lines of the proof of Lemma 3.3, one easily check that the assumption α < 2 leads to (1 -ψ ε )w ε 2 ⇀ 0. Now, using the test function ε(1 -ψ ε )φ(x, Z, x/ε 2 ) instead of εφ(x, Z, x/ε 2 ) in the latter derivation of div X u 1 , we state that

ε(1 -ψ ε )divu ε 2 ⇀ 1 h 1 div X h 2 • Z∂ Z ξ 0 .
Integrating over T d-1 , we recover once again L = 0.

Momentum equation

Lemma 3.6 Let q 0 be the "strange" function defined by the following weak convergence in L 2 (ω; H -2 (0, 1)):

∇ ε 2 (1 -ψ ε )h ε 2 2 Z∂ Z (u ε -u 0 ) + Z∂ Z Z∂ Z 1 2 ε 2 p ε ∇(ε 2 |(1 -ψ ε )h ε 2 | 2 ) ⇀ q 0 .
The limit of the first momentum equation is

N ψ Z∂ Z u 0 -∂ 2 ZZ u 0 + h 2 1 ∇ x p 0 + q 0 = 0, where N ψ (x) = T d-1 |∇ X (1 -ψ 0 (x, X))h 2 (X) | 2 dX, x ∈ ω.
Remark 3.3 Assuming moreover (H1) or (H2), that is ∇(ψ ε h ε 2 ) strongly two-scale converges to ∇ X (ψ 0 h 2 ) or ψ ε strongly two-scale converges to ψ 0 , and using ∂ Z p 0 = 0, one checks that q 0 = 0.

Proof: Let us pass to the limit ε → 0 in the momentum equation. We multiply Equation ( 5) by ε 2 φ(x, Z) where φ ∈ H 1 0 (ω; H 2 0 (0, 1)) and we obtain:

Ω ε 2 ∇ x u ε • ∇ x φ - Ω ε h 1 + ε(1 -ψ ε )h ε 2 (∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε ) • ε∇ x u ε ∂ Z (Zφ) - Ω ε 2 h 1 + ε(1 -ψ ε )h ε 2 (∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε )Z∂ Z u ε ∇ x φ + Ω ε 2 |∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε | 2 (h 1 + ε(1 -ψ ε )h ε 2 ) 2 ∂ Z u ε • ∂ Z (Z 2 φ) + Ω 1 (h 1 + ε(1 -ψ ε )h ε 2 ) 2 ∂ Z u ε • ∂ Z φ - Ω ε 2 p ε div x φ + Ω 1 h 1 + ε(1 -ψ ε )h ε 2 ε 2 p ε ∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε • ∂ Z (Zφ) = 0.
Passing to the limit ε → 0, we get

-lim ε→0 Ω ε 2 ∇ (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) • ε∇ x u ε + lim ε→0 Ω |(1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε | 2 h 2 1 ∂ Z u ε • ∂ Z (Z 2 φ) + Ω 1 h 2 1 ∂ Z u 0 • ∂ Z φ - Ω p 0 div x φ + lim ε→0 Ω ε 3 p ε div (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) = 0. ( 23
)
We express the first term of the left hand side of ( 23) using once again ( 5), but for the test function

ε 3 (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ).
For this choice of test function, we obtain:

Ω ∇ x u ε • ε 3 ∇ (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) = Ω ε 3 h 1 + ε(1 -ψ ε )h ε 2 ∇ x h 1 + ε -1 ((1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε ) • Z∂ Z u ε ∇ (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) + Ω ε 2 h 1 + ε(1 -ψ ε )h ε 2 ∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • ε∇ x u ε ∂ Z Z (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) - Ω ε 2 |∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε | 2 (h 1 + ε(1 -ψ ε )h ε 2 ) 2 ε∂ Z u ε • ∂ Z Z 2 (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) - Ω 1 ε 2 (h 1 + ε(1 -ψ ε )h ε 2 ) 2 ε 3 ∂ Z u ε • ∂ Z (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) + Ω ε 3 p ε div (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) - Ω 1 h 1 + ε(1 -ψ ε )h ε 2 ε 2 p ε ε ∇ x h 1 + ε -1 (1 -ψ ε )∇ X h ε 2 -ε 1-α h ε 2 ∇ x ψ ε • ∂ Z Z (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) .
Passing to the limit ε → 0 in the latter relation, we get:

lim ε→0 Ω ∇ x u ε • ε 3 ∇ (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) = lim ε→0 Ω ε 2 h 1 (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • Z∂ Z u ε ∇ (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) + lim ε→0 Ω ε 3 p ε div (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) -lim ε→0 Ω 1 h 1 ε 2 p ε (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ x ψ ε • ∂ Z Z (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) . (24)
Inserting the previous computation in [START_REF] Zhou | Fractal geometry model for wear prediction[END_REF], we obtain:

-lim ε→0 Ω 1 h 1 (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε • Z∂ Z u ε ∇ ε 2 (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) -lim ε→0 Ω ε 3 p ε div (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) + lim ε→0 Ω 1 h 1 ε 2 p ε (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε (1 -ψ ε )h ε 2 h 1 ∂ Z (Z∂ Z (Zφ)) + lim ε→0 Ω |(1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε | 2 h 2 1 ∂ Z u ε • ∂ Z (Z 2 φ) + Ω 1 h 2 1 ∂ Z u 0 • ∂ Z φ - Ω p 0 div x φ + lim ε→0 Ω ε 3 p ε div (1 -ψ ε )h ε 2 h 1 ∂ Z (Zφ) = 0.
We thus note that the combined use of ( 23) and ( 24), that is the use of the momentum equation ( 5) for two different ad hoc test functions, is sufficient to pass to the limit without studying the limit behavior of the non-bounded sequence (εp ε ). This is a subsequent improvement of [START_REF] Bresch | Roughness-Induced Effect at Main order on the Reynolds Approximation[END_REF]. At this step, we have proven that:

-lim ε→0 Ω 1 h 2 1 (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε 2 Z∂ Z u ε ∂ Z (Zφ) + lim ε→0 Ω 1 h 1 ε 2 p ε (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε (1 -ψ ε )h ε 2 h 1 ∂ Z (Z∂ Z (Zφ)) + lim ε→0 Ω |(1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε | 2 h 2 1 ∂ Z u ε • ∂ Z (Z 2 φ) + Ω 1 h 2 1 ∂ Z u 0 • ∂ Z φ - Ω p 0 div x φ = 0,
that is, combining the first and third lines of the latter relation,

lim ε→0 Ω 1 h 1 ε 2 p ε (1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε (1 -ψ ε )h ε 2 h 1 ∂ Z (Z∂ Z (Zφ)) + lim ε→0 Ω |(1 -ψ ε )∇ X h ε 2 -ε 2-α h ε 2 ∇ψ ε | 2 h 2 1 Z∂ Z u ε • φ + Ω 1 h 2 1 ∂ Z u 0 • ∂ Z φ - Ω p 0 div x φ = 0. ( 25
)
It remains to exhibit the partial differential equation corresponding to the latter weak formulation. Lemma 3.6 is proven.

Remark 3.4 Assuming α < 2, the definition of the "strange" function q 0 becomes:

∇ ε 2 (1 -ψ ε )h ε 2 2 Z∂ Z (u ε -u 0 ) ⇀ q 0 weakly in L 2 (ω; H -2 (0, 1)). Indeed, if α < 2, lim ε 2 p ε ∇(ε 2 |(1-ψ ε )h ε 2 | 2 ) = lim ε 2 p ε (1-ψ ε ) 2 ∇(ε 2 |h ε 2 | 2 ) = T d-1 η 0 ∇ X (|h 2 | 2
)dX where η 0 is the two-scale limit of the sequence (ε 2 p ε (1 -ψ ε ) 2 ). This limit is computable without assuming the strong L 2 -convergence of (ε 2 p ε ) as in [START_REF] Bayada | Homogenization of the Stokes system in a thin film flow with rapidly varying thickness[END_REF]. A first proof is in [START_REF] Mikelić | Remark on the result on homogenization in hydrodynamical lubrication by G. Bayada and M. Chambat[END_REF]. Here we simply use the two-scale convergence. With the slight modification of the proof of Lemma 3.2, we obtain

lim ε→0 Ω ε 2 p ε (1 -ψ ε ) div X φ ε dx dZ - Ω ε 2 p ε ε 2-α ∇ψ ε φ ε dx dZ = 0,
for any admissible two-scale test function φ ε . Since α < 2, it follows that ∇ X η 0 = 0. Thus, by the periodicity of h 2 ,

T d-1 η 0 ∇ X (|h 2 | 2 )dX = η 0 T d-1 ∇ X (|h 2 | 2 )dX = 0.
3.4 A modified Reynolds approximation: proof of Theorem 0.1

Let us check that the system formed by the equations presented in Lemmas 3.4 and 3.6 corresponds to a "modified Reynolds" system. The limit momentum equation is (see lemma 3.6)

N ψ Z∂ Z u 0 -∂ 2 ZZ u 0 + h 2 1 ∇ x p 0 + q 0 = 0.
We integrate it in the variable Z (for each fixed x) and find, with the boundary conditions:

u 0 (x, Z) = Z 0 s 0 e N ψ (x)(s 2 -t 2 )/2 (∇ x p 0 (x) + q 0 (x, t) h 1 (x) 2 ) dtds - 1 0 s 0 e N ψ (x)(s 2 -t 2 )/2 (∇ x p 0 (x) + q 0 (x, t) h 1 (x) 2 ) dtds Z 0 e N ψ (x)s 2 /2 ds 1 0 e N ψ (x)s 2 /2 ds h 1 (x) 2 + 1 - Z 0 e N ψ (x)s 2 /2 ds 1 0 e N ψ (x)s 2 /2 ds U b . (26)
Next, integrating with respect to Z the divergence equation (see lemma 3.4)

div x (h 1 u 0 ) + ∂ Z (w 1 + L -Z∇ x h 1 • u 0 ) = 0,
and taking into account that the velocity w 1 + L -Z∇ x h 1 • u 0 cancels for Z = 0 and for Z = 1 we obtain

div x 1 0 h 1 u 0 dZ = 0.
With the previous expression (26) for the velocity the following pressure equation is obtained

div x h 3 1 A∇ x p 0 = div x h 1 (BU b -Q 0 )
where A and B are two functions defined in ω by

A(x) = 12 N ψ (x) e N ψ (x)/2 1 0 e -N ψ (x)t 2 /2 dt -1 - 12 N ψ (x) (e N ψ (x)/2 -1) 1 0 s 0 e N ψ (x)(s 2 -t 2 )/2 dtds 1 0 e N ψ (x)s 2 /2 ds , B(x) = 1 N ψ (x) (e N ψ (x)/2 -1) 1 1 0 e N ψ (x)s 2 /2 ds , x ∈ ω,
assuming that function y → (e y/2 -1)/y is continuously extended to R + and Q 0 is defined by

Q 0 (x) = 1 0 Z 0 s 0 e N ψ (x)(s 2 -t 2 )/2 q 0 (x, t) dtds - 1 0 s 0 e N ψ (x)(s 2 -t 2 )/2 q 0 (x, t) dtds Z 0 e N ψ (x)s 2 /2 ds 1 0 e N ψ (x)s 2 /2 ds dZ, x ∈ ω.
Part (i) of Theorem 0.1 is proven. Part (ii) is straightforward since the limit momentum equation is now given by:

N ψ Z∂ Z u 0 -∂ 2 ZZ u 0 + h 2 1 ∇ x p 0 = 0.
We thus have q 0 = 0 and then Q 0 = 0.

Some numerical illustrations

In this section, we present some numerical results. The goal is to quantify the effect of roughness on a portion of a surface. So we compare the solutions of the classical Reynolds equation (corresponding to the smooth domain) and those obtained from the equation with partial roughness, see Theorem 0.1 on page 3 and Remark 0.2 after. Physically, we are in the framework of lubrication. Consequently, we are interestd in the pressure forces in a flow between two surfaces whose relative velocity is imposed. As usual, all the geometry of the gap between the two surfaces is taking into account by the upper surface (it corresponds to the height h 1 ) and the relative velocity is taking account by the lower surface (its imposed velocity is denoted by U b , while the upper surface is at rest). Moreover, we know that the problem is well posed if and only if we give boundary conditions: either by imposing Dirichlet condition on the pressure, either by imposing Neumann type flow. A physically interesting situation is the following: we give a Neumann condition at the "entrance" of the channel (that is a flux denoted by Q e ), and we impose a pressure at the other boundaries. More precisly, for the next simulations we use the following data (see also Fig. 2):

ω = [0, 1] × [0, 1], h 1 (x, y) = (2x -1) 2 + 0.5, U b = (1, 0)
and Q e = 0.5

Without rugosities

To situate the next results, the first figure that we present (see Fig. 2, on right) corresponds to the case of flow in a smooth domain. In that case, we solve the classical Reynolds equation, see the Remark 0.3. We use the Freefem++ language (see http://www.freefem.org/ff++/ ) and consequently a variational form of the Reynolds equation: for all test function q vanishing on the boudaries {x = 1}, {y = 0} and {y = 1}, we get

ω h 3 1 12 ∇p • ∇q - ω h 1 2 U b • ∇q + {x=0} Q e q = 0.
Finally, we use the Scilab software (http://www.scilab.org/ ) to view results.

Some rugosities

From the results of Theorem 0.1, it seems simple to simulate the rugosities effets on the upper surface. Morally, it is sufficient to specify first: on which area of the surface roughness ago, and secondly: quantify the roughness using their form. These data allow to evaluate the function N ψ , and therefore the coefficients A and B involved in the statement of the Theorem 0.1. For standard rugosities, for instance described by a function of the form α cos(βX), it is reasonable to take the following examplarity values We then present three cases where the domain ω is partially rough (see Fig. 3, 4 and 5). We can note two remarkable points. First, the effect induced by the roughness is not a local effect. Thus, although roughness occupy only a portion of the domain, then the pressure is disrupted throughout the domain, including upstream roughness. Secondly, the effect of roughness is quite significant even if these wrinkles are present only a small part of the domain (see Fig. 5).

NFigure 2 :

 2 Figure 2: On the left figure, we draw the three-dimensional domain and indicate the main data. On the right figure, we plot the pressure profile without rugosity, that is the solution to the classical Reynolds equation.

Figure 3 :Figure 4 :

 34 Figure 3: On the left figure, we indicate the part of the top boundary which is rough. On the right figure, we compare the pressure profile without rugosity (grid) and the pressure profile obtain with the corresponding rugosities (plain).

Figure 5 :

 5 Figure 5: On the left figure, we indicate the part of the top boundary which is rough. On the right figure, we compare the pressure profile without rugosity (grid) and the pressure profile obtain with the corresponding rugosities (plain).

Faculté des Sciences et Techniques de Saint-Jérôme -Université P. Cézanne -LATP UMR 6632, 13397 Marseille Cedex

20, France Email: c.choquet@univ-cezanne.fr 2 Université de Lyon -INSA de Lyon -Pôle de Mathématiques -CNRS, UMR5208, Institut Camille Jordan -21 av. Jean Capelle, 69621 Villeurbanne Cedex, France Email: laurent.chupin@insa-lyon.fr 3 LAMA -CNRS UMR5127 -Université de Savoie, 73376 Le Bourget du Lac, France Email: gisclon@univ-savoie.fr

In all this document, the operators (like ∆) without index denote the operators with respect all the variables. To specify the operators with respect only one variable we use subscript notations, see for instance ∆x or ∂z.

Rigorously, the function u ε defined in the previous Stokes system (1)-(4) is not egal to the function u ε used in this rescale domain. However, to not introduce numerous notations, we denote always the same.

Acknowledgments -The second author of this work has been partially supported by the ANR project n • ANR-08-JCJC-0104-01 : RUGO (Analyse et calcul des effets de rugosités sur les écoulements).