Clément De

Seguins Pazzis

Peter Šemrl
email: peter.semrl@fmf.uni-lj.si

Continuous space-time transformations *

Keywords: AMS classification: 15A86, 55E40, 83A05 Minkowski space, space-time event, hermitian matrix, coherency, adjacency, homotopy

We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincaré similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form. In the presence of the continuity assumption the main tool in the proof is a basic result from the homotopy theory of spheres.

1 Introduction and statement of the main result

The problem

Throughout the paper R 4 = {(x, y, z, t) : x, y, z, t ∈ R} will be equipped with the standard topology. At the foundations of relativity theory lies the Lorentz-Minkowski indefinite inner product defined by (x, y, z, t), (u, v, w, s) = -xuyvzw + ts.

A fundamental result due to Alexandrov [START_REF] Alexandrov | Seminar Report, Uspehi Mat[END_REF][START_REF] Alexandrov | A contribution to chronogeometry[END_REF][START_REF] Alexandrov | Mappings of spaces with families of cones and space-time transformations[END_REF] (see also [START_REF] Benz | Real Geometries, BI Wissenschaftsverlag[END_REF][START_REF] Zeeman | Causality implies the Lorentz group[END_REF]) states that every bijective map φ : R 4 → R 4 with the property that for every pair of space-time events (r 1 , r 2 ) ∈ (R 4 ) 2 we have r 1r 2 , r 1r 2 = 0 ⇐⇒ φ(r 1 )φ(r 2 ), φ(r 1 )φ(r 2 ) = 0 [START_REF] Alexandrov | Seminar Report, Uspehi Mat[END_REF] is of the form

φ : r ∈ R 4 → k Qr + a, (2)
where k is a positive real number, a is a vector in R 4 , and Q is a 4 × 4 Lorentz matrix, that is, a matrix satisfying QM Q t = M , where

M =    -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 1    .
For a physical interpretation of the above theorem we refer to [11, p.691]. Let us just mention that for a given r ∈ R 4 the set {s ∈ R 4 : sr, sr = 0} is called the light cone with vertex r.

Because of its importance in the mathematical foundations of relativity theory, Alexandrov's theorem has been improved in several directions. Instead of the four-dimensional Minkowski space one can consider Minkowski spaces of any dimension ≥ 3 or even infinite-dimensional Minkowski spaces, maps can be defined on domains of Minkowski spaces (then, besides Lorentz maps composed with dilations, inversions and singular double inversions satisfy property (1) ), such maps were considered with the bijectivity assumption relaxed to injectivity only or to surjectivity only, the relation (1) can be replaced by a similar one with "=" replaced by either "≥", or ">", etc. An interested reader can find this kind of results in [START_REF] Borchers | The structure of space-time transformations[END_REF][START_REF] Chubarev | Linearity of space-time transformations without the one-to-one, line-onto-line, or constancy-of-speed-of-light assumptions[END_REF][START_REF] Lester | A physical characterization of conformal transformations of Minkowski spacetime[END_REF][START_REF] Lester | On Null-Cone Preserving Mappings[END_REF][START_REF] Pfeffer | Lorentz transformations of a Hilbert space[END_REF][START_REF] Popovici | Characterizing the conformality in a Minkowski space[END_REF] and the references therein.

When such maps are defined on the whole Minkowski space (not just on some open domain), these results can be considered as various possible derivations of Lorentz transformations under weak mathematical properties and the proofs are often based on certain improvements of the fundamental theorem of affine geometry under weak assumptions.

In this paper we will describe the general form of continuous maps φ : R 4 → R 4 with the property that for every pair of space-time events r 1 , r 2 ∈ R 4 we have

r 1 -r 2 , r 1 -r 2 = 0 ⇒ φ(r 1 ) -φ(r 2 ), φ(r 1 ) -φ(r 2 ) = 0. (3)
We assume neither injectivity nor surjectivity and we assume that the condition r 1r 2 , r 1r 2 = 0 is preserved in one direction only. Still, under these very weak assumptions (besides the weak preserving property our only assumption is continuity) we get almost the same conclusion as in Alexandrov's classical result. Namely, such maps are either Poincaré similarities, that is, Lorentz transformations composed with dilations as in (2), or they are of a very special degenerate form that can be completely described.

As far as we know our approach is completely different from known proof techniques. It has been known before [START_REF] Hua | Starting with the unit circle[END_REF] that our problem has an equivalent reformulation as the one of characterizing coherency preserving maps on 2 by 2 hermitian matrices. Combining elementary linear algebra and geometric techniques with topological tools (we will use one basic result from the homotopy theory of spheres) we will show that continuous coherency preservers on 2 by 2 hermitian matrices are either degenerate, or they preserve adjacency. It has been known before (here, nontrivial methods from geometry are needed) that non-degenerate adjacency preservers are of the standard form. First, let us make clear what we mean by a degenerate coherency preserver: Definition 1.1 A continuous coherency preserver φ : R 4 → R 4 is called degenerate when its range is included in a light cone. Now, we can state our main result: Theorem 1.1 Let φ : R 4 → R 4 be a continuous map satisfying [START_REF] Alexandrov | Mappings of spaces with families of cones and space-time transformations[END_REF]. Then either φ is a Poincaré similarity, or φ is degenerate.

A constructive description of all degenerate continuous coherency preservers is given in Section 3.

Restatement in terms of hermitian matrices

Our problem can be put into the broader context of real quadratic spaces: let (V, q) be a real quadratic space, i.e. V is a real vector space and q is a quadratic form on V . The polar form (x, y) → 1 2 (q(x + y)q(x)q(y)) of q is denoted by b q . The quadratic space (V, q) is called a Minkowski space when V is finite-dimensional and q has signature (1, n -1), where n := dim V is greater than 1 (in that case, q is non-degenerate).

In that context, two points x, y in V are called coherent whenever q(x-y) = 0, and they are called adjacent whenever they are coherent and x = y. Given a ∈ V , the coherency cone of a is defined as

C(a) := {m ∈ V : q(m -a) = 0}. Note that C(a) = a + C(0). A coherency preserver is a map φ : V → V that satisfies ∀(a, b) ∈ V 2 , q(b -a) = 0 ⇒ q(φ(b) -φ(a)) = 0;
such a map is called degenerate when the range of φ is included in a coherency cone.

A linear similarity of (V, q) is a bijective linear transformation u for which there is a nonzero scalar λ such that ∀x ∈ V, q(u(x)) = λ q(x). An affine similarity of (V, q) is the composite of a linear similarity with a translation.

It is known that the Poincaré similarities are the affine similarities of (R 4 , r → r, r ). Thus, our main theorem can be reformulated as follows:

Theorem 1.2 Let (V, q) be a 4-dimensional Minkowski space and φ : V → V be a continuous coherency preserver. Then φ is an affine similarity or it is degenerate.

As all Minkowski spaces with a given dimension are isometric, proving Theorem 1.2 only requires that we prove it for a particular 4-dimensional Minkowski space.

An important example of such a space consists of H 2 , the space of all 2 by 2 hermitian matrices, equipped with the quadratic form det. For any such matrix

M = t + z x + iy x -iy t -z , we have det M = -x 2 -y 2 -z 2 + t 2 ,
and it is then easily seen that (H The optimal version of his theorem for hermitian matrices has been obtained only very recently [START_REF] Huang | Adjacency preserving maps on hermitian matrices[END_REF]. It reads as follows.

Theorem 1.3 Let φ : H 2 → H 2 be an adjacency preserver. Then one of the following holds.

1. There exist S ∈ H 2 , a rank one orthogonal projection R, and a function

ρ : H 2 → R such that ∀A ∈ H 2 , φ(A) = ρ(A)R + S.
2. There exist c ∈ {-1, 1}, an invertible 2 by 2 complex matrix T , and S ∈ H 2 such that either

∀A ∈ H 2 , φ(A) = c T A T * + S, (4)
or ∀A ∈ H 2 , φ(A) = c T A t T * + S. (5)
Clearly, the map A → ρ(A)R + S preserves adjacency if and only if we have ρ(A) = ρ(B) whenever A and B are adjacent. An example of such a function ρ is ρ(A) = trace (A). Indeed, if A and B are adjacent, then B -A is of rank one and as it is diagonalizable we find trace (B -A) = 0.

Every map of the form (4) or of the form (5) will be called a standard adjacency preserver.

In fact, in [START_REF] Huang | Adjacency preserving maps on hermitian matrices[END_REF] adjacency preservers from the set of all m by m hermitian matrices into the set of all n by n hermitian matrices, where m and n are arbitrary positive integers, were described. We have formulated the result only for the 2 by 2 matrices as only this low-dimensional case is needed to solve our problem.

Clearly, every adjacency preserver is a coherency preserver. The first impression might be that the problems of characterizing adjacency preservers and coherency preservers are almost the same. But it turns out that the second one is much more difficult. A first evidence might be that degenerate adjacency preserving maps are of an extremely simple form, while the structure of non-standard coherency preservers is more complicated. We have a complete description of adjacency preservers between matrix spaces of arbitrary dimensions while at present the description of coherency preservers seems to be out of reach even in the simplest 2 by 2 case.

It is well-known that the standard adjacency preservers as described in ( 4) and ( 5) are the affine similarities of (H 2 , det). It follows that in order to establish Theorem 1.2 it suffices to prove the following result: Theorem 1.4 Let (V, q) be a Minkowski space whose dimension is greater than 3, and let φ : V → V be a continuous coherency preserver. Then either φ preserves adjacency, or it is degenerate.

Structure of the article

In Section 2, we state a few basic results on Minkowski spaces that will be used throughout the rest of the article. Section 3 features a constructive view of continuous degenerate coherency preservers on 4-dimensional Minkowski spaces. The last section, which is independent on the results of Section 3, consists of the proof of Theorem 1.4. This proof is mostly self-contained, with the notable exception of the use of some tools from the homotopy theory of spheres.

Basic lemmas 2.1 Maximal coherent sets

The following results are well-known but we reprove them for the sake of completeness.

Lemma 2.1 Let (V, q) be a Minkowski space. Let a, b, c be pairwise coherent points of V . Then, a, b, c belong to the same affine line.

Proof. Setting x := ba and y := cb, we have q(x) = q(y) = q(x + y) = 0. Thus, b q (x, y) = 0. If x and y were linearly independent, Span(x, y) would be a two-dimensional totally isotropic subspace of V , contradicting the fact that the Witt index of q equals 1.

✷

A coherent set in a Minkowski space (V, q) is a subset of V whose elements are pairwise coherent. As an obvious corollary to the above lemma, we get:

Corollary 2.2
The maximal coherent subsets of a Minkowski space V are the lines of the form a + Rx in which a ∈ V and x ∈ C(0) \ {0}.

With a similar way of reasoning as above, one shows that every line that is included in C(a) must go through a.

Intersection of coherency cones

Throughout the rest of the section, (V, q) denotes an arbitrary Minkowski space.

Lemma 2.3 Letbe an arbitrary norm on the Minkowski space V . There exists a constant M > 0 such that, for all x and y in V , there exists z ∈

C(x) ∩ C(y) such that z -x ≤ M y -x .
Proof. Denote by n the dimension of V . No generality is lost in assuming that

V = R n , q : (x 1 , . . . , x n ) → x 2 n - n-1 k=1 x 2 k
, and the norm under consideration is the standard Euclidean norm (x 1 , . . . , x n ) → n k=1 x 2 k . Letting (x, y) ∈ V 2 , we prove that there is some z ∈ C(x) ∩ C(y) such that zx ≤ yx . In this respect, one sees that by applying a translation followed with a transformation of the form ( -→ u , t) → (Q -→ u , t) with Q ∈ O(R n-1 ), no generality is lost in assuming that x = (0, . . . , 0) and y = (0, . . . , 0, a, b) for some (a, b) ∈ R 2 , in which case one checks that z := 0, . . . , 0, a+b 2 , a+b 2 has the required properties.

✷

The following result is probably known but we have not been able to find any reference to it in the literature. Lemma 2.4 Let (V, q) be a Minkowski space whose dimension is greater than 3. Let a, b, c be pairwise non-collinear vectors in C(0). Then there exists d ∈ V \C(0) that is coherent with a, b and c.

Proof. Denote by B the polar form of q and by ⊥ the orthogonality relation that is associated with it. Then, the problem amounts to finding d ∈ V such that q(d) = 0 and 2B(a,

d) = 2B(b, d) = 2B(c, d) = q(d).
The space P := {ab, b -c} ⊥ has dimension at least 2. Note that a, b, c are linearly independent: indeed, the contrary would yield a 2-dimensional subspace W of V that contains a, b, c, and hence the quadratic form q would vanish on three distinct 1-dimensional subspaces of W : it would follow that q vanishes everywhere on W , contradicting the fact that the Witt index of q equals 1. Then, a, ab, bc are also linearly independent.

As q is non-degenerate, this leads to P ⊂ {a} ⊥ ; in other words d → B(d, a) does not vanish everywhere on P . Moreover, as the Witt index of q equals 1 the quadratic form q does not vanish everywhere on P either. Thus, the product of the polynomial maps d → B(d, a) and d → q(d) cannot vanish everywhere on P , which yields a vector d 0 ∈ P such that B(d 0 , a) = 0 and q(d 0 ) = 0.

It follows that the polynomial q(t d 0 -a) = t 2 q(d 0 )-2tB(d 0 , a) in the variable t has exactly two real roots: we denote by t 0 the nonzero one. Then, with d := t 0 d 0 , we find that q(d) = t 2 0 q(d 0 ) = 0 and q(da) = 0, the latter of which leads to q(d)

Q := (x, y, z, 1) | (x, y, z) ∈ R 3 , x 2 + y 2 + z 2 = 1 .
Note that Q ⊂ C(0) and that every nonzero element of C(0) can be written in a unique way as t s with t ∈ R * and s ∈ Q. In particular, space-time events in Q are coherent if and only if they are equal.

Using this notation, we can give a constructive description of all degenerate continuous coherency preserving maps on R 4 : Proposition 3.1 Let φ : R 4 → R 4 be an arbitrary map. The following conditions are equivalent:

(a) φ is a degenerate continuous coherency preserver.

(b) There exist a countable set J, a family (U j ) j∈J of pairwise disjoint open subsets of R 4 , a family (s j ) j∈J of vectors of Q, a space-time event s ′ and a continuous map f : R 4 → R such that:

(i) For all distinct j and j ′ in J and all r ∈ U j and r ′ ∈ U j ′ , one has rr ′ , rr ′ = 0;

(ii) The map f vanishes everywhere on R 4 \ j∈J U j ;

(iii) For all r ∈ R 4 ,

φ(r) = s ′ + f (r).s j if r ∈ U j s ′ otherwise.
In (b), note that in the case when J is the empty set the map φ is a constant map sending all space-time events to s ′ .

Proof. Assume first that condition (b) holds. Obviously, the range of φ is included in C(s ′ ) and hence φ is degenerate. Next, we show that φ is continuous and that it satisfies [START_REF] Alexandrov | Mappings of spaces with families of cones and space-time transformations[END_REF]. If the sequence of space-time events (r n ) converges to r and r ∈ U j for some j, then r n ∈ U j for all n that are large enough, and hence φ(r n ) converges to φ(r) because f is continuous. If, on the other hand, r does not belong to any of the sets U j , then f (r n ) → f (r) = 0 as n tends to infinity, and because Q is bounded we see that φ(r n ) → s ′ = φ(r) as n → ∞. Therefore, φ is continuous. Let r 1 and r 2 in R 4 be such that φ(r 1 )φ(r 2 ), φ(r 1 )φ(r 2 ) = 0.

As ∀j ∈ J, s j , s j = 0, we deduce that r 1 ∈ U j and r 2 ∈ U j ′ for some distinct j and j ′ in J and it follows from our assumptions that r 1r 2 , r 1r 2 = 0. Therefore, φ preserves coherency.

Conversely, assume that condition (a) holds. The range of φ is then included in C(s ′ ) for some space-time event s ′ . We can write φ(r)s ′ = (?, ?, ?, f (r)) for all r ∈ R 4 . The map f : R 4 → R is then continuous and hence

U := f -1 (R * )
is an open subset of R 4 (possibly empty). The map

g : r ∈ U → 1 f (r) (φ(r) -s ′ )
is obviously continuous and its range is included in Q. For s ∈ Q, set U s := g -1 ({s}), so that the subsets U s are pairwise disjoint and their union equals U . Next, we prove that each subspace U s is open in U . To do so, we prove that g is locally constant on U . Let r 0 ∈ U . By Lemma 2.3, we can find a neighborhood U ′ of r 0 in U such that for all r ∈ U ′ , there exists r ′ ∈ U that is coherent with both r 0 and r. Let r ∈ U ′ . Then, we obtain r ′ ∈ U ∩ C(r 0 ) ∩ C(r). The space-time events φ(r 0 ) and φ(r ′ ) are coherent and belong to C(s ′ )\{s ′ }. By Lemma 2.1, it follows that g(r 0 ) and g(r ′ ) are collinear, and hence g(r 0 ) = g(r ′ ). Similarly, we obtain g(r) = g(r ′ ), and hence g(r 0 ) = g(r). Thus, g is locally constant on U . It follows that U s is an open subset of U , and hence of R 4 , for all s ∈ Q.

Finally, set J := {s ∈ Q : U s = ∅}. As R 4 is separable the set J is countable. Then, we know that (U s ) s∈J is an open cover of U and that f vanishes everywhere on R 4 \ U , and it is straightforward to check that

∀r ∈ R 4 , φ(r) = s ′ + f (r).s if r ∈ U s s ′ otherwise.
Finally, let j and j ′ be different elements of J, and let r ∈ U j and r ′ ∈ U j ′ . Then, as g(r) = g(r ′ ), the events φ(r) and φ(r ′ ) are non-coherent, and hence r and r ′ are non-coherent. Thus, all the requirements in (b) are fulfilled.

✷

We next show that it is easy to construct such degenerate maps. All we need to do is to find a family of open sets U n , n ∈ N, and a continuous function f with the above properties. Let U n , n ∈ N, be an ε-neighbourhood (with respect to the usual Euclidean distance) of the space-time event (n, 0, 0, 0). By choosing ε small enough we find that rs, rs = 0 for every pair of space-time events (r, s) ∈ U m × U n with distinct non-negative integers m and n. If we choose arbitrary nonzero continuous functions f n : R 4 → R with support in U n , for n ∈ N, then it is straightforward to check that the function

f : r → f n (r) if r ∈ U n 0 otherwise is continuous.
4 From coherency preservers to adjacency preservers

General considerations

In proving Theorem 1.4 for all n-dimensional Minkowski spaces, it suffices to address the situation of a particular one. Thus, we fix an integer n ≥ 4 and we consider the space V = R n equipped with the quadratic form

q : (x 1 , . . . , x n ) → x 2 n - n-1 k=1 x 2 k .
The polar form of q is denoted by B. In V , we have a particular subset

Q = (x 1 , . . . , x n-1 , 1) | (x 1 , . . . , x n-1 ) ∈ R n-1 , n-1 k=1 x 2 k = 1
and the linear form η : (x 1 , . . . , x n ) → x n .

Every nonzero vector x of C(0) then splits in a unique fashion as x = t p for some t ∈ R * and some p ∈ Q, and more precisely we have t = η(x) and p = 1 η(x) x. For any a ∈ V , we define

π a : C(a) \ {a} → Q m → 1 η(m-a) (m -a)
and we note that it is an open mapping. Obviously, Q is to the (n -2)-dimensional sphere S n-2 . In particular, it is compact and connected. Clearly, if φ : V → V is a coherency preserver, then φ maps every coherent line into some coherent line. Using π a , the set of all coherent lines through the point a can be identified with Q.

We finish with the simple observation -that will be used repeatedly in the rest of the section -that composing a map φ : V → V with translations on both sides affects neither the assumptions nor the conclusion of Theorem 1.4.

Images of coherent lines

Proposition 4.1 Let φ : V → V be a continuous coherency preserver. Assume that φ is constant on some coherent line. Then φ is degenerate.

Proof. No generality is lost in assuming that φ is constant on Ra for some

a ∈ C(0) \ {0}. Let x ∈ V be such that B(a, x) = 0. As ∀t ∈ R, q(ta -x) = -2B(a, x)t+q(x), some point b ∈ Ra is coherent with x, so that φ(x) is coherent with φ(b) = φ(0).
The set {x ∈ V : B(a, x) = 0} is a linear hyperplane of V and hence {x ∈ V : B(a, x) = 0} is dense in V . Since C(φ(0)) is a closed subset of V and φ is continuous, we conclude that the range of φ is included in C(φ(0)).

✷

We will assume from now on that the continuous coherency preserver φ : V → V is not degenerate. Then, by Proposition 4.1, for every point a ∈ V and every vector p ∈ Q, there exists a unique p ′ ∈ Q such that φ maps a + Rp into φ(a) + Rp ′ . Hence, for each a ∈ V the map φ induces a map Proof. Assume that a sequence (a n ) ∈ V N converges to a and a sequence (p n ) ∈ Q N converges to p. We choose a real number t 0 such that φ(a + t 0 p) = φ(a) + λ ϕ a (p) for some nonzero real number λ. By the continuity of φ we know that φ(a

ϕ a : Q → Q such that ∀a ∈ V, ∀p ∈ Q, φ(a + Rp) ⊂ φ(a) + Rϕ a (p).
n + t 0 p n ) = φ(a n ) + s n ϕ an (p n ) → φ(a) + λ ϕ a (p)
as n → ∞, and since φ(a n ) converges to φ(a), we have

s n ϕ an (p n ) → λ ϕ a (p)
as n → ∞. By taking the last coordinate, it follows that lim s n = λ, and as λ = 0 we conclude that lim ϕ an (p n ) = ϕ a (p), as desired.

✷

As Q is compact, the compact-open topology on C(Q, Q) -the space of all continuous functions from Q to itself -coincides with the topology of uniform convergence. A straightforward consequence of the above lemma is that the map from V to C(Q, Q) given by a → ϕ a is continuous. and as they are constant we deduce that they are equal. In the general case, Lemma 2.3 yields a point c that is coherent with both a and b, and we deduce that ϕ a = ϕ c = ϕ b . Now, we have a vector p ∈ Q such that φ(b) ∈ φ(a) + Rp for every coherent pair (a, b). Let a ∈ V . Then there is a point b that is coherent with both 0 and a, and hence there are scalars α and β such that φ(a)φ(b) = α p and φ(b)φ(0) = β p. Thus, φ(a) = φ(0) + (α + β) p. Therefore, the range of φ is included in the coherent line φ(0) + Rp, which is itself included in C(φ(0)). This contradicts the assumption that φ is non-degenerate. The aim of this section is to prove the following statement, which is the main part of our proof of Theorem 1.4. Proposition 4.4 Let φ : V → V be a continuous coherency preserver. Assume that φ is not degenerate, φ(0) = 0 and ϕ 0 is nonconstant. Assume further that 0 is generic. Then φ preserves adjacency.

The proof will be carried out in a few steps that will be formulated as lemmas. Proof. Assume on the contrary that φ(a) = 0 for some a ∈ C(0) \ {0}. Since ϕ 0 is continuous and nonconstant, its range contains at least three different points. It follows that there exist vectors b, c in C(0) \ {0} such that a, b, c are pairwise non-coherent and φ(b) and φ(c) are non-coherent elements of C(0). According to Lemma 2.4 there exists d ∈ V \ C(0) that is coherent with a, b, and c. We claim that φ(m) = 0 for all m in some neighborhood of d, contradicting the assumption that 0 is generic.

Among the points of Rb, we see that b is coherent with d but 0 is not. Finally, Lemma 2.3 yields a neighborhood U ′′ of d such that U ′′ ⊂ U ′ and for every m ∈ U ′′ , there exists m ′ ∈ U ′ that is coherent with both d and m. Let then m ∈ U ′′ , and take m ′ as above. As φ(m ′ ) is coherent with φ(d) = 0, we find that φ(m ′ ) = 0, and then applying the same principle once more leads to φ(m) = 0. Thus, φ vanishes everywhere on U ′′ , contradicting the assumption that 0 is generic. The assumption that ϕ 0 is nonconstant yields the existence of some c ∈ C(0) \ {0} such that φ(c) is not coherent with φ(a). Then clearly, c is coherent neither with a, nor with b. Lemma 2.4 yields some d ∈ V \ C(0) that is coherent with a, b, and c. But then φ(d) is coherent with φ(a) and φ(b), and thus φ(d) belongs to the coherent line through these two points, which goes through 0. The coherency of φ(d) with φ(c) then yields that φ(d) = 0.

As in the previous lemma we can find a neighborhood U of d such that for every m ∈ U the point φ(m) is coherent with t φ(a) and s φ(a) for some nonzero real numbers t, s with t = s, and φ(m) is coherent also to r φ(c) for some nonzero real number r. But then, as before, φ(m) = 0 for all m ∈ U , contradicting our assumption that 0 is generic. ✷ Lemma 4.7 Let φ : V → V be as in Proposition 4.4. Then the map ϕ 0 is a homeomorphism.

Proof. The map ϕ 0 is injective and continuous from Q to itself. By the invariance of domain theorem ϕ 0 (Q) is open in Q. On the other hand ϕ 0 (Q) is compact in Q. As Q is connected we deduce that ϕ 0 (Q) = Q. ✷ Lemma 4.8 Let φ : V → V be as in Proposition 4.4. Then for every a ∈ V the map ϕ a is nonconstant.

Proof. Here we will apply the homotopy theory of spheres. The range of the continuous map a ∈ V → ϕ a is arcwise connected in C(Q, Q). Since ϕ 0 is a homeomorphism, every map in its arcwise component in C(Q, Q) is of the same degree which is either 1 or -1, and hence such a map must be nonconstant. ✷ Lemma 4.9 Let φ : V → V be as in Proposition 4.4. Then every a ∈ V is generic for φ.

Proof. Assume this is not true. Then there exists a nonempty open subset U ⊂ V on which φ is constant. Let D be an arbitrary coherent line through some point a of U . We know that φ is continuous and nonconstant on D and hence the range of the restriction of φ to D is uncountable. Therefore we can find b ∈ D such that φ(b) ∈ W, the set of all points in V whose φ-preimages have nonempty interior in V . In particular, a = b and b is generic. We have ab = s p for some nonzero real number s and some vector p ∈ Q.

Conclusion

Now, we can finish the proof of Theorem 1.4. Let φ : V → V be a continuous coherency preserver. Assume that φ is non-degenerate. By Proposition 4.3, there exists a generic point a such that ϕ a is nonconstant. Then, the continuous coherency preserver m → φ(m+a)-φ(m) satisfies the assumptions of Proposition 4.4, and hence it preserves adjacency. Therefore, φ preserves adjacency.

  = 2B(d, a). As d belongs to {ab, b -c} ⊥ , we have 2B(d, a) = 2B(d, b) = 2B(d, c). Therefore, d has the required properties. ✷ 3 A constructive view of degenerate maps

Set

Lemma 4 . 2

 42 The map Φ :V × Q → Q defined by Φ : (a, p) → ϕ a (p)is continuous.

✷ 4 . 4

 44 When 0 is a non-degenerate generic point

Lemma 4 . 5

 45 Let φ : V → V be as in Proposition 4.4. Then φ maps C(0) \ {0} into itself.

  Thus, the affine map t → q(dtb) = q(d) -2B(d, b)t is nonconstant, and hence B(d, b) = 0. Similarly B(d, c) = 0. Thus, as B is continuous we can find a neighborhood U of d such that ∀m ∈ U, B(m, b)B(m, c) = 0. Let m ∈ U . One checks that the sole point of Rb that is coherent with m is q(m) 2B(m,b) b, and ditto for b replaced with c. As φ is continuous and φ(b) = 0 and φ(c) = 0, we deduce that there is a neighborhood U ′ of d such that U ′ ⊂ U and every m ∈ U ′ is coherent with some b ′ ∈ Rb for which φ(b ′ ) = 0 and with some c ′ ∈ Rc for which φ(c ′ ) = 0. Let m ∈ U ′ be such that φ(m) is coherent with 0, and let b ′ and c ′ be as above with respect to m. Then, φ(m), φ(0) = 0 and φ(b ′ ) are pairwise coherent, and hence φ(m) ∈ Rφ(b ′ ). Similarly φ(m) ∈ Rφ(c ′ ) and hence φ(m) = 0 as φ(b ′ ) and φ(c ′ ) are non-collinear. In particular φ(d) = 0 since φ(d) is coherent with φ(a) = 0.

✷ Lemma 4 . 6

 46 Let φ : V → V be as in Proposition 4.4. Then the map ϕ 0 is injective. Proof. Assume on the contrary that there exist non-collinear vectors a and b in C(0) \ {0} such that φ(a) and φ(b) are coherent. The map t → φ(t b) is continuous from the unit interval and takes different values φ(0) and φ(b) on its boundary points. Hence, replacing b by t b for a suitable nonzero t from the unit interval, we may assume further that φ(a) = φ(b).

  If p ′ = p is a vector of Q close enough to p, then b + s p ′ ∈ U . Both lines, one going through b and a, and the other one going through b and b + s p ′ are mapped into the line going through φ(b) and φ(a) = φ(b + s p ′ ). Thus, the map ϕ b is not injective. This contradicts Lemma 4.6 applied to the map m → φ(m + b)φ(b).

✷

  We are now ready to complete the proof of the main result of this section. Proof of Proposition 4.4. Let a and b be adjacent points of V . Then b is generic and ϕ b is nonconstant. Applying Lemma 4.5 to the map m → φ(m + b)φ(b), we conclude that φ((ab) + b)φ(b) is a nonzero element of C(0), as desired.

✷

  Two matrices A and B of H 2 are coherent if and only if A -B is singular, and they are adjacent if and only if A -B has rank 1.Note that this corresponds to the usual notion of adjacency between matrices with the same format. A map φ : H 2 → H 2 is called an adjacency preserver if φ(A) and φ(B) are adjacent whenever A and B are adjacent. Since A and B are coherent if and only if A = B or A and B are adjacent, the problem of describing the general form of coherency preservers is closely related to the one of characterizing adjacency preservers. The study of adjacency preservers was initiated by Hua seven decades ago.

	2 , det) is a Minkowski space (it is isomet-ric to the standard Minkowski space (R 4 , r → r, r ) through (x, y, z, t) →
	t + z x + iy x -iy t -z	).

grant from ARRS, Slovenia.

The existence of a non-degenerate generic point

Let φ : V → V be a non-degenerate continuous coherency preserver. We denote by W ⊂ V the set of all points b ∈ V such that φ -1 ({b}) has nonempty interior in V . By separability of V , the set W is countable. We will call a point a ∈ V generic if φ(a) ∈ W. Proposition 4.3 Let φ : V → V be a non-degenerate continuous coherency preserver. Then there exists a generic point a such that ϕ a is nonconstant.

Proof. We perform a reductio ad absurdum by assuming that ϕ a is constant for every generic point a. There are three main steps: we shall successively prove:

• That ϕ a is constant for all a ∈ V ;

• That a → ϕ a is constant on V ;

• And finally that φ maps V into a coherent line.

Let a ∈ V . We shall prove that ϕ a is locally constant. Let p ∈ Q. As φ is continuous and nonconstant on the line a + Rp, it takes uncountably many values on it, and hence we can find b ∈ a + Rp that is generic and such that

Let us consider an arbitrary normon V , and denote by M ≥ 1 a constant satisfying the conclusion of Lemma 2.3 for that norm. As φ is continuous at b, we can choose ε > 0 such that φ(m) = φ(a) for all m ∈ V such that m-b ≤ ε. We conclude that ϕ a (π a (b ′ )) = ϕ a (π a (b)) for every generic point b ′ in ∆. However, the set of all vectors ϕ a (π a (b ′ )) = π φ(a) (φ(b ′ )), with non-generic b ′ ∈ ∆, is countable. It follows that ϕ a ( ∆) is countable. On the other hand, as ϕ a is continuous and ∆ is connected the space ϕ a ( ∆) is connected. We conclude that ϕ a ( ∆) consists of a single point, which proves our claim.

Thus, ϕ a is locally constant. As Q is connected and ϕ a is continuous we conclude that ϕ a is constant. Now, we move on to our second step. Let a and b be two points of V . Assume first that a and b are adjacent. Denoting by p the sole vector in Q such that φ((ab)) ⊂ φ(a) + Rp, we see that both maps ϕ a and ϕ b take the value p,


Vahagn Nersesyan
email: vahagn.nersesyan@uvsq.fr

A proof of approximate controllability of the 3D Navier-Stokes system via a linear test

Keywords: AMS subject classifications: 35Q30, 35Q31, 35Q35, 93B05, 93B18 Navier-Stokes system, linearised Euler system, approximate controllability, return method, linear test, saturation property

We consider the 3D Navier-Stokes system driven by an additive finitedimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearised around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity.

Introduction

In this paper, we consider the 3D Navier-Stokes (NS) system for incompressible viscous fluids on the torus T 3 = R 3 /2πZ 3 :

∂ t u -ν∆u + u, ∇ u + ∇p = f (t, x), div u = 0, (0.1)
where ν > 0 is the viscosity of the fluid, u = (u 1 (t, x), u 2 (t, x), u 3 (t, x)) and p = p(t, x) are the unknown velocity field and pressure, and f is an external force.

We fix any T > 0 and assume that the force is of the form

f (t, x) = h(t, x) + η(t, x), t ∈ J T = [0, T ], x ∈ T 3 ,
where h : J T × T 3 → R 3 is a given smooth function and η is a control taking values in some subspace H ⊂ H k (T 3 , R 3 ), k ≥ 3. The subspace H incorporates different constraints that might be imposed on the control; in the examples considered in this paper, it gives the Fourier modes that are directly perturbed by the control force. We are mostly interested here by the situation when H is a finite-dimensional subspace not depending on the viscosity. Projecting Eq. (0.1) to the space H of divergence-free vector fields with zero mean value (see (0.9)) and assuming that h(t) and η(t) belong to H for any t ∈ J T , we eliminate the pressure term from the problem and obtain an evolution equation for the velocity field:

u + νLu + B(u) = h + η. (0.2)
This equation is supplemented with the initial condition u(0) = u 0 . (0.3)

Recall that, for any u 0 ∈ H k = H k (T 3 , R 3 ) ∩ H, problem (0.2), (0.3) has a unique local-in-time strong solution (see Section 1.1).

In this introduction, we formulate a simplified version of our main result, assuming that the subspace H is given by H = span{l(ℓ) sin ℓ, x , l(ℓ) cos ℓ, x : |ℓ| ≤ 2, ℓ ∈ Z 3 * }, (0.4)

where {l(ℓ), l(-ℓ)} is an arbitrary orthonormal basis in {x ∈ R 3 : ℓ, x = 0}.

Main Theorem. Eq. (0.2) is approximately controllable in small time by Hvalued controls, i.e., for any initial condition u 0 ∈ H k+1 , any target u 1 ∈ H k+1 , and sufficiently small δ > 0, there is a control η δ ∈ L 2 (J T δ , H) and a strong solution u of problem (0.2), (0.3) defined on J T δ such that u(T δ) → u 1 in H k as δ → 0 + . (0.5)

Moreover, the control η δ can be chosen in the form

η δ = R δ (u 0 , u 1 ) + ζ δ , (0.6)
where R δ : H k × H k → L 2 (J T δ , H) is a linear bounded operator with a finitedimensional range and ζ δ ∈ L 2 (J T δ , H), both R δ and ζ δ do not depend on (u 0 , u 1 ). Limit (0.5) is uniform with respect to u 0 and u 1 in a bounded set of H k+1 .

A more general version of this result is given in Section 2. In particular, we define there a saturation property that implies small time approximate controllability for different subspaces H spanned by eigenfunctions of the Stokes operator. As a consequence of the Main Theorem, we obtain the following approximate controllability property in fixed time.

Corollary. Eq. (0.2) is approximately controllable in time T > 0 by H-valued controls, i.e., for any ε > 0 and any u 0 , u 1 ∈ H k , there is a control η ∈ L 2 (J T , H) and a strong solution u of Eq. (0.2) defined on J T such that

u(T ) -u 1 H k < ε.
Roughly speaking, this result is obtained by applying the Main Theorem on a small time interval, then by forcing the trajectory to remain near u 1 for sufficiently long time.

The problem of controllability of PDEs with an additive finite-dimensional force has been studied by many authors in the recent years. Agrachev and Sarychev [AS05, AS06, AS08] were the first who considered this problem; they established the approximate controllability of the NS and Euler systems on the 2D torus. Shirikyan generalised their approach to study the NS system on the 3D torus [START_REF] Shirikyan | Approximate controllability of three-dimensional Navier-Stokes equations[END_REF][START_REF] Shirikyan | Exact controllability in projections for threedimensional Navier-Stokes equations[END_REF] and the Burgers equation on the real line [START_REF] Shirikyan | Approximate controllability of the viscous Burgers equation on the real line[END_REF] and on a bounded interval with Dirichlet boundary conditions [START_REF] Shirikyan | Control theory for the Burgers equation: Agrachev-Sarychev approach[END_REF]. Rodrigues and Phan [START_REF] Rodrigues | Navier-Stokes equation on the rectangle: Controllability by means of low mode forcing[END_REF][START_REF] Phan | Approximate controllability for Navier-Stokes equations in 3D rectangles under Lions boundary conditions[END_REF] considered the 2D and 3D NS systems on rectangles with Lions boundary conditions. Compressible and incompressible 3D Euler systems were studied by Nersisyan [START_REF] Nersisyan | Controllability of 3D incompressible Euler equations by a finite-dimensional external force[END_REF][START_REF] Nersisyan | Controllability of the 3D compressible Euler system[END_REF], and the 2D cubic Schrödinger equation by Sarychev [START_REF] Sarychev | Controllability of the cubic Schrödinger equation via a low-dimensional source term[END_REF]. More recently, the author considered the approximate controllability of Lagrangian trajectories of the 3D NS system [START_REF] Nersesyan | Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force[END_REF] and parabolic PDEs with polynomially growing nonlinearities [START_REF] Nersesyan | Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension[END_REF]. Boulvard et al. [START_REF] Boulvard | Controllability and ergodicity of 3D primitive equations driven by a finite-dimensional force[END_REF] considered the 3D system of primitive equations of meteorology and oceanology with control acting directly only on the temperature equation. The proofs of these papers are based on infinite-dimensional extensions of Lie algebraic methods. Most of them provide sharp results, in the sense that they give necessary and sufficient conditions on the Fourier modes that should be perturbed by the control in order to ensure approximate controllability.

In this paper, we take a different route. We proceed by developing an approach by Coron [START_REF] Coron | On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions[END_REF], who considered the approximate controllability of the 2D NS system with Navier slip boundary conditions and used control forces that are localised in the physical space or on the boundary. That approach, called return method, has been later used by Coron and Fursikov [CF96] to study the global exact controllability to trajectories of the 2D NS system on manifolds without boundary, by Coron and Glass [Cor96a,[START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF] to consider the global exact boundary controllability of the 2D and 3D Euler systems, by Fursikov and Imanuilov [START_REF] Fursikov | Exact controllability of the Navier-Stokes and Boussinesq equations[END_REF] to study the global exact controllability to trajectories of the 3D Boussinesq system, and by many other authors. We refer the reader to the Chapter 6 of the book [START_REF] Coron | Control and Nonlinearity[END_REF] for a detailed discussion of the return method, for applications to different control problems, and for more references.

The present paper is the first to extend this method to the case of forces that are localised in the Fourier space. The configuration we use here does not provide sharp results1 in terms of the number of Fourier modes directly perturbed by the control, but gives new and simpler proof with new information about the structure of the control. Roughly speaking, the idea of the proof consists in developing u(t) as follows:

u(t) = δ -1 w(δ -1 t) + v(δ -1 t) + r δ (t) for small δ > 0,
where w(t) is a suitable solution of the Euler system (cf. (1.5)) and v(t) is a solution of the Euler system linearised around w(t) (cf. (1.6)); both correspond to some controls taking values in the subspace H defined by (0.4). We take w(t) in the form

w(t) = ℓ∈Z 3 * ,|ℓ|≤1 (ψ c ℓ (t)l(ℓ) cos ℓ, x + ψ s ℓ (t)l(ℓ) sin ℓ, x ) , (0.7)
where the functions {ψ c ℓ , ψ s ℓ } ⊂ W 1,2 (J T , R) are chosen such that the boundary conditions

ψ c ℓ (0) = ψ c ℓ (T ) = ψ s ℓ (0) = ψ s ℓ (T ) = 0 (0.8)
are satisfied and the derivatives { ψc ℓ , ψs ℓ } form an observable family. Replacing the expression (0.7) of the function w(t) into the left hand side of the Euler system, we infer that w(t) is indeed a solution corresponding to some H-valued control. The observability property implies that the linearised Euler system is approximately controllable by H-valued controls. Furthermore, choosing η in the form (0.6), we show that sup t∈J T δ r δ (t) H k → 0 as δ → 0. In view of (0.8), this implies that u(t) behaves like v(t) at the endpoints 0 and T δ as δ → 0. Then the approximate controllability of the linearised Euler equation allows to conclude (0.5). The operator R δ in (0.6) is an approximate right inverse of the resolving operator of the linearised Euler system and ζ δ is explicitly given in terms of the solution w and the corresponding control.

The proof of the Main Theorem is general enough and can be applied to many other equations, such as the complex Ginzburg-Landau equation, the Euler system, and parabolic PDEs with polynomial nonlinearities.

This paper is organised as follows. In Section 1, we formulate a perturbative result on solvability of the 3D NS system and explain under what conditions its approximate controllability can be derived from that of the linearised Euler system. In Section 2, we show that the conditions in Section 1 are satisfied when a saturation property holds for the set of controlled Fourier modes. Finally, in Section 3, we discuss the validity of the saturation property.

Acknowledgement

This research was supported by the ANR through the grant NONSTOPS ANR-17-CE40-0006-02.

Notation

Here we collect some notation used in this paper. Z 3 is the integer lattice in R 3 , Z 3 * = Z 3 \{0}, and T 3 is the torus R 3 /2πZ 3 . L p (T 3 , R 3 ), 1 ≤ p < ∞ is the Lebesgue space endowed with the norm • L p . H k (T 3 , R 3 ) is the Sobolev space of order k ≥ 1 endowed with the scalar product (•, •) k and the corresponding norm

• k . H k = H k (T 3 , R 3 ) ∩ H, where H = u ∈ L 2 (T 3 , R 3 ) : div u = 0 in T 3 , T 3 u(x)dx = 0 . (0.9)
H is endowed with the L 2 scalar product •, • and the corresponding norm • .

Let X be a Banach space with the norm • X . Then B X (a, R) denotes the closed ball in X of radius R > 0 centred at a ∈ X. C(J T , X) is the space of continuous functions u :

J T = [0, T ] → X endowed with the norm u C(JT ,X) = max t∈JT u(t) X .
L p (J T , X), 1 ≤ p < ∞ is the space of measurable functions u : J T → X with the norm

u L p (JT ,X) = T 0 u(t) p X dt 1/p . L p loc (R + , X)
is the space of measurable functions u : R + → X whose restriction to J T belongs to L p (J T , X) for any T > 0.

W m,p (J T , X), m ≥ 1 is the space of functions u : J T → R such that d i dt i u ∈ L p (J T , X) for 0 ≤ i ≤ m.
Throughout this paper, the same letter C is used to denote unessential positive constants that may change from line to line.

1 Linear test for approximate controllability 1.1 Perturbative result Projecting the NS system to the space H, we rewrite it in the following equivalent form without pressure term

u + νLu + B(u) = f, (1.1) u(0) = u 0 , (1.2)
where L = -∆ is the Stokes operator, B(u) = Π( u, ∇ u), and Π is the Leray orthogonal projection onto H in L 2 . In this section, we recall a perturbative result for problem (1.1), (1.2). Let us take any integer k ≥ 3 and define the space

X T,k = C(J T , H k ) ∩ L 2 (J T , H k+1 ) endowed with the norm u X T ,k = u C(JT ,H k ) + u L 2 (JT ,H k+1 ) . Proposition 1.1. Let û0 ∈ H k and f ∈ L 2 loc (R + , H k-1
). There is a maximal time T * = T * (û 0 , f ) > 0 and a unique solution û of problem (1.1), (1.2) with u 0 = û0 and f = f whose restriction to the interval J T belongs to X T,k for any

T < T * . If T * < ∞, then û(t) k → +∞ as t → T - * . For any T < T * , there are numbers κ = κ(T, Λ) > 0 and C = C(T, Λ) > 0, where Λ = û X T ,k + f L 2 (JT ,H k-1 ) , such that (i) for any u 0 ∈ H k and f ∈ L 2 (J T , H k-1 ) satisfying u 0 -û0 k + f -f L 2 (JT ,H k-1 ) < κ, (1.3)
there is a unique solution u ∈ X T,k of problem (1.1), (1.2);

(ii) let S be the resolving operator of problem (1.1), (1.2), i.e., the mapping taking (u 0 , f ) satisfying (1.3) to the solution u. Then

S(u 0 , f ) -S(û 0 , f ) X T ,k ≤ C u 0 -û0 k + f -f L 2 (JT ,H k-1 ) .
Moreover, the problem is regularising in the sense that, when f is smooth, the restriction of S(û 0 , f ) to the interval (0, T * ] belongs to C ∞ ((0, T * ] × T 3 ).

See, for exemple, Chapter 17 in [START_REF] Taylor | Partial Differential Equations[END_REF] for the proof of local well-posedness and regularising property. The properties (i) and (ii), under these regularity assumptions, are proved in Theorem 1.3 in [START_REF] Nersesyan | Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force[END_REF], using standard arguments.

In what follows, we fix any time T > 0, any integer k ≥ 3, and any function h ∈ L 2 (J T , H k-1 ), and assume that

f = h + η. Let us denote by Θ(u 0 , h, T ) the set of functions η ∈ L 2 (J T , H k-1 ) such that problem (1.1), (1.2) has a solution u ∈ X T,k . In view of Proposition 1.1, the set Θ(u 0 , h, T ) is open in L 2 (J T , H k-1
). We denote by S t (u 0 , h + η) the restriction of the solution at time t < T * (u 0 , h + η).

Formulation and proof

By developing the arguments of [START_REF] Coron | On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions[END_REF], we show in this section how the approximate controllability of the NS system u + νLu + B(u) = h + η

(1.4) can be derived from the approximate controllability of the linearised Euler system. More precisely, we consider the Euler system

ẇ + B(w) = ζ, (1.5) and its linearisation v + Q(v, w) = g, (1.6) where Q(v, w) = B(v, w) + B(w, v), B(v, w) = Π( v, ∇ w).
(1.7)

The functions η, ζ, and g are considered as controls taking values in the same (finite or infinite-dimensional) subspace H of H k+1 . We will use the following two conditions.

(C 1 ) There is a function ζ ∈ L 2 (J T , H) and a solution w ∈ C(J T , H k+2 ) ∩ W 1,2 (J T , H k+1 ) of Eq. (1.5) such that w(0) = w(T ) = 0, (1.8) Lw(t) ∈ H for t ∈ J T .
(1.9) (C 2 ) The linear Eq. (1.6), with a reference trajectory w as in Condition (C 1 ), is approximately controllable in time T > 0, i.e., for any ε > 0 and any v 1 ∈ H k+1 , there is a control g ∈ L 2 (J T , H) such that the solu-

tion v ∈ C(J T , H k+1 ) ∩ W 1,2 (J T , H k ) of Eq. (1.6) with initial condition v(0) = 0 satisfies v(T ) -v 1 k+1 < ε.
Proposition 1.2. Let H be a subspace of H k+1 such that Condition (C 1 ) is satisfied. Then for any u 0 ∈ H k+1 , any g ∈ L 2 (J T , H), and sufficiently small δ > 0, there is a control

η δ ∈ Θ(u 0 , h, T δ) ∩ L 2 (J T δ , H) such that S T δ (u 0 , h + η δ ) → v(T ) in H k as δ → 0 + , (1.10) where v ∈ C(J T , H k+1 ) ∩ W 1,2 (J T , H k ) is the solution of Eq. (1.6) with initial condition v(0) = u 0 . Moreover, η δ is given explicitly by η δ = δ -1 g(δ -1 t) + δ -2 ζ(δ -1 t) + νδ -1 Lw(δ -1 t), t ∈ J T δ , (1.11)
and limit (1.10) is uniform with respect to u 0 in a bounded set of H k+1 .

Proof.

Step 1. Preliminaries. Let us take any M > 0, any u 0 ∈ B H k+1 (0, M ), and any η ∈ L 2 loc (R + , H) and denote by u(t) = S t (u 0 , h + η), t < T * = T * (u 0 , h + η) the solution of problem (1.4), (1.2). Following [START_REF] Coron | On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions[END_REF], we make a time substitution and consider the functions

v δ (t) = v(δ -1 t), g δ (t) = δ -1 g(δ -1 t), w δ (t) = δ -1 w(δ -1 t), ζ δ (t) = δ -2 ζ(δ -1 t), r(t) = u(t) -v δ (t) -w δ (t), t < T δ = min{T δ, T * }.
(1.12)

Assume that we have found a control η = η δ ∈ L 2 (J T , H) such that

T δ < T δ * = T * (u 0 , h + η δ ) for small δ > 0. (1.13)
Then, in view of the equalities (1.8), (1.12), and v(0) = u 0 , we have

r(0) = 0, r(T δ) = u(T δ) -v(T ).
(1.14) Thus, we need to choose η δ ∈ L 2 (J T , H) such that, in addition to (1.13), also the following limit holds

r(T δ) k → 0 as δ → 0 + , (1.15)
uniformly with respect to u 0 ∈ B H k+1 (0, M ). This will imply limit (1.10).

Step 2. Proof of (1.13) and (1.15). The functions v δ (t) and w δ (t), t ∈ J T satisfy the equations

vδ + Q(v δ , w δ ) = g δ , ẇδ + Q(w δ ) = ζ δ .
This implies that r is a solution of the equation

ṙ + νLr + B(r, r + v δ + w δ ) + B(v δ + w δ , r) = ξ δ , t < T δ , (1.16)
where

ξ δ = h + η δ -νLv δ -νLw δ -B(v δ ) -g δ -ζ δ .
Choosing

η δ = g δ + ζ δ + νLw δ ∈ L 2 (J T δ , H) (cf.
(1.11)), we get

ξ δ = h -νLv δ -B(v δ ).
(1.17)

Taking the scalar product in H of Eq. (1.16) with L k r, integrating by parts, then integrating in time, and using the first equality in (1.14), we obtain

1 2 r 2 k + ν t 0 r 2 k+1 ds = t 0 ξ δ , L k r ds - t 0 B(r, r + v δ + w δ ), L k r ds - t 0 B(v δ + w δ , r), L k r ds = I 1 + I 2 + I 3 . (1.18)
To estimate I 1 , we integrate by parts and use (1.17) and the inequalities of Cauchy-Schwarz and Young:

|I 1 | ≤ t 0 ξ δ k-1 r k+1 ds ≤ C t 0 h 2 k-1 + νLv δ 2 k-1 + B(v δ ) 2 k-1 ds + ν 4 t 0 r 2 k+1 ds.
By a change of variable, we have

t 0 h 2 k-1 ds ≤ T δ 0 h 2 k-1 ds, t 0 Lv δ 2 k-1 ds ≤ δ T 0 v 2 k+1 ds, t 0 B(v δ ) 2 k-1 ds ≤ δ T 0 B(v) 2 k-1 ds ≤ Cδ T 0 v 2 k ds, t ∈ J T δ .
Thus, there is

ε δ = ε δ (M ) > 0 not depending on t ∈ J T δ and u 0 ∈ B H k+1 (0, M ) such that ε δ → 0 as δ → 0 + and |I 1 | ≤ ε δ + ν 4 t 0 r 2 k+1 ds, t < T δ . (1.19)
We estimate I 2 and I 3 as follows:

|I 2 | ≤ t 0 | B(r, r + v δ + w δ ), L k r | ds ≤ C t 0 B(r) 2 k-1 + | B(r, v δ + w δ ), L k r | ds + ν 4 t 0 r 2 k+1 ds, |I 3 | ≤ t 0 | B(v δ + w δ , r), L k r | ds. Note that t 0 B(r) 2 k-1 ds ≤ C t 0 r 4 k ds, t 0 | B(r, v δ + w δ ), L k r | ds ≤ C t 0 ( v δ k+1 + w δ k+1 ) r 2 k ds, t 0 | B(v δ + w δ , r), L k r | ds ≤ C t 0 ( v δ k + w δ k ) r 2 k ds,
where we used the inequalities (see [START_REF] Constantin | Navier-Stokes equations[END_REF])

| B(a, b), L k b | ≤ C a k b 2 k , | B(a, b), L k c | ≤ C a k b k+1 c k , a, b ∈ H k , c ∈ H k+1 . Thus |I 2 + I 3 | ≤ C t 0 ( v δ k+1 + w δ k+1 ) r 2 k ds + C t 0 r 4 k ds + ν 4 t 0 r 2 k+1 ds.
Combining this with (1.18) and (1.19), we obtain

r 2 k ≤ ε δ + C t 0 ( v δ k+1 + w δ k+1 ) r 2 k ds + C t 0 r 4 k ds, t < T δ .
By the Gronwall inequality,

r 2 k ≤ ε δ + C t 0 r 4 k ds exp C t 0 ( v δ k+1 + w δ k+1 ) ds .
For δ ≤ δ 0 (M ) and t ∈ J T δ , we have

t 0 ( v δ k+1 + w δ k+1 ) ds = tδ 0 (δ v k+1 + w k+1 ) ds ≤ 1. Thus r 2 k ≤ ε δ + C t 0 r 4 k ds, t < T δ , (1.20)
where C = C(M ) > 0 does not depend on t, δ, and u 0 , and by the same letter ε δ we denote e C ε δ . Let us set

Φ(t) = ε δ + C t 0 r 4 k ds. Inequality (1.20) implies that ( Φ) 1/2 ≤ C Φ, which is equivalent to Φ/Φ 2 ≤ C.
Integrating the latter, we obtain

Φ(t) ≤ ε δ (1 -Cε δ t) -1 , t < T δ .
Choosing δ sufficiently small, we see that

Φ(t) ≤ 2ε δ < 1, t < T δ .
This implies both assertions (1.13) and (1.15) and completes the proof of the proposition.

The following is the main result of this section.

Theorem 1.3. Let H be a subspace of H k+1 such that Conditions (C 1 ) and (C 2 ) are satisfied. Then Eq. (1.4) is approximately controllable in small time by Hvalued controls, i.e., for any u 0 , u 1 ∈ H k+1 and sufficiently small δ > 0, there is a control η δ ∈ Θ(u 0 , h, T δ) ∩ L 2 (J T δ , H) such that

S T δ (u 0 , h + η δ ) → u 1 in H k as δ → 0 + . (1.21)
Moreover, the control η δ can be chosen in the form

η δ = R δ (u 0 , u 1 ) + ζ δ , (1.22)
where R δ :

H k × H k → L 2 (J T δ , H
) is a linear bounded operator with a finitedimensional range and ζ δ ∈ L 2 (J T δ , H), both R δ and ζ δ do not depend on (u 0 , u 1 ). Limit (1.21) is uniform with respect to u 0 and u 1 in a bounded set of H k+1 .

Proof. Let us denote by

A :

H k+1 × L 2 (J T , H) → C(J T , H k+1 ) ∩ W 1,2 (J T , H k ), (v 0 , g) → v
the resolving operator of Eq. (1.6) with the initial condition v(0) = v 0 , and let A t be its restriction at time t. By Condition (C 2 ), the image of the mapping

A T (0, •) : L 2 (J T , H) → H k+1
is dense in H k+1 . Hence, we can construct an approximate right inverse for A T (0, •). More precisely, by Proposition 2.6 in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], for any ε > 0, there is a linear bounded operator R ε :

H k → L 2 (J T , H) such that A T (0, R ε f ) -f k ≤ ε f k+1 for f ∈ H k+1 .
Now let us take any M > 0 and any u 0 , u 1 ∈ B H k+1 (0, M ). Applying the previous inequality with f = u 1 -A T (u 0 , 0), we get

A T (u 0 , g ε ) -u 1 k ≤ ε u 1 -A T (u 0 , 0) k+1 ≤ εC,
where g ε = R ε (u 1 -A T (u 0 , 0)) and C = C(M ) > 0 is a constant. Combining this with Propositions 1.1 and 1.2, we complete the proof of the theorem.

We close this section with the following result.

Corollary 1.4. Assume that the conditions of Theorem 1.3 are satisfied. Then Eq. (1.4) is approximately controllable in time T > 0 by H-valued controls, i.e., for any ε > 0 and any u

0 , u 1 ∈ H k , there is a control η ∈ Θ(u 1 , h, T ) ∩ L 2 (J T , H) such that S T (u 0 , h + η) -u 1 k < ε.
Proof. By the regularising property of the NS system (see Proposition 1.1), the solution corresponding to initial point u 0 ∈ H k and control η = 0 becomes instantaneously smooth. Combining this with Theorem 1.3, we see that Eq. (1.4) is approximately controllable in small time in the sense that, for any

u 0 , u 1 ∈ H k , there is a control ηδ ∈ Θ(u 0 , h, T δ) ∩ L 2 (J T δ , H) such that S T δ (u 0 , h + ηδ ) → u 1 in H k as δ → 0 + . (1.23)
Thus, to prove approximate controllability in fixed time, it suffices to show that, for any T, ε > 0 and any u 1 ∈ H k , there is a control

η 1 ∈ Θ(u 1 , h, T ) ∩ L 2 (J T , H) such that S T (u 1 , h + η 1 ) -u 1 k < ε,
where the initial condition and the target coincide with u 1 . By Proposition 1.1, there is a time τ > 0 such that the control η = 0 is in the set Θ(u 1 , h, τ ) and the function S • (u 1 , h) : J τ → H k is continuous. Taking τ sufficiently small and using the property (ii) in Proposition 1.1, we find a number r ∈ (0, ε) such that η = 0 belongs to Θ(v, h, τ ) for any v ∈ B H k (u 1 , r) and

S t (v, h) -u 1 k < ε, t ∈ J τ .
Thus starting from any initial point v ∈ B H k (u 1 , r), the solution corresponding to η = 0 remains in the ball B H k (u 1 , ε) on the time interval J τ . If τ > T , then the proof is complete. Otherwise, applying (1.23) with initial point u ′ 0 = S τ (v, h) and target u 1 , we find a small time

T ′ < T -τ and a control η 2 ∈ Θ(u ′ 0 , h, T ′ ) ∩ L 2 (J T ′ , H) such that S T ′ (u ′ 0 , h + η 2 ) -u 1 k < r.
By the choice of r and τ , if 2τ + T ′ > T , then again the proof is complete. Otherwise, we complete the proof by iterating the above argument finitely many times.

Remark 1.5. In this corollary, the control η is not of the form (1.22). The affine dependence on (u 0 , u 1 ) is lost after the first application of zero control in the r-neighborhood of u 1 . Indeed, this comes from the fact that S t (u 1 , 0) is nonlinear in u 1 . Analysing the above proof, we easily see that for given ε, M > 0 and any u 0 , u 1 ∈ B H k+1 (0, M ), the restriction η| [0,T δ] of the control is of the form (1.22), while the restriction η| [T δ,T ] does not depend on u 0 .

Proof of the Main Theorem

The goal of this section is to show that Conditions (C 1 ) and (C 2 ) are verified for different subspaces H spanned by a finite number of eigenfunctions of the Stokes operator. Also we prove the Main Theorem formulated in the Introduction.

More general formulation

Let us fix any numbers α i > 0, i = 1, 2, 3 and endow the space R 3 with the scalar product

x, y α = 3 i=1 α -1 i x i y i .
For any ℓ ∈ Z 3 * , let us denote

c ℓ (x) = l(ℓ) cos ℓ, x α , s ℓ (x) = l(ℓ) sin ℓ, x α ,
where {l(ℓ), l(-ℓ)} is any orthonormal basis in the hyperplane

ℓ ⊥α = {x ∈ R 3 : x, ℓ α = 0}.
The family {c ℓ , s ℓ } ℓ∈Z 3 * is a complete orthogonal system in H k composed of eigenfunctions of the Stokes operator. Let K ⊂ Z 3 * be a finite symmetric set (i.e., K = -K). We associate with K a non-decreasing sequence of finite-dimensional subspaces by

H 0 (K) = span{c ℓ , s ℓ : ℓ ∈ K},
(2.1)

H i (K) = span{η 1 + Q(η 2 , ξ) : η 1 , η 2 ∈ H i-1 (K), ξ ∈ H 0 (K)}, i ≥ 1, (2.2)
where Q is the bilinear form defined by (1.7).

Definition 2.1. We say that

K ⊂ Z 3 * is saturating if the subspace ∪ ∞ i=1 H i (K) is dense in H k .
The following theorem is proved in the next two subsections.

Theorem 2.2. Assume that K ⊂ Z 3 * is a saturating set. Then Conditions (C 1 ) and (C 2 ) are satisfied for the subspace H = H 1 (K), and therefore the conclusions of Theorem 1.3 and Corollary 1.4 hold.

The following theorem provides a practical way for constructing saturating sets. Recall that K ⊂ Z 3 * is a generator if any vector of Z 3 is a finite linear combination of vectors of K with integer coefficients.

Theorem 2.3. If a finite symmetric set K ⊂ Z 3
* is a generator, then it is saturating.

See Section 3 for a proof of this result. Now we turn to the proof of the results formulated in the Introduction.

Proof of the Main Theorem and the Corollary. For any ℓ ∈ R 3 * , we denote by P ℓ the orthogonal projection in R 3 onto the hyperplane ℓ ⊥α . Then, for any a ∈ R 3 , we have the equalities Π(a cos ℓ, x α ) = (P ℓ a) cos ℓ, x α , Π(a sin ℓ, x α ) = (P ℓ a) sin ℓ, x α .

These equalities and some simple trigonometric computations show that

2Q(a cos ℓ 1 , x α , b sin ℓ 2 , x α ) = cos ℓ 1 -ℓ 2 , x α P ℓ1-ℓ2 ( a, ℓ 2 α b -b, ℓ 1 α a) + cos ℓ 1 + ℓ 2 , x α P ℓ1+ℓ2 ( a, ℓ 2 α b + b, ℓ 1 α a) , (2.3)
2Q(a cos ℓ 1 , x α , b cos ℓ 2 , x α ) = sin ℓ 1 -ℓ 2 , x α P ℓ1-ℓ2 ( a, ℓ 2 α b -b, ℓ 1 α a) -sin ℓ 1 + ℓ 2 , x α P ℓ1+ℓ2 ( a, ℓ 2 α b + b, ℓ 1 α a) , (2.4)
2Q(a sin ℓ 1 , x α , b sin ℓ 2 , x α ) = sin ℓ 1 -ℓ 2 , x α P ℓ1-ℓ2 ( a, ℓ 2 α b -b, ℓ 1 α a) + sin ℓ 1 + ℓ 2 , x α P ℓ1+ℓ2 ( a, ℓ 2 α b + b, ℓ 1 α a) (2.5) for any ℓ 1 , ℓ 2 ∈ Z 3 * , a ∈ ℓ ⊥α 1 , and b ∈ ℓ ⊥α 2 .
Let us consider the set K = {(±1, 0, 0), (0, ±1, 0), (0, 0, ±1)} which is, clearly, a generator. Due to identities (2.3)-(2.5), the subspace H 1 ( K) is contained in the subspace defined by (0.4). Applying Theorems 2.2 and 2.3 with the set K and taking α = (1, 1, 1), we obtain the Main Theorem and the Corollary.

Remark 2.4. The papers [START_REF] Shirikyan | Approximate controllability of three-dimensional Navier-Stokes equations[END_REF][START_REF] Shirikyan | Exact controllability in projections for threedimensional Navier-Stokes equations[END_REF][START_REF] Nersesyan | Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force[END_REF] provide a sharp version of the Corollary regarding the dimension of the control space. In these papers, a nonlinear saturation property is defined for the 3D NS system (1.4), and in the case h ≡ 0 and α = (1, 1, 1), the system is proved to be approximately controllable in time T > 0 by H 0 (K)-valued controls if and only if K is a generator (see Theorem 4.5 in [START_REF] Nersesyan | Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force[END_REF]). The subspace H 1 (K) is strictly larger than H 0 (K).

Checking Condition (C 1 )

Let us denote H = H 1 (K) and consider the function

w(t) = ℓ∈K (ψ c ℓ (t) c ℓ + ψ s ℓ (t) s ℓ ) , (2.6)
where {ψ c ℓ , ψ s ℓ } ℓ∈K are any functions in W 1,2 (J T , R) verifying the boundary conditions

ψ c ℓ (0) = ψ c ℓ (T ) = ψ s ℓ (0) = ψ s ℓ (T ) = 0.
As c ℓ and s ℓ are eigenfunctions of the Stokes operator, we have Lw(t) ∈ H for t ∈ J T . Let us denote ζ = ẇ + B(w) and show that ζ ∈ L 2 (J T , H). Indeed, we have ẇ ∈ L 2 (J T , H) by the construction. Moreover, the equality

B(w) = ℓ1,ℓ2∈K Q(ψ c ℓ1 (t)c ℓ1 + ψ s ℓ1 (t)s ℓ1 , ψ c ℓ2 (t)c ℓ2 + ψ s ℓ2 (t)s ℓ2 )
implies that B(w) ∈ C(J T , H). Thus, Condition (C 1 ) is satisfied.

Checking Condition (C 2 )

Condition (C 2 ) is more subtle and is satisfied under additional hypotheses on the functions {ψ c ℓ , ψ s ℓ } ℓ∈K entering (2.6). We shall use the notion of observable family of functions from [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF].

To prove the approximate controllability of Eq. (1.6), we shall use some arguments close to the ones in Section 4 in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], where the controllability of the linearised 2D NS system is established. An important difference is that there is no diffusion term in Eq. (1.6), so we do not have a parabolic regularisation property. As a consequence, we cannot use the L 2 -adjoint problem and the H k -adjoint is of rather complicated form and seems to be ill-suited for our purposes. Instead of using the adjoint problem, we use the equation satisfied by the derivative of the resolving operator with respect to the initial condition (see (2.13)).

Step 1. Observable family. A family {φ i } n i=1 ⊂ L 2 (J T , R) is said to be observable 2 if for any subinterval J ⊂ J T , any continuous function b : J → R, and any C 1 -functions a i : J → R the equality

b(t) + n i=1 a i (t)φ i (t) = 0 in L 2 (J, R) (2.7)
implies that a i ≡ b ≡ 0, 1 ≤ i ≤ n on J. An example of observable family can be constructed as follows. Let φ i : J T → R be bounded measurable functions having left and right limits at any point of J T . Moreover, let there be disjoint countable dense sets {D i } n i=1 in J T such that φ i is discontinuous on D i and continuous on J T \D i . Then the family {φ i } n i=1 is observable. Indeed, take any 1 ≤ i ≤ n and any s ∈ D i . All the functions φ j , j = i are continuous at s, so the jump at s of the function on the left-hand side of (2.7) is equal to a i (s)(φ i (s + )-φ i (s -) = 0. It follows that a i (s) = 0 for any s ∈ D i , hence a i ≡ 0 on J, by density and continuity. By (2.7), we have also b ≡ 0 on J.

Let us now fix an observable family of functions {φ c ℓ , φ s ℓ } ℓ∈K ⊂ L 2 (J T , R) and denote

ψ c ℓ (t) = φ(t) t 0 φ c ℓ (τ ) dτ, ψ s ℓ (t) = φ(t) t 0 φ s ℓ (τ ) dτ, t ∈ J T ,
where φ : J T → R is a C 1 -function such that φ(t) = 0 if and only if t = T . Of course, Condition (C 1 ) remains true in this case.

Step 2. Reduction. Let us fix any k ≥ 3 and denote by R(t, τ ) :

H k → H k , 0 ≤ τ ≤ t ≤ 1 the two-parameter resolving operator of the linearised problem v + Q(w, v) = 0, v(τ ) = v 0 .
(2.8)

Then A : L 2 (J T , H k ) → H k , g → T 0 R(T, τ )g(τ ) dτ,
is the resolving operator of Eq. (1.6) with initial condition v(0) = 0. Denote by P H : H k → H k the orthogonal projection onto H in H k . Our goal is to show that the image of the linear operator

A 1 : L 2 (J T , H k ) → H k , A 1 = AP H is dense in H k .
It is equivalent to show that the kernel of the adjoint operator

A * 1 : H k → L 2 (J T , H k ), z → P H R(T, τ ) * z is trivial, where R(T, τ ) * : H k → H k is the H k -adjoint of R(T, τ ).
Step 3. Triviality of ker A * 1 . Let us take any z ∈ ker A * 1 and show that z = 0. Indeed, for any g ∈ H, we have (g, R(T, τ ) * z) k = 0 for a.e. τ ∈ J T . This implies that (R(T, τ )g, z) k = 0 for any τ ∈ J T ,

(2.9) by continuity in τ of R(T, τ )g. Let fix any T 1 ∈ (0, T ) and rewrite this equality as follows:

(R(T 1 , τ )g, z 1 ) k = 0 for any τ ∈ J T1 , (2.10)

where z 1 = R(T, T 1 ) * z. Taking τ = T 1 , we obtain (g, z 1 ) k = 0, (2.11)

i.e., z 1 is orthogonal to H = H 1 (K) in H k . Let us show that z 1 is orthogonal also to H 2 (K). To this end, let us denote y(t, τ ) = R(τ + t, τ )g, (2.12) and note that y(t, τ ) is the solution of the problem

ẏ(t, τ ) + Q(w(τ + t), y(t, τ )) = 0, t ∈ (0, T -τ ), y(0, τ ) = g. It follows that Y (t, τ ) = ∂ ∂τ y(t, τ ) is the solution of Ẏ (t, τ ) + Q(w(τ + t), Y (t, τ )) + Q( ẇ(τ + t), y(t, τ )) = 0, t ∈ (0, T -τ ), (2.13) Y (0, τ ) = 0.
(2.14)

On the other hand, taking the derivative in τ of (2.12) and choosing t = T 1 -τ , we obtain

∂ ∂τ R(T 1 , τ )g = Y (T 1 -τ, τ ) -Ṙ(T 1 , τ )g = Y (T 1 -τ, τ ) + Q(w(T 1 ), R(T 1 , τ )g).
(2.15)

In the last equality, we used Eq. (2.8). Taking the derivative of (2.10) in τ and using equalities (2.13)-(2.15), we arrive at

0 = T1-τ 0 (Q(w(τ + t), Y (t, τ )) + Q( ẇ(τ + t), y(t, τ )), z 1 ) k dt -(Q(w(T 1 ), R(T 1 , τ )g), z 1 ) k = T1-τ 0 (Q(w(τ + t), Y (t, τ )), z 1 ) k dt + T1 τ (Q( ẇ(t), R(t, τ )g), z 1 ) k dt -(Q(w(T 1 ), R(T 1 , τ )g), z 1 ) k .
Differentiating this in τ , we get

b(τ ) + ℓ∈K (a c ℓ (τ )φ c ℓ (τ ) + a s ℓ (τ )φ s ℓ (τ )) = 0 for τ ∈ J T1 , where b(τ ) = ∂ ∂τ T1-τ 0 (Q(w(τ + t), Y (t, τ )), z 1 ) k dt -Q( φ(τ ) w(τ ), g), z 1 k + T1 τ Q( ẇ(t), ∂ ∂τ R(t, τ )g), z 1 k dt - ∂ ∂τ (Q(w(T 1 ), R(T 1 , τ )g), z 1 ) k , a c ℓ (τ ) = -φ(τ ) (Q(c ℓ , g), z 1 ) k , a s ℓ (τ ) = -φ(τ ) (Q(s ℓ , g), z 1 ) k , w(τ ) = ℓ∈K τ 0 (φ c ℓ (t) c ℓ + φ s ℓ (t) s ℓ ) dt.
The functions {a c ℓ , a s ℓ } ℓ∈K are continuously differentiable and b is continuous on J T1 . By observability of {φ c ℓ , φ s ℓ } ℓ∈K , we have thus a c ℓ ≡ a s ℓ ≡ 0 on J T1 for any ℓ ∈ K. As a consequence, (Q(c ℓ , g), z 1 ) k = (Q(s ℓ , g), z 1 ) k = 0, which, combined with (2.11), implies that z 1 is orthogonal to H 2 (K). Recalling the definition of z 1 , we conclude that (R(T, T 1 )g, z) k = 0 for any T 1 ∈ (0, T ) and g ∈ H 2 (K).

Denoting T 1 by τ , we obtain (2.9), but now for any g in H 2 (K). Iterating this argument, we prove (2.9) for any g ∈ ∪ ∞ i=1 H i (K). Taking τ = T and using the saturation hypothesis, we get that z = 0. This completes the proof of Condition (C 2 ) and that of Theorem 2.2.

Saturation property

In this section, we prove Theorem 2.3. In what follows, we write ℓ 1 ∦ ℓ 2 to indicate that the vectors ℓ 1 , ℓ 2 ∈ R 3 are non-parallel.

Step 1. Reduction. Let us define a sequence of finite symmetric sets in Z 3 as follows:

K 0 = K, K j = K j-1 ∪ {ℓ 1 + ℓ 2 : ℓ 1 ∈ K j-1 , ℓ 2 ∈ K, ℓ 1 ∦ ℓ 2 }, j ≥ 1.
As K is a generator, this sequence is strictly increasing and

∪ ∞ j=1 K j = Z 3 . (3.1)
Let us assume that we have shown the inclusion

H i (K j ) ⊂ H i+3 (K j-1 ) for any i ≥ 0, j ≥ 1, (3.2)
where H i (K j ) are the subspaces defined by (2.1) and (2.2) with K = K j and c 0 = s 0 = 0. Then (3.2) implies that

H 0 (K j ) ⊂ H 3 (K j-1 ) ⊂ H 6 (K j-2 ) ⊂ . . . ⊂ H 3j (K).
Combining this with (3.1), we see that the subspace ∪ ∞ j=1 H j (K) is dense in H k , i.e., K is saturating. Thus we need to prove (3.2).

Step 2. Proof of (3.2). We first consider a particular case.

Step 2.1. Let us take any ℓ 1 ∈ K j-1 and ℓ 2 ∈ K such that ℓ 1 ∦ ℓ 2 and denote by δ = δ(ℓ 1 , ℓ 2 ) one of two unit vectors in ℓ ⊥α 1 ∩ ℓ ⊥α 2 . In this step, we show that δ cos ℓ 1 + ℓ 2 , x α , δ sin ℓ 1 + ℓ 2 , x α ∈ H i+1 (K j-1 ).

(3. This implies that δ cos ℓ 1 + ℓ 2 , x α ∈ H i+1 (K j-1 ). The second inclusion in (3.3) is proved in a similar way.

Step 2.2. Now we prove (3.2). Let us take any r ∈ K such that the family E = {ℓ 1 , ℓ 2 , r} is a linearly independent. This is possible since K is a generator. To simplify notation, let us denote (x, y, z) E = xℓ 1 + yℓ 2 + zr for any x, y, z ∈ R. Then the family {(1, -1, 1) E , (-1, 1, 1) E , (1, 1, -1) E } is linearly independent, so the intersection of the hyperplanes (1, -1, 1) ⊥α E , (-1, 1, 1) ⊥α E , and (1, 1, -1) ⊥α E is {0}. To fix the ideas, let us assume that (1, 1, 1) E / ∈ (1, 1, -1) ⊥α E , (3.6) the cases (1, 1, 1) E / ∈ (-1, 1, 1) ⊥α E and (1, 1, 1) E / ∈ (1, -1, 1) ⊥α E are treated in a similar way. By (3.3), we have δ(ℓ 1 , ℓ 2 ) cos (1, 1, 0) E , x α , δ(ℓ 1 , ℓ 2 ) sin (1, 1, 0) E , x α ∈ H i+1 (K j-1 ).

(3.7)

Now writing (1, 1, 1) E = (1, 1, 0) E + (0, 0, 1) E , recalling that (0, 0, 1) E = r ∈ K, and using (3.5) and (3.7), we obtain cos (1, 1, 1) E , x α P (1,1,1)E ( δ(ℓ 1 , ℓ 2 ), (0, 0, 1) E α b + b, (1, 1, 0) E α δ(ℓ 1 , ℓ 2 )) = Q(δ(ℓ 1 , ℓ 2 ) cos (1, 1, 0) E , x α , b sin (0, 0, 1) E , x α ) + Q(b cos (0, 0, 1) E , x α , δ(ℓ 1 , ℓ 2 ) sin (1, 1, 0) E , x α ) ∈ H i+2 (K j-1 ) (3.8)

for any b ∈ (0, 0, 1) ⊥α E . As the family E is a linearly independent, we have that δ(ℓ 1 , ℓ 2 ), (0, 0, 1) E α = 0. Hence, G = { δ(ℓ 1 , ℓ 2 ), (0, 0, 1) E α b + b, (1, 1, 0) E α δ(ℓ 1 , ℓ 2 ) : b ∈ (0, 0, 1) ⊥α E } is a two-dimensional subspace of R 3 contained in (1, 1, -1) ⊥α E , i.e., G = (1, 1, -1) ⊥α E . Then (3.6) implies that P (1,1,1)E G = (1, 1, 1) ⊥α E . Combining this with (3.8), we derive that c ±(1,1,1)E ⊂ H i+2 (K j-1 ).

(3.9)

In a similar way, one proves that s ±(1,1,1)E ⊂ H i+2 (K j-1 ).

(3.10)

Applying the result of Step 2.1 to the difference (1, 1, 0) E = (1, 1, 1) E -(0, 0, 1) E and using (3.9) and (3.10), we obtain δ((1, 1, 1) E , (0, 0, 1) E ) cos (1, 1, 0) E , x α ∈ H i+3 (K j-1 ).

Combining this with the fact that δ((1, 1, 1) E , (0, 0, 1) E ) ∦ δ(ℓ 1 , ℓ 2 ) and (3.3), we obtain that c ±(ℓ1+ℓ2) ∈ H i+3 (K j-1 ). The proof of s ±(ℓ1+ℓ2) ∈ H i+3 (K j-1 ) is similar. This completes the proof of (3.2).

  3)Indeed, by identity (2.3), we have2Q(b cos ℓ 2 , x α , a sin ℓ 1 , x α ) = -cos ℓ 1 -ℓ 2 , x α P ℓ1-ℓ2 ( a, ℓ 2 α b -b, ℓ 1 α a) + cos ℓ 1 + ℓ 2 , x α P ℓ1+ℓ2 ( a, ℓ 2 α b + b, ℓ 1 α a) (3.4)for any a ∈ ℓ ⊥α 1 , and b ∈ ℓ ⊥α 2 . Summing (2.3) and (3.4), we obtaincos ℓ 1 + ℓ 2 , x α P ℓ1+ℓ2 ( a, ℓ 2 α b + b, ℓ 1 α a) = Q(a cos ℓ 1 , x α , b sin ℓ 2 , x α ) + Q(b cos ℓ 2 , x α , a sin ℓ 1 , x α ). (3.5)We take a = δ and b such that b, ℓ 1 α = 1. This choice is possible since ℓ 1 ∦ ℓ 2 . Then (3.5) becomesδ cos ℓ 1 + ℓ 2 , x α = Q(δ cos ℓ 1 , x α , b sin ℓ 2 , x α ) + Q(b cos ℓ 2 , x α , δ sin ℓ 1 , x α ).

A sharp version of Corollary is obtained in the papers[START_REF] Shirikyan | Approximate controllability of three-dimensional Navier-Stokes equations[END_REF][START_REF] Shirikyan | Exact controllability in projections for threedimensional Navier-Stokes equations[END_REF][START_REF] Nersesyan | Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force[END_REF]. The results of these papers imply, in particular, the approximate controllability in fixed time T > 0 by controls taking values in the smaller subspace span{l(ℓ) sin ℓ, x , l(ℓ) cos ℓ, x : |ℓ| ≤ 1, ℓ ∈ Z 3 * }; see Remark

2.4 for more details.

Note that, the observability property we use here is stronger than the one introduced in Definition 4.1 in[START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF].