Stéphane Simon

S T Éphane

Simon And

Patrick Verovic
email: patrick.verovic@univ-savoie.fr

FUNCTIONS WITH STRICTLY CONVEX EPIGRAPH

Keywords: Mathematics Subject Classification. Primary: 52A07, Secondary: 52A05. Key words and phrases. Convexity, affine geometry, strict convexity, topological vector spaces

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Introduction

This paper is concerned with the relationship between strict convexity of functions defined over a domain in a topological real vector space and strict convexity of their epigraphs. A subset of a topological real vector space is said to be strictly convex if it is convex and if in addition there is no non-trivial segment in its boundary. This notion will be made more precise in Section 2.

Even though strict convexity is less studied than convexity in the literature, there are nevertheless many fields where strict convexity of a subset of a topological real vector space is used. We give here three different examples which illustrate this geometric property.

The first example, which is actually the starting point of the present work, concerns epigraphs of functions. It is a well-known result that a function has a convex epigraph if and only if it is convex. This is a bridge between geometric convexity and analytic convexity (see for example [START_REF] Rockafellar | Convex analysis[END_REF]Theorem 4.1, page 25]). Therefore, a natural question is to know whether the same equivalence holds when replacing "convexity" by "strict convexity". To the best of our knowledge, nothing has been studied about this issue in the litterature, even in the case when the domain of f lies in R n . This is why we propose to fill the gap in the present paper, not only in R n but in the general framework of topological real vector spaces. This is done in the Main Theorem that we state in Section 1.

The second example deals with strict convexity of the unit ball in a normed real vector space (in that case, the norm itself is sometimes called strictly convex, which is unfortunate). This property is equivalent to saying that there exists a real number p > 1 such that the p-th power of the norm is a strictly convex function (we may see Theorem 11.1 in [3, page 110]), and this is actually equivalent to the strict convexity of the epigraph of this function as the Main Theorem will show. It is important to work with such norms since they yield interesting properties in functional analysis. For instance, given a Banach space E with strictly convex unit ball, any non-empty family of commutative non-expansive mappings from a non-empty closed convex and weakly compact subset of E into itself has a common fixed point (see for example [START_REF] Belluce | Normal structure in Banach spaces[END_REF]). Nevertheless, if the unit ball of a normed vector space is not strictly convex, this may be offset in at least two different ways. Indeed, any reflexive Banach space can be endowed with an equivalent norm whose unit ball is strictly convex (see for example [START_REF] Lindenstrauss | On nonseparable reflexive Banach spaces[END_REF]). On the other hand, any separable Banach space can be endowed with an equivalent norm which is smooth and whose unit ball is strictly convex (see for example [6, page 33]).

The third example concerns optimization problems-more precisely, the relationship between strict convexity and uniqueness of minimizers. When dealing with an optimization problem on a topological real vector space, the search for a value of the variable where the cost function achieves a minimum is much more easier in case this function and the constraint set are both convex (see for example [START_REF] Ekeland | Convex analysis and variational problems[END_REF]). Moreover, if the cost function is strictly convex, such a minimizer is then unique. On the other hand, if the constraint set is both strictly convex and given by the epigraph of a function (we shall see in which case this is possible owing to the Main Theorem), and if the cost function has no minimum over the whole space, then such a minimizer is unique too.

As we may notice throughout these three examples, it is of great importance to know whether the epigraph of a function is strictly convex or not.

Of course, for a function defined over R n , the strict convexity of its epigraph is merely equivalent to being strictly convex. But what happens for a function with an arbitrary domain which lies in an arbitrary topological real vector space?

In order to give a complete answer to this question and deduce some of its consequences in Section 1, we shall examine two issues in Section 2: topological aspects of epigraphs of functions defined on any topological space on the one hand, and the notion of strict convexity for sets in general topological real vector spaces (no matter which dimension they have or whether they are Hausdorff) on the other hand.

Finally, Section 3 is devoted to the proofs of all the results we mention in the previous sections.

Motivations, Main Theorem and consequences

The relationship between convexity of sets and convexity of functions is given by the following well-known result that is quite easy to prove. Proposition 1.1. Let C be a subset of a real vector space V and f : C -→ R a function. Then we have the equivalence

C and f are both convex ⇐⇒ Epi(f ) is convex .
This is a geometric way of characterizing the convexity of a function by looking at its epigraph.

Such a property naturally raises the issue of studying what happens when convexity is replaced by strict convexity whose meaning will be given in Definition 2.5 (real vector spaces being of course replaced by arbitrary-possibly non-Hausdorff-topological real vector spaces).

At first sight, we may believe that for a function defined over a general topological real vector space, strict convexity of its epigraph is merely equivalent for the function to be strictly convex in the usual sense. But this is false as we can observe in the following example.

Example 1.1. Consider the real vector space

V := C 0 (R, R) ∩ L 2 (R, R) ⊆ R R
endowed with the topology T of pointwise convergence (this is nothning else than the product topology, which is therefore Hausdorff), and let f :

C := V -→ R be the function defined by f(u) := u 2 2 .
On the one hand, f is strictly convex since for any u ∈ V its Hessian at u with respect to the norm • 2 on V is equal to 2 • , • , and hence positive definite.

On the other hand, whereas (0, 0) and (0, 2) are in the epigraph of f, their midpoint (0, 1) does not belong to ri(Epi(f )) =

• ˚ Epi(f ), that is, Epi(f ) is not a neighborhood of (0, 1) for the product topology on V ×R as we can check with the sequence (u n ) n 1 of V defined by

u n (x) :=            2 » nx -n 2 + 1/n for x ∈ [n -1/n 2 , n] , 2/ √ n for x ∈ [n , 2n] , 2 » 2n 2 -nx + 1/n for x ∈ [2n , 2n + 1/n 2 ]
, and 0 for x n -1/n 2 or x 2n + 1/n 2 , which converges to zero with respect to T but satisfies (u n , 1) ∈ Epi(f ) for any n 1 since one has

f(u n ) f(u n ×1 [n , 2n] ) = 4 > 1.
This proves that Epi(f ) ⊆ V ×R is not strictly convex.

Even in the Hausdorff finite-dimensional case, things are not as simple as they seem. Indeed, if we consider the open disc

C := {(x, y) ∈ R 2 | x 2 + y 2 < 1}, the function f : C -→ R defined by f(x, y) := x 2 + y 2 is strictly convex but its epigraph is not.
Moreover, either in the non-Hausdorff finite-dimensional case or in the infinite-dimensional case, convexity does not always implies continuity. In contrast and among other things, we shall see that strict convexity of the epigraph does always insure continuity of the function.

So, the question of finding a complete characterization of functions with domain in a topological real vector space whose epigraph is strictly convex does deserve our attention.

This characterization is described as follows, where rb stands for the relative boundary (see Definition 2.4).

Main Theorem. Let C be a subset of a topological real vector space V and f : C -→ R a function. Then we have the following equivalence:

Ü C is convex and open in Aff(C) ,
f is strictly convex and continuous ,

∀ x 0 ∈ rb(C) ⊆ C, f(x) -→ +∞ as x -→ x 0 ê ⇐⇒ Epi(f ) is strictly convex .
Remark. It is to be noticed that this equivalence is still true if "continuous" is changed into "locally bounded from above".

The proof, that we postpone until Section 3, splits into the direct implication and its converse. The direct implication is the consequence of three main facts. The first one is the convexity of Epi(f ) given by the convexity of f. The second one is the closeness of the epigraph of f in Aff(C)×R due to both the continuity of the f and its behavior near the boundary of C, which insures that any segment whose end points are in the boundary of Epi(f ) is contained in Epi(f ). The third one is the property that any open segment whose endpoints are in the boundary of Epi(f ) actually lies inside the interior of Epi(f ) in Aff(C)×R as a result of the strict convexity of f and the two previous facts.

As for the converse implication, there are four main things to be used. The first one is the convexity of both C and f given by the convexity of Epi(f ). The second one is the openness of C in Aff(C) as a consequence for the epigraph not to contain vertical segments in its boundary. The third one is the fact that the interior of Epi(f ) in Aff(C)×R lies inside the strict epigraph of f. The fourth one is the property for f to be locally bounded on some non-empty open set in C as a result of the non-emptyness of the interior of Epi(f ) in Aff(C)×R. All these properties yield the continuity of f and give the behavior of f near the boundary of C.

Before giving some consequences of this result, all of whose will also be proved in Section 3, let us just show on a simple example how it may be usefull for checking strict convexity of the epigraph of a function.

Example 1.2. Consider the open convex subset

C := ]-1, 1[×]-1, 1
[ of the topological real vector space V := R 2 (with its usual topology), and let f : C -→ R be the smooth function defined by f(x, y)

:= 1/[(1 -x 2 )(1 -y 2 )].
For any point (x, y) ∈ C, we then compute

∂ 2 f ∂x 2 (x, y) = 2× 1 + 3x 2 (1 -x 2 )(1 -y 2 )
> 0, and the Hessian matrix of f at (x, y) has a determinant which is equal to

4(5x 2 y 2 + 3y 2 + 3x 2 + 1)/[(x -1) 4 (x + 1) 4 (y -1) 4 (y + 1) 4 ] > 0 .
The function f is therefore strictly convex and hence the Main Theorem insures that its epigraph is strictly convex since we have Aff(C) = R 2 and f(x, y) -→ +∞ as (x, y) converges to any point (x 0 , y 0 ) ∈ ∂C.

The first consequence of the Main Theorem is obtained by taking C := V.

Proposition 1.2. Given a strictly convex function f : V -→ R defined on a topological real vector space V, we have the equivalence

Epi(f ) is strictly convex ⇐⇒ Epi(f ) has a non-empty interior in V ×R .
Another use of the Main Theorem is related to the property for a subset C of a real vector space V to be convex if and only if all its intersections with the straight lines of V are convex. Indeed, let us recall the following easy-to-prove result about convex functions.

Proposition 1.3. For any subset C of a real vector space V and any function f :

C -→ R, we have (1) ⇐⇒ (2) ⇐⇒ (3) with (1) Epi(f ) is convex, (2) Epi Ä f |C∩G ä is convex for any affine subspace G of V, and (3)
Epi Ä f |C∩L ä is convex for any straight line L of V.
Then, a natural question is to know whether these equivalences are still true when replacing convexity by strict convexity.

Here is the answer.

Proposition 1.4. For any subset C of a topological real vector space V and any function

f : C -→ R, we have (1) ⇐⇒ (2) =⇒ (3) with (1) Epi(f ) is strictly convex, (2) Epi Ä f |C∩G ä
is strictly convex for any affine subspace G of V, and

(3) Epi Ä f |C∩L ä is strictly convex for any straight line L in V.
It is to be noticed that the implication (3) =⇒ (1) in Proposition 1.4 is not true. Indeed, the function f : V -→ R that we considered in Example 1.1 has an epigraph which is not strictly convex whereas for any vectors u 0 , w ∈ V with w = 0 the function ϕ : R -→ R defined by ϕ(t)

:= f(u 0 + tw) = w 2 2 t 2 + 2 u 0 , w t + u 0 2
2 is obviously strictly convex and continuous. Therefore, Epi Ä f |u 0 +Rw ä is strictly convex according to the Main Theorem and since the map γ : R -→ u 0 + Rw defined by γ(t) := u 0 + tw is a homeomorphism. This last point is a consequence of Theorem 2 in [2, Chapitre I, page 14] since u 0 + Rw is a finite-dimensional affine space whose subspace topology is Hausdorff (as is the topology on V ). Nevertheless, in case V is equal to the canonical topological real vector space R n , this implication is true as a consequence of the Main Theorem.

Proposition 1.5. Given a subset C of R n and a function f : C -→ R, the following properties are equivalent:

(1) Epi(f ) is strictly convex. (2) Epi Ä f |C∩L ä is strictly convex for any straight line L in R n .
As a straightforward consequence of Proposition 1.5, we obtain in particular the following classical result.

Consequence. Any function

f : R n -→ R satisfies the equivalence Epi(f ) is strictly convex ⇐⇒ f is strictly convex .

Preliminaries

In order to make precise the terms used in the previous section, and before proving in Section 3 the Main Theorem and its consequences, we have to give here some definitions and properties about the epigraph of a function and the notion of strict convexity.

2.1. Epigraphs. We begin by recalling the definitions of the epigraph and the strict epigraph for a general function.

Definition 2.1. Given a set X and a function f :

X -→ R, the epigraph of f is defined by Epi(f ) := {(x, r) ∈ X×R | f(x) r} ,
whereas its strict epigraph is defined by

Epi s (f ) := {(x, r) ∈ X×R | f(x) < r} .
Remark 2.1. It is straightforward that these two sets satisfy the relation

Epi s (f ) = (X×R) σ(Epi(-f )) ,
where σ denotes the involution of X×R defined by σ(x, r) := (x, -r).

From now on, X will denote a topological space and we shall give a list of useful properties of the epigraph of a function with domain in X (Proposition 2.1 to Proposition 2.5) that we will need in the sequel (we may refer to [5, pages 34 and 123]).

Proposition 2.1. Given a subset S of a topological space X and a function f : S -→ R such that Epi(f ) has a non-empty interior in X×R, there exists a non-empty open set U in X with U ⊆ S on which f is bounded from above.

Proof.

Given (x 0 , t 0 ) ∈ • ˚ Epi(f ),
there exists an open set U in X such that one has x 0 ∈ U and U×{t 0 } ⊆ Epi(f ) since Epi(f ) is a neighborhood of (x 0 , t 0 ) in X×R. Therefore, for any x ∈ U, we get f(x) t 0 . Proposition 2.2. Given a function f : X -→ R defined on a topological space X, a point x 0 ∈ X and a number r 0 ∈ R, the following properties are equivalent:

(1) (x 0 , r 0 ) ∈ • ˚ Epi(f ). (2) ∃ r < r 0 , x 0 ∈ • ˇ f -1 (-∞, r). (3) ∃ s < r 0 , {x 0 }×[s, +∞) ⊆ • ˚ Epi(f ).
This is a characterization of the interior of the epigraph of a function.

Proof. Point 1 =⇒ Point 2. Assume we have (x 0 , r 0 ) ∈ • ˚ Epi(f ).
Then there exist a neighborhood V of x 0 in X and a number ε > 0 that satisfy the inclusion V ×[r 0 -2ε , r 0 + 2ε] ⊆ Epi(f ), from which we get f(x) r 0 -2ε for any x ∈ V, or equivalently

V ⊆ f -1 (-∞, r) with r := r 0 -ε < r 0 . This proves that x 0 belongs to the interior of f -1 (-∞, r) in R. Point 2 =⇒ Point 3. Assume that we have x 0 ∈ • ˇ f -1 (-∞, r) for some r < r 0 . Therefore, there is a neighborhood V of x 0 in X that satisfies V ⊆ f -1 (-∞, r) ⊆ f -1 (-∞, r],
which yields V ×[r, +∞) ⊆ Epi(f ), and hence the inclusion {x 0 }×[s, +∞) ⊆ • ˚ Epi(f ) holds with s := (r + r 0 )/2 since we have s ∈ (r, r 0 ) and since the interval [r, +∞) is a neighborhood in R of any number τ ∈ [s, +∞).

Point 3 =⇒ Point 1. This is clear. Proposition 2.3. Any function f : X -→ R defined on a topological space X satisfies the following properties:

(1)

• ˚ Epi(f ) = • ¸ Epi s (f ).
(2)

• ˚ Epi(f ) ∩ ({x}×R) ⊆ {x}×(f(x), +∞) for any x ∈ X.
This property gives a topological relationship between the epigraph and the strict epigraph of a function.

Proof. Point 1. Given a point (x, r) ∈ • ˚ Epi(f ), there exists r 0 < r that satisfies {x}×[r 0 , +∞) ⊆ Epi(f ), which yields in particular f(x) r 0 , and hence f(x) < r. This proves

• ˚ Epi(f ) ⊆ Epi s (f ), which implies • ˚ Epi(f ) ⊆ • ¸
Epi s (f ) by taking the interiors in the product space X×R.

Conversely, the obvious inclusion Epi

s (f ) ⊆ Epi(f ) yields • ¸ Epi s (f ) ⊆ • ˚ Epi(f ).
Point 2. By Point 1 above, we have

• ˚ Epi(f ) ⊆ Epi s (f ), and hence • ˚ Epi(f ) ∩ ({x}×R) ⊆ Epi s (f ) ∩ ({x}×R) = {x}×(f(x), +∞) for any x ∈ X.
Proposition 2.4. Given a function f : X -→ R defined on a topological space X and a point x 0 ∈ X, the following properties are equivalent:

(1) f is upper semi-continuous at x 0 .

(2

) {x 0 }×(f(x 0 ), +∞) = • ˚ Epi(f ) ∩ ({x 0 }×R).
This is a geometric characterization of the upper semi-continuity of a function in terms of its epigraph.

Proof.

Point 1 =⇒ Point 2. Given an arbitrary ε > 0, there exists a neighborhood V of x 0 in X that satisfies f(x) f(x 0 ) + ε/2 for any x ∈ V. Thus, we have

V ×[f(x 0 ) + ε/2 , +∞) ⊆ Epi(f ), and hence (x 0 ,f(x 0 ) + ε) ∈ • ˚ Epi(f ).
So we proved the direct inclusion ⊆.

The reverse inclusion ⊇ is straightforward by Point 2 in Proposition 2.3.

Point 2 =⇒ Point 1. Conversely, given an arbitrary number ε > 0, we have (x 0 ,f(x 0 ) + ε) ∈ {x 0 }×(f(x 0 ), +∞) ⊆ • ˚ Epi(f ).
Thus, there exists a neighborhood V of x 0 in X satisfying V ×{f(x 0 ) + ε} ⊆ Epi(f ), which yields f(x) f(x 0 ) + ε for any x ∈ V. Proposition 2.5. Given a subset A of a topological space X, a point x 0 ∈ A and a function f : A -→ R, we have the equivalence

Ä f(x) -→ +∞ as x -→ x 0 ä ⇐⇒ Ä (x 0 , r) ∈ Epi(f ) for any r ∈ R ä .
This is a characterization of the closure of the epigraph of a function.

Proof. * ( =⇒ ) Given r ∈ R, there exists a neighborhood V of x 0 in X such that we have in particular the inclusion f(V ) ⊆ [r + 2 , +∞). Thus, we get (V ×(-∞ , r + 1]) ∩ Epi(f ) = ∅, which shows that (x 0 , r) does not belong to Epi(f ) since V ×(-∞ , r + 1] is a neighborhood of (x 0 , r) in X×R. * ( ⇐= ) Given r ∈ R, there exist a neighborhood V of x 0 in X and a real number ε > 0 such that V ×[r -ε , r + ε] does not meet Epi(f ).

Since we have r ∈

[r -ε , r + ε], this yields f(V ) ∩ (-∞, r) = ∅, which is equivalent to the inclusion f(V ) ⊆ (r, +∞). So, we have proved f(x) -→ +∞ as x -→ x 0 .

Strict convexity.

In this subsection, we first recall the definitions of the relative interior and the relative closure of a set in a topological real vector space since they underly strict convexity, and then we establish a couple of useful properties needed in Section 3.

We begin with two basic notions in affine geometry: the affine hull and convex sets (see for example [START_REF] Rockafellar | Convex analysis[END_REF] and [START_REF] Webster | [END_REF]).

Definition 2.2. The affine hull Aff(S) of a subset S of a real vector space V is the smallest affine subspace of V which contains S.

So, for any subsets A and B of V satisfying A ⊆ B, we obviously have Aff(A) ⊆ Aff(B). Proposition 2.6. Given real vector spaces V and W, any subsets A ⊆ V and B ⊆ W satisfy

Aff(A×B) = Aff(A)×Aff(B) .

Proof. * For any x, y ∈ B with x = y, the straight line L passing through x and y lies in Aff(B), and hence for any a ∈ A the straight line {a}×L of V ×W lies in Aff(A×B) since it contains the points (a, x), (a, y) ∈ A×B. Therefore, we get A×Aff(B) ⊆ Aff(A×B).

On the other hand, we also have Aff(A)×B ⊆ Aff(A×B) by the same reasoning. A subset C of V is said to be convex if we have [x, y] ⊆ C for all x, y ∈ C.

These two inclusions yield

In other words, C is convex if and only if its intersection with any straight line L in V is an "interval" of L.

From now on and throughout the section, V will denote a topological real vector space.

Before we go on, let us just point out some facts. Remark 2.2. 1) Given a neighborhood U of the origin in V, the following properties hold: (a) For any vector x ∈ V, there exists ε > 0 such that we have [-ε, ε]x ⊆ U.

(b) For any vector x ∈ V, there exists λ > 0 such that we have x ∈ λU (the set U is then said to be absorbing).

(c) We have Vect(U) = V.

Indeed, Point a is easy to prove and the implications a =⇒ b =⇒ c are straightforward.

2) Given a finite-dimensional real vector space W, there exists a unique topological real vector space structure on W which is Hausdorff. Endowed with this structure, W is then isomorphic to the canonical topological real vector space R n , where n denotes the dimension of W (see Theorem 2 in [2, Chapitre I, page 14]).

Definition 2.4. Let S be a subset of a topological real vector space V.

(1) The relative interior ri(S) of S is the interior of S with respect to the relative topology of Aff(S). ( 2) The relative closure rc(S) of S is the closure of S with respect to the relative topology of Aff(S). ( 3) The relative boundary rb(S) of S is the boundary of S with respect to the relative topology of Aff(S) (so we have rb(S) = rc(S) ri(S)).

Proposition 2.7. Let V be a topological real vector space.

(1) For any subsets A and B of V, we have the implication A ⊆ B =⇒ rc(A) ⊆ rc(B).

(2) For any subset A of V, we have

(A) ∩ Aff(A ∩ W ) ⊆ A ∩ W holds. Therefore, we get ri(A) ∩ Aff(A ∩ W ) ⊆ ri(A ∩ W ) since ri(A ∩ W ) is the largest open set in Aff(A ∩ W ) which is contained in A ∩ W.
(A ∩ W ) = rc(A ∩ W ) ri(A ∩ W ) ⊆ rc(A ∩ W ) [ri(A) ∩ Aff(A ∩ W )] = [rc(A ∩ W ) ri(A)] ∪ [rc(A ∩ W ) Aff(A ∩ W )] = rc(A ∩ W ) ri(A) (use rc(A ∩ W ) ⊆ Aff(A ∩ W )) ⊆ rc(A) ri(A) = rb(A) .
Proposition 2.8. Let X be a subset of a topological real vector space V and A a subset of X×R such that they satisfy Aff(A) = Aff(X)×R. Then we have the following properties:

(1) The interior of A in X×R contains ri(A).

(2) The closure of A in X×R lies in rc(A).

Proof. Point 1. Given (x 0 , r 0 ) ∈ ri(A), there exist a neighborhood U of x 0 in V and a neighborhood W of r 0 in R such that we have [U ∩ Aff(X)]×W ⊆ A, which implies

(U ∩ X)×W = (X×R) ∩ [U ∩ Aff(X)]×W ⊆ (X×R) ∩ A = A .
Then, since U ∩ X is a neighborhood of x 0 in X, we get that (x 0 , r 0 ) is in the interior of A with respect to X×R.

Point 2. From X×R ⊆ Aff(X)×R = Aff(A), we deduce that the closure A X×R of A in X×R satisfies A X×R = A ∩ (X×R) ⊆ A ∩ Aff(A) = rc(A) .
Now, here is the definition of a strictly convex set, which is the key notion of the present work.

Definition 2.5. A subset C of a topological real vector space V is said to be strictly convex if for any two distinct points x, y ∈ rc(C) one has ]x, y[ ⊆ ri(C).

Remark.

1) It is to be noticed that strict convexity is a topological property whereas convexity is a mere affine property. 2) A strictly convex set is of course convex.

3) According to the common geometric intuition, saying that a subset C of V is strictly convex means that C is convex and that there is no non-trivial segment in the relative boundary of C. Owing to Proposition 16 in [2, Chapitre II, page 15], this is an easy consequence of the very definition of strict convexity. 4) This definition coincides with the usual one when V is the canonical topological real vector space R n since in this case the closeness of Aff(C) in V yields rc(C) = C.

Proposition 2.9. For any strictly convex subset C of a topological real vector space V, we have the implication

C = ∅ =⇒ ri(C) = ∅.
Proof.

There are two cases to be considered, depending on whether rc(C) is a single point or not.

If we have rc(C) = {x}, then C also reduces to {x} since we have ∅ = C ⊆ rc(C). Therefore, we obtain Aff(C) = {x}, and this implies ri(C) = {x} = ∅.

On the other hand, if we have x, y ∈ rc(C) with x = y, then the inclusion ]x, y[ ⊆ ri(C) holds by strict convexity of C, and hence one has ri(C) = ∅ since ]x, y[ is not empty.

Remark 2.3. When dealing with a single strictly convex subset C of a general topological real vector space V, we will always assume in the hypotheses that C has a non-empty interior in V in order to insure Aff(C) = V, and this makes sense by Proposition 2.9 and Point 2.b in Proposition 2.7.

We shall now prove that strict convexity is a two-dimensional (topological) notion whereas convexity is-by its very definition-a one-dimensional (affine) notion.

Proposition 2.10. Given a topological real vector space V with dim(V ) 2, any subset C of V whose interior is not empty satisfies the equivalence Remark. In case V is one-dimensional but its topology is not Hausdorff, the strictly convex subsets of V, unlike those of R (endowed with its usual topology), do not coincide with its convex subsets. Indeed, if the topology of V is for example trivial, then the only strictly convex subsets of V are the empty set and V itself.

C is strictly convex ⇐⇒ C ∩ P is
Finally, in order to be complete, let us recall the definition of a (strictly) convex function.

Definition 2.6. Given a convex subset C of a real vector space V, a function f : C -→ R is said to be (1) convex if we have f((1 -t)x + ty) (1 -t)f(x) + tf(y) for any points x, y ∈ C and any number t ∈ (0, 1), and (2) strictly convex if we have f((1-t)x + ty) < (1-t)f(x)+ tf(y) for any distinct points x, y ∈ C and any number t ∈ (0, 1).

Remark 2.4.

1) It is to be noticed that both convexity and strict convexity of functions are mere affine notions.

2) A strictly convex function is of course convex.

3) Given a convex subset C of a real vector space V, a function f : C -→ R is convex if and only if one has

f Ñ n i=1 λ i x i é n i=1 λ i f(x i ) (Jensen's inequality)
for any integer n 1, any points x 1 , . . . , x n ∈ C, and any numbers λ 1 , . . . , λ n ∈ [0, +∞) which satisfy

n i=1 λ i = 1.
This is obtained by induction on n 1.

4) Given a convex subset C of a real vector space V, a convex function f : C -→ R, a subset A ⊆ C and a real number M > 0, we have the following equivalence:

Ä ∀ x ∈ A, f(x) M ä ⇐⇒ Ä ∀ x ∈ Conv(A), f(x) M ä ,
where Conv(A) stands for the convex hull of A, i. e., the smallest convex subset of V which contains A.

Indeed, it is a well-known fact that each x ∈ Conv(A) writes x = n i=1 λ i x i for some integer n 1, some points x 1 , . . . , x n ∈ A and some numbers λ 1 , . . . , λ n ∈ [0, +∞) which satisfy

n i=1 λ i = 1. Therefore, this implies f(x) n i=1 λ i f(x i ) n i=1
λ i M = M by using Point 3 above.

Proofs

This section is devoted to the proofs of the Main Theorem and of all its consequences that we mentionned in Section 1.

Let us first begin with the following affine property.

Lemma 3.1. For any function f : S -→ R defined on a subset S of a real vector space, we have Aff(Epi(f )) = Aff(S)×R.

Finally, given x 0 ∈ rb(C), we have ({x 0 }×R) ∩ rc(Epi(f )) = ∅ by using the second implication in Lemma 3.3, and hence f(x) -→ +∞ as x -→ x 0 by Proposition 2.5.

Proof of Proposition 1.2. * ( =⇒ ) Since Epi(f ) is not empty, the same holds for

Ä f |C∩G ä = Epi(f ) ∩ (G×R) ⊆ Epi(f ) yields rc Ä Epi Ä f |C∩G ää = Epi Ä f |C∩G ä ∩ (Aff(C ∩ G)×R) ⊆ Epi Ä f |C∩G ä ∩ (Aff(C)×R) ⊆ Epi(f ) ∩ (Aff(C)×R) = rc(Epi(f )) .
Therefore, any two distinct points (x, r), (y, s) ∈ rc {-e i , e i }.

Ä Epi Ä f |C∩G ää are in rc(Epi(f )), which implies that each (z, t) ∈ ](x, r) , (y, s)[ is in ri(Epi(f )) since Epi(f ) is strictly convex. So, there exist ε > 0 and a neighborhood Ω of z in V such that the set (Ω∩Aff(C))×(t-ε , t+ ε) is included in Epi(f ), from which we get [Ω ∩ Aff(C ∩ G)]×(t -ε , t + ε) ⊆ (Ω ∩ Aff(C) ∩ G)×(t -ε , t + ε) = [(Ω ∩ Aff(C))×(t -ε , t + ε)] ∩ (G×R) ⊆ Epi(f ) ∩ (G×R) = Epi Ä f |C∩G ä since one has Aff(C ∩ G) ⊆ Aff(C) ∩ G and Epi(f ) ∩ (G×R) = Epi Ä f |C∩G ä , proving that (z, t) is in rc Ä Epi Ä f |C∩G ää .

This shows that Epi

But (v 1 , . . . , v d ) is a basis of W since B is, which implies that f satisfies Im(f ) = W and is a linear isomorphism onto its image. Therefore, x 0 + f is a homeomorphism from R d (endowed with its usual topology) onto x 0 + W = Aff(C). As a consequence, we then get that U is a neighbourhood of x 0 in Aff(C) since Ω is a neighborhood of the origin in R d , and hence C is itself a neighborhood of x 0 in Aff(C). * On the other hand, f is strictly convex since its restriction to any straight line in R n is strictly convex and since strict convexity is an affine notion. * Moreover, the convexity of f on the open convex subset C of the finite-dimensional affine space Aff(C) equipped with the topology induced from that of R n implies that f is continuous according to the corollary given in

  Aff(A)×Aff(B) ⊆ Aff(Aff(A)×B) ⊆ Aff(Aff(A×B)) = Aff(A×B) . * Conversely, since Aff(A)×Aff(B) is an affine subspace of V ×W which contains A×B, we obviously have Aff(A×B) ⊆ Aff(A)×Aff(B). Definition 2.3. Given points x and y in a real vector space V, the set [x, y] := {(1 -t)x + ty | t ∈ [0, 1]} is called the (closed) line segment between x and y, whereas the set ]x, y[ := [x, y] {x, y} is called the open line segment between x and y (the latter set is therefore empty in case one has x = y).

.( 3 )Point 1 .

 31 (a) Aff(rc(A)) = Aff(A), and (b) ri(A) = ∅ ⇐⇒ Ä A = ∅ and Aff(ri(A)) = Aff(A) ä For any subset A of V and any affine subspace W of V, we have (a) ri(A) ∩ Aff(A ∩ W ) ⊆ ri(A ∩ W ), and (b) rb(A ∩ W ) ⊆ rb(A) ∩ W.Proof. Given subsets A and B of V with A ⊆ B, we can writerc(A) = A ∩ Aff(A) ⊆ B ∩ Aff(B) = rc(B) since we have Aff(A) ⊆ Aff(B) and A ⊆ B.Point 2.a. Using A ⊆ rc(A), we first get Aff(A) ⊆ Aff(rc(A)). Conversely, we have rc(A) ⊆ Aff(A) by the very definition of rc(A), and hence one obtains the inclusion Aff(rc(A)) ⊆ Aff(Aff(A)) = Aff(A).Point 2.b. If the open set ri(A) in Aff(A) is not empty, then we have Aff(ri(A)) = Aff(A) by Point 1.c in Remark 2.2, and A is not empty since one has ri(A) ⊆ A.Conversely, if we have A = ∅ and Aff(ri(A)) = Aff(A), then we get Aff(ri(A)) = Aff(A) = ∅, and hence ri(A) = ∅ by using the obvious equality Aff(∅) = ∅. Point 3.a. Since ri(A) is open in Aff(A), the intersection ri(A) ∩ Aff(A ∩ W ) is open in the subspace Aff(A ∩ W ) ⊆ Aff(A). On the other hand, we have ri(A) ⊆ A and Aff(A ∩ W ) ⊆ Aff(W ) = W, and hence the inclusion ri

Point 3 .

 3 b. We first have Aff(A ∩ W ) ⊆ Aff(W ) = W, and hence rb(A ∩ W ) lies in W. On the other hand, combining Point 1 and Point 3.a yields rb

  strictly convex for any affine plane P in V . Proof. * ( =⇒ ) Let x, y ∈ rc(C ∩ P ) with x = y. Then, we first have x, y ∈ C since the inclusion rc(C ∩ P ) ⊆ C holds according to Point 1 in Proposition 2.7, and this yields ]x, y[ ⊆ • C by strict convexity of C. On the other hand, we have x, y ∈ P from rc(C ∩ P ) ⊆ Aff(C ∩ P ) ⊆ Aff(P ) = P, and hence we get ]x, y[ ⊆ • C ∩ P by convexity of P. In particular, the open set • C ∩ P of P is not empty, and this implies Aff

= P by Point 3 in

 3 Remark 2.2, which yields P ⊆ Aff(C ∩ P ) since we have • C ⊆ C. Therefore, we get • C ∩ P ⊆ • C ∩ Aff(C ∩ P ) ⊆ ri(C ∩ P ) by Point 3.a in Proposition 2.7, which gives ]x, y[ ⊆ ri(C ∩ P ). This proves that C ∩ P is strictly convex. * ( ⇐= ) Let x, y ∈ C with x = y. Since the dimension of V = Aff(C) is greater than one, C does not lie in a line, and hence there exists a point z ∈ • C such that x, y, z are not collinear. Therefore, P := Aff({x, y, z}) is an affine plane in V. Then, Proposition 16 in [2, Chapitre II, page 15] implies that the open segments ]z, x[ and ]z, y[ are in • C ∩ P, and hence in C ∩ P. This insures that rc(]z, x[) and rc(]z, y[) are in rc(C ∩ P ). But we have x ∈ rc(]z, x[) since any neighborhood U of x in V satisfies ]z, x[ ∩ U = ∅ by Point 1.a in Remark 2.2. And the same argument yields y ∈ rc(]z, y[). So, we have obtained x, y ∈ rc(C ∩ P ), and hence ]x, y[ ⊆ ri(C ∩ P ) since C ∩ P is strictly convex. Now, if we pick u ∈ ]x, y[ and define v := u -z, then there exists a number t > 0 that satisfies w := u + tv ∈ C ∩ P since u + Rv is the straight line passing through u, z ∈ C ∩ P ⊆ Aff(C ∩ P ) and since C ∩ P is a neighborhood of u ∈ ri(C ∩ P ) in Aff(C ∩ P ). Therefore, the segment [z, w] lies in the convex set C ∩ P, and hence in C, which yields [z, u] ⊆ [z, w] {w} ⊆ • C by Proposition 16 in [2, Chapitre II, page 15]. This proves that C is strictly convex.

  by Proposition 2.9. * ( ⇐= ) By Proposition 2.1 with X := V and S := V, we get that f is bounded from above on a non-empty open set in V, and hence it is continuous by Proposition 21 in [2, Chapitre II, page 20]. Therefore, the Main Theorem with C := V implies that Epi(f ) is strictly convex. Proof of Proposition 1.4. Point 1 =⇒ Point 2. Given an affine subspace G of V, the inclusion Epi

Point 2 =⇒ Point 1 .Lemma 3 . 4 .Point 1 =⇒ Point 2 . 4 . 2 =⇒

 21341242 This is clear.Point 2 =⇒ Point 3. This is straightforward.Before proving Proposition 1.5, we need two key lemmas. Given a convex subset C of a topological real vector space V and a straight line L in V, we have the implicationÜ C is open in Aff(C) , C ∩ L = ∅ , the subspace topology on L is Hausdorff ê =⇒ rb(C ∩ L) = rb(C) ∩ L .Proof of Proposition 1.5.Since the equivalence is obvious when C is empty or reduced to a single point, we may assume that C has at least two distinct points. This implication has already been proved in Proposition 1.PointPoint 1. * First of all, the intersection of C with any straight line L in R n is convex by applying the converse part of the Main Theorem to the function f |C∩L . Hence, C is convex. * Let us then show that C is open in Aff(C). Fix x 0 ∈ C, and consider the subsetG := {x -x 0 | x ∈ C} of V.Then the vector subspaceW := {v -x 0 | v ∈ Aff(C)} of V is generated by G since C is agenerating set of the affine space Aff(C). Hence, there exists a subset B of G which is a basis of W. Denoting by d ∈ {1, . . . , n} the dimension of the affine subspace Aff(C) of R n , we have dim(W ) = d, and hence there are vectors x 1 , . . . , x d ∈ C such that we can write B = {x i -x 1 i d}. Now, for each i ∈ {1, . . . , d}, if L i denotes the straight line in R n passing through x 0 and x i , the strict convexity of Epi Äf |C∩L i ä implies that C ∩ L i is open in Aff(C ∩ L i ) = L i = Aff({x 0 , x i }) by Lemma 3.3, which insures the existence of a number r i > 0 such that the vectorv i := r i (x i -x 0 ) satisfies {x 0 -v i , x 0 + v i } ⊆ C ∩ L i ⊆ C .Therefore, the convex hull U ofd i=1 {x 0 -v i , x 0 + v i }lies in the convex set C. Since the map f : R d -→ R n defined by f (λ 1 , . . . , λ d ) := d i=1 λ i v i is linear and sends the canonical ordered basis (e 1 , . . . , e d ) of R d to the family (v 1 , . . . , v d ) of R n , we can write U = x 0 + f(Ω), where Ω := {(λ 1 , . . . , λ d ) ∈ R d | |λ 1 | + • • • + |λ d | 1} is the convex hull of d i=1

[ 2 ,

 2 Chapitre II, page 20]. * Finally, given any point a ∈ rb(C) and any straight line L in R n passing through a, we have either C ∩ L = ∅, which obviously yields lim x→a f(x) = +∞ with x ∈ C ∩ L, or C ∩ L = ∅, which first implies a ∈ rb(C ∩ L) by Lemma 3.4, and then lim x→a f(x) = +∞ with x ∈ C ∩ L by the Main Theorem since Epi Ä f |C∩L ä is strictly convex. In both cases, we obtain lim x→a f(x) = +∞ by Lemma 3.5.

Proof.

We obviously have Epi(f ) ⊆ S ×R ⊆ Aff(S)×R, and hence Aff(Epi(f )) ⊆ Aff(S)×R. Conversely, given x ∈ S, we have {x}×[f(x),f(x) + 1] ⊆ Epi(f ) by the very definition of Epi(f ), which gives {x}×R = Aff({x}×[f(x),f(x) + 1]) ⊆ Aff(Epi(f )) by Proposition 2.6. Therefore, we get S ×R ⊆ Aff(Epi(f )), and this yields Aff(S)×R = Aff(S ×R) ⊆ Aff(Epi(f )) by Proposition 2.6 once again.

Then, let us establish two technical but useful topological properties. Lemma 3.2. Given a subset S of a topological real vector space V and a function f : S -→ R, we have the implication

Proof.

Assume that all the hypotheses are satisfied, and let (a, α) ∈ rc(Epi(f )). First of all, notice that a is in rc(S) since the projection of V ×R onto V is continuous and since we have Aff(Epi(f )) = Aff(S)×R by Lemma 3.1. If we had a ∈ S = ri(S) (remember that S is open in Aff(S)), then we would get a ∈ rb(S), and hence f(x) -→ +∞ as x -→ x 0 := a, which yields (a, α) ∈ rc(Epi(f )) by Proposition 2.5, a contradiction. Therefore, the point a is necessarily in S. Now, assume that we have f(a) > α, and let ε := (f(a) -α)/2 > 0. Since f is lower semi-continuous at a, there exists a neighborhood U of a in S that satisfies the inclusion f(U) ⊆ [f(a) -ε , +∞) = [α + ε , +∞). But S is a neighborhood of a in Aff(S), and hence so is U. Thus, combining (a, α) ∈ rc(Epi(f )) and Aff(Epi(f )) = Aff(S) × R, we can find x ∈ U and λ ∈ (α -ε , α + ε) with f(x) λ, which yields f(x) ∈ (-∞ , α + ε), contradicting the inclusion above. So, we necessarily have f(a) α, or equivalently (a, α) ∈ Epi(f ). Conclusion: we proved rc(Epi(f )) ⊆ Epi(f ), which means that Epi(f ) is closed in Aff(Epi(f )) = Aff(S)×R. Lemma 3.3. For any function f : C -→ R defined on a convex subset C of a topological real vector space V, we have (1) =⇒ (2) =⇒ (3) =⇒ (4) with

where for each s ∈ R the map τ s : V ×R -→ V ×R is defined by τ s (x, r) := (x, r + s).

Proof.

Point 1 =⇒ Point 2. Since for each s ∈ R the map τ s is an affine homeomorphism, we have the equality τ s (rc(Epi(f ))) = rc(τ s (Epi(f ))). Then, for any s 0, we obtain τ s (rc(Epi(f ))) ⊆ rc(Epi(f )) since one has τ s (Epi(f )) ⊆ Epi(f ). Hence, given s > 0 and (x, r) ∈ rc(Epi(f )), we can write (x, r + 2s) = τ 2s (x, r) ∈ rc(Epi(f )), which implies that the midpoint τ s (x, r) = (x, r + s) is in ri(Epi(f )) since Epi(f ) is strictly convex. This proves the first inclusion. The second inclusion is straightforward since we have Aff(Epi(f )) = Aff(C)×R by Lemma 3.1. Proof of the Main Theorem. We may assume that C is not empty since this equivalence is obviously true otherwise. * ( =⇒ ) Let (x, r), (y, s) ∈ rc(Epi(f )) with (x, r) = (y, s), fix t ∈ (0, 1), and define

By Lemma 3.2, we already have (x, r), (y, s) ∈ Epi(f ). Then, since C and f are both convex, Epi(f ) is convex by Proposition 1.1, which implies (a, α) ∈ Epi(f ).

There are now two cases to be considered.

• Case x = y and r < s.

Here we have a = x = y, which yields

In both cases, we get (a, α) ∈ ri(Epi


Simon Thalabard

Bruce Turkington

Optimal thermalization in a shell model of homogeneous turbulence

We investigate the turbulence-induced dissipation of the large scales in a statistically homogeneous flow using an "optimal closure," which one of us (BT) has recently exposed in the context of Hamiltonian dynamics. This statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. We focus on a simple problem for which the optimal closure can be fully and exactly worked out: the relaxation arbitrarily far-from-equilibrium of a single energy shell towards Gibbs equilibrium in an inviscid shell model of 3D turbulence. The predictions of the optimal closure are validated against DNS and contrasted with those derived from EDQNM closure.

Introduction

The concept of a turbulent cascade -be it direct or inverse -is tied to non-Gaussianity. In the case of 3D homogeneous isotropic turbulence, this is patent from Kolmogorov 4/5 law, which states that the rate of energy dissipation is proportional to the skewness of the velocity increment. Be the deviations from Gaussianity weak (as in the 2D inverse cascade of energy) or strong (as in the 3D direct cascade), Gaussian models of turbulence are therefore paradoxical, in that they cannot fully represent turbulent fluctuations and interactions. The paradox is most evident in the so-called Quasi-Normal (QN) cumulant discard closure proposed by [1], which relies on a Gaussian Ansatz to close the cumulant hierarchy. There, in order to provide a non-trivial closed equation for the energy spectrum, Gaussianity needs to be used inconsistently, that is, for the fourth cumulant only and not for the third. Still, the QN theory is non-realizable, as it leads to the development of non-spurious negative energy spectra (see [2,[START_REF] Orszag | Lectures on the Statistical Theory of Turbulence[END_REF]). Orszag showed that the QN theory can be patched into the now-celebrated Eddy Damped Quasi Normal Markov theory (EDQNM), by taking the Markovian limit and introducing phenomenological eddy damping. While EDQNM provides a practical theory that has been widely used in the past decades for the purpose of subgrid modeling (see [START_REF] Lesieur | Turbulence in Fluids[END_REF][START_REF] Baerenzung | [END_REF]), it remains conceptually unsatisfactory as it relies on an ad-hoc modeling of the statistics to correct the defects of the QN theory. Besides, EDQNM does not completely resolve the realizability issue : counter-examples can be constructed in dimension d ≤ 2, where a negative energy spectrum develops for arbitrarily small times (see [6]).

The reason why the QN theory fails so badly remains unclear. Is it due to the Gaussian Ansatz or rather its inconsistent use ? More generally, is the choice of a Gaussian as a proxy for the true ensemble really bound to yield either a trivial or an ad-hoc closure ? The purpose of this paper is to answer this question in the negative. To this end, we rely on a unconventional "optimal closure framework" recently proposed by one of us in the context of Hamiltonian systems ( [7]). When applied to turbulence problems, it allows a consistent use of a Gaussian Ansatz (and corresponding vanishing third cumulant) without loss of the intrinsic dissipation due to turbulence. This apparent paradox is resolved by making a clear distinction between the true ensembles and their proxies. While this distinction is not transparent in the cumulant hierarchy, it is revealed when one works directly with the Liouville equation. Schematically said, any attempt to model the unresolved turbulent scales comes with a cost, which can be quantified via a residual in the Liouville equation. The idea of the optimal closure is to make this cost minimal in an appropriate sense.

In this paper, the exposition of this closure will be practical rather than general. While there exist intimate connections between the optimal closure and general outof-equilibrium frameworks, those will not be discussed here, and we refer the reader to [7] for a thorough exposition. For the sake of clarity, we study a simple test problem, for which closure can be entirely implemented, namely, the relaxation arbitrarily far from equilibrium of an energy shell towards Gibbs equilibrium in a Galerkin-truncated, inviscid, statistically homogeneous fluid. The one-mode relaxation has been considered by [8] for Galerkin-truncated 3D dynamics. They showed that the relaxation could be understood in terms of a two-fluid model, where thermalization is caused by the small-scale dynamics, and that insightful predictions could be obtained by EDQNM. To emphasize the key issues in the closure theory, without obscuring them in the mathematical analysis, here we focus on a shell model dynamics, which mimics the nonlinearity of the 3D truncated Euler dynamics. In this specific problem, not only does the optimal strategy yield a consistent closure, it also predicts explicitly the relaxation profile and the time-scales involved.

From a physical point of view, inviscid Galerkin-truncated dynamics have been invoked to describe the interactions between thermalized and non-thermalized degrees of freedom. In this sense such systems are now known to provide minimal models of turbulence, with the highest wavenumbers typically acting as a bath of energy (see [9,10,11]). This feature was in fact early noticed by Kraichnan (see [12,[START_REF] Eyink | Kraichnan A Voyage Through Turbulence[END_REF]), and lies at the heart of his Fluctuation-Dissipation theorem (see [START_REF] Kraichnan | [END_REF]), later exploited by Leith for the purpose of climate modeling (see [15,16]). Truncated dynamics also appear in the modeling of quantum turbulence, where bottlenecks of energy between quantum and classical degrees of freedom yield a partial thermalization of an intermediate range of scales (see [17]). While the long-time properties of truncated dynamics can be described by standard equilibrium statistical mechanics, the non-equilibrium transients are far less understood, as they involve the interplay between conservative and dissipative dynamics [18,19]. In recent years, the increase of computer of power and systematic use of direct numerical simulations has reinvigorated the subject and motivated new research directions. They have been used to probe the relevance of renormalization group techniques and associated perturbation expansions [20,21], and they have also revealed intriguing decaying phenomenologies : from "Tyger-resonances" that pre-curse thermalization [22,23] to anomalous spectral laws found in so-called Leith models of turbulence [24,25].

The paper is organized as follows. We first recall a few salient points about the shell dynamics that we use, and expose the general out-of-equilibrium set-up that we consider. Second, we outline the optimal closure procedure, and solve the one-mode relaxation problem. We then contrast its predictions with those obtained from the EDQNM framework. The skill of the optimal closure is tested against numerical simulations. We end by briefly suggesting how this approach could be extended to more complex problems.

Test problem setup

Shell dynamics

In order to keep the algebra involved in the derivation of the optimal closure as simple as possible, we study a class of "shell dynamics", which slightly extends the popular GOY dynamics initially proposed by [START_REF] Gledzer | System of hydrodynamic type admitting two quadratic integrals of motion[END_REF]27]. This model is known to reproduce many of the statistical signatures of high-reynolds turbulent cascades (see [28,29]), among which is intermittency. The dynamical variables v l (1 ≤ l ≤ N ) represent the typical velocity for the wavelength k(l). They are taken to obey a very general nonlinear quadratic

dynamics vl = C lmn γ n v m v n = Def F l [v]. (1)
Here and later, stars denote complex conjugation, and summations over pairs of dummy indices are implied. The coefficients γ n and C lmn are yet to be prescribed. We first require that the nonlinear dynamics (1) preserves the two following quadratic invariants :

E[v] = N l=1 |v l | 2 and H[v] = N l=1 γ l |v l | 2 . ( 2
)
To make E and H broad analogues of the kinetic energy and the helicity, preserved by the 3D Euler dynamics, dimensional analysis suggests setting γ l = (-1) l k(l). Independently of this choice, the conservation of both the energy and the helicity is achieved by requiring the coefficients C lmn to be fully anti-symmetric, namely,

C lmn + C lnm = 0 and C lmn = C mnl = C nlm for any l, m, n. (3)
To specify dynamics completely, we now make the standard assumption that the triadic interactions between the shells are local: the variables v l (1 ≤ l ≤ N ) are nonlinearly coupled to their nearest and next-nearest neighboring shells only. The coefficients C lmn are then zero except when {l, m, n} is a set of consecutive integers. The remaining coefficients are then entirely determined by the value of the coefficients C (l-1)l(l+1) , using the anti-symmetry properties (3). We choose

C (l-1)l(l+1) = k(l + 1) γ l -γ l-1 if 1 < l < N, and 0 otherwise. (4)
By further choosing logarithmically spaced shells k(l) = k 0 λ l , one retrieves exactly the standard formulation of GOY model found for example in [29]; that is, up to a global rescaling v → -iv/λ 3 that we omit here.

Relaxation Towards Equilibrium

In addition to conserving the two quadratic invariants (2), the dynamics (1) trivially satisfies a detailed Liouville theorem. In particular, this implies that Gibbs ensembles are statistical invariants for the dynamics, the Gibbs distribution for (1) being defined in terms of the "inverse temperatures" β l as

ρ G = 1 Z e -βE-ζH = 1≤l≤N ρ G,l , with ρ G,l = β l π e -β l |v l | 2 , and β l = β + ζγ l .
(

) t = 0 1 l 0 N 0 1/N 2/N "bath" "perturbed" l E(l) -→ t = ∞ 1 l 0 N 0 1/N 2/N "bath" "perturbed" l E(l) Figure 1. 5
Cartoon of the out-of-equilibrium setup, with total energy E = 1. At time t = 0, the mode l 0 is set away from equipartition:

E[l 0 ]
1/N . The excess energy is distributed among the modes l = l 0 , which constitute the "bath" of energy. The averaged helicity is H = H equi (see definition in text).

In principle, we expect that any non-degenerate ensemble of solutions with prescribed ensemble-averaged energies and helicities converges towards the corresponding Gibbs ensemble in a statistical sense, no matter how far the initial ensemble is from equilibrium. In particular, we focus our attention on nonequilibrium initial ensembles generated at time t = 0 + by taking the inverse temperature of one specific perturbed mode l 0 far away from its equilibrium Gibbs value. In this set-up, the N -1 other modes can be thought of as a bath of energy and helicity. The total energy (resp. the total helicity) is then the sum of two terms : The perturbed energy E l 0 (resp. the perturbed helicity H l 0 ) and the bath energy E B (resp. the bath helicity H B )). For example, we may choose to make the system relax towards the equipartition ensemble, corresponding to the case ζ = 0 in (5) and corresponding total helicity

H equi = -E λ + (-λ) N +1 [1 + λ] -1 . Setting H B = H equi -H l 0 and E B = 1 -E l 0 at
initial time, we then expect the system to relax towards the equipartition ensemble, a situation cartooned in Figure 1. This is precisely the relaxation process that we plan to approximate using a closed and tractable set of equations.

The optimal closure framework

General philosophy

Fully and accurately describing the relaxation of the probability density ρ(t, •) is clearly out of reach. It would require solving the following Liouville equation,

[∂ t + L] log ρ(t, •) = 0 with L = F l ∂ z l + F l ∂ z l , (6)
a task that involves as much (if not more) analytical and computational effort as solving the original dynamics. A more accessible path is to look for approximate solutions of the Liouville equation, a strategy that does not require solving the entire dynamics, but instead relies on a closure hypothesis. More specifically, we make the following seemingly naive Ansatz, that throughout the relaxation the probability densities can be approximated by time-dependent Gaussians,

ρ(b(t), •) = 1≤l≤N ρl (b l (t), •), with ρl (b l (t), •) = b l (t) π e -b l (t)|v l | 2 , (7)
that are entirely determined by the inverse temperature vector b = (b 1 , ..., b N ). We will write • ρ for averages with respect to the distribution ρ.

It is obvious, yet crucial, to note that those tilded averages do not in general match the true ensemble averages • , that is, averages with respect to the density ρ. While valid for the specific times t = 0 + and t → +∞, a crude identification between • and • ρ is certainly not justified for intermediate times. This is patent from the equation for the energy shell evolution, viz.,

d dt |v l | 2 = 2C lmn γ n [ v l v m v n ] , (8)
whose r.h.s would then vanish, preventing any relaxation! The essential observation is that the densities ρ do not necessarily satisfy the Liouville equation ( 6), and that their error can be quantified in terms of the Liouville Residual :

R[b(t), •] = [∂ t + L] log ρ = 0. ( 9
)
This statistical quantity can be interpreted as the local rate of information loss for proxying ρ by ρ ( [7]). Therefore, using ρ as a proxy for ρ does not necessarily make the r.h.s of disappear, the averages being formally related through

• = e -t 0 e -τ L R(t-τ )dτ • ρ = • ρ. ( 10
)
Quite generally, the Liouville Residual can be written as the sum of two contributions, which we label R r and R i . The subscripts stand for "r eversible" and "i rreversible". For the dynamics (1), the Liouville residual associated to the Gaussian Ansatz ( 7) is readily obtained from the definition (9) and a few lines of algebra. It reads

R(b, ḃ, v) = ḃl 1 b l -|v l | 2 Rr + (-2)b l [C lmn γ n v l v m v n ] R i . ( 11
)
The philosophy of the optimal closure is to pick the inverse temperature profile b(t) whose dynamics provide a closest match to the actual Liouville dynamics. That is, the optimal profile minimizes the rate of information loss over the relaxation. To achieve this, we associate to each Ansatz a lack-of-fit Lagrangian density, which we define up to a weighing constant κ 2 > 0 as

L lof [b, ḃ] = 1 2 [R r + κR i ] 2 ρ = 1 2 R 2 r ρ Lr + κ 2 2 R 2 i ρ L i . ( 12
)
The closure is then defined by solving the optimization problem:

min b(t) ∞ 0 dt L lof [b, ḃ] over paths b(t) subject to b l (0 + ) = b 0 l b l (∞) = β l . ( 13
)
The lack-of-fit Lagrangian density is expressed as the sum of a "reversible" density L r with the "irreversible" density L i . By analogy to standard Hamiltonian mechanics, the reversible contribution is akin to a kinetic energy, while the irreversible contribution may be thought of minus the potential energy. The nonequilibrium thermodynamics associated with this optimization formulation is discussed by [7] and [30].

3.2. The quasi-normal optimal closure.

The lack-of-fit Lagrangian

We use the general expression (11) for the Liouville Residual, together with the Gaussian Ansatz ( 7) to obtain the lack-of-fit Lagrangian. Some lines of algebra then yield specifically :

L r = 1 2 ḃ2 l b 2 l and L i = κ 2 2 g(l) b l + f (l, m, n) b l b m b n . (14)
for functions f and g that are found to be

f (l, m, n) = C 2 lmn γ n (γ n -γ m ) and g(l) = C 2 lmn 2γ m γ l -γ m γ n -γ 2 l . (15)

Single mode relaxation towards equipartition

We now consider the specific scenario cartooned in Figure 1, where a single mode l 0 relaxes towards the equipartition value β = N/E, obtained by setting ζ = 0 in the definition (5) of Gibbsian ensembles. We further assume that the Lagrangian depends only on two temperatures: the temperature associated to the perturbed mode 1/b l 0 and the bath temperature 1/b B , so that b l (t) = b B (t) for all l = l 0 .

For this single-mode relaxation we can entirely determine the temperature profile that solves the optimization problem [START_REF] Eyink | Kraichnan A Voyage Through Turbulence[END_REF]. From Equations ( 16) and ( 14), we find that the Lagrangian density can be recast in terms of a single characteristic parameter φ κ (l 0 ) as follows:

L r = 1 2 ḃ2 l 0 b 2 l 0 + N -1 2 ḃ2 B b 2 B and L i = φ κ (l 0 ) 2 - 2 b B + 1 b l 0 + b l 0 b 2 B , with φ κ (l 0 ) = κ 2 1≤m,n≤N f (l 0 , m, n). (17)
The solution to the optimization problem ( 13) is obtained from the Euler-Lagrange equations associated to the Lagrangian density (17). To this end, it is easier to work with the following thermodynamic variables:

σ l 0 = log b l 0 β and σ B = √ N -1 log b B β , (18)
so that the Lagrangian density now reads

L lof = σ2 l 0 2 + σ2 B 2 + φ κ (l 0 )E 2N -2e - σ B √ N -1 + e -σ l 0 + e σ l 0 -2 σ B √ N -1 . (19)
Assuming N 1, we obtain the temperature dynamics to first order in 1/N as σB = 0 and σl 0 = -2

φ κ (l 0 )E N sinh σ l 0 2 . ( 20
)
The relaxation profile is then given by tanh

σ l 0 4 = tanh σ l 0 (0 + ) 4 e -2t/τκ(l 0 ) with τ κ (l 0 ) = 4N φ κ (l 0 )E . ( 21
)
In terms of the inverse temperatures,

b l 0 (t) = β    b 0 l 0 /β + tanh t/τ κ (l 0 ) b 0 l 0 /β tanh t/τ κ (l 0 ) + 1    2 . (22)
The optimal closure therefore predicts a tanh-decay of the energy perturbation E l 0 (t) = 1/b l 0 (t), namely :

E l 0 (t) = (1/β) tanh ±2 (∆ 0 + t/τ κ (l 0 )) with ∆ 0 = atanh b 0 l 0 /β ∓1/2 ,
and ± = sign b 0 l 0 /β -1 .

(

) 23
We finally note that for the GOY dynamics, the constants φ κ (l) can be explicitly determined from Equations ( 4) and (17) as

φ κ (l) = κ 2 k 2 0 λ 2l • λ 6 + λ 2 (λ -1) 2 + 1 . (24)

Turbulent dissipation and asymptotics

The set of predictions ( 20)-( 24) is not trivial. In particular, it is worth emphasizing that the tanh "optimal decay" is very different from standard exponential damping that would be obtained using EDQNM. In our case, only small perturbations are damped exponentially. The initial decay of large perturbations is algebraic.

To see this, let us first follow the optimal decay of an initially large energy perturbation, say E l 0 (t) = (1 + δ l 0 (t))/β with δ l 0 (0 + ) 1 (or alternatively β 0 l 0 /β 1). At early times (t τ κ (l 0 )), the tanh decay follows a power-law dependence : δ l 0 (t) ∝ t -2 k -2 (l 0 ). At late times, the damping becomes exponential δ l 0 (t) ∼ 2e -t/τκ(l 0 ) . Similarly, any initially small perturbation, say 0 < δ l 0 (0 + ) 1, also essentially decays exponentially : δ l 0 (t) ∼ δ l 0 (0 + )e -t/τκ(l 0 ) . In both two regimes, the net effect of the bath could be modeled in terms of a scale-dependent eddy viscosity ν opt (l 0 ) = k -2 (l 0 )τ -1 κ (l 0 ), which yields the scaling ν opt (k) ∝ k -1 E/N , where k denotes the magnitude of the k th wavenumber. In this very simple setting, the optimal closure clearly identifies the relevant time scale as the equilibrium time associated to the bath, without any external modeling. In this sense, the perturbed energy is dissipated without requiring any artificial eddy damping.

By contrast, were we to tackle the damping from an EDQNM point of view, the answer would be less definitive. The result depends on the eddy damping introduced to model the transfer of energy between the perturbed shell and its nearest and next-nearest neighbors. On the one hand, if the eddy viscosities are built upon the typical "equilibrium time" τ κ (l 0 ), we recover the damping rate associated to the small perturbation limit of the optimal framework. On the other hand, we may as well assume that the transfers of energy between the perturbed shell and the bath are mediated by a constant flux , in accordance with the standard phenomenology of turbulence. We then obtain the scalings v[k] ∝ k -1/3 , and corresponding turnover time τ [k] ∝ k -2/3 . The eddy viscosity would then scale as k -4/3 . It is thus hard to decide a priori which timescale is the most appropriate in this specific test problem, and hence the EDQNM closure does not produce a clear result. The reader interested in the details of the EDQNM closure for the shell dynamics can refer to Appendix A.

Thermodynamic interpretation

We conclude this section with a comment on the "thermodynamic variable" σ l 0 defined in (18). A direct calculation shows that σ l 0 = log ρ ρ + const.; σ l 0 can therefore be interpreted as minus the entropy of the proxy density ρ. Its time derivative is akin to an entropy production rate, which is directly related to the energy dissipation rate of the perturbed mode. In our problem, the energy dissipation rate can be identified as

l 0 = R|v l 0 | 2 ρ.
In turn, one can check that R|v l 0 | 2 ρ = -ḃl 0 /b 2 l 0 , so thatσl 0 = b l 0 l 0 : the rate of energy dissipation is the instantaneous temperature times the entropy production. When t → ∞, the energy dissipation rate becomes zero because the system has thermalized, while the entropy production rate tends to zero because the true ensemble becomes Gaussian.

Numerics

Protocol

In order to test the predictions of the optimal closure, we perform two series of numerical experiments for the shell dynamics (1), using N = 23 (set A) and N = 31 wavenumbers (set B) with intershell ratio λ = 2, so that k(n) = k 0 2 n ∈ [k min ; k max ]. We simulate relaxations towards the energy equipartition ensemble, which we recall is obtained by setting β = N and ζ = 0 in Equation ( 5). The corresponding averaged energies and helicities are then set to E equi = 1 and H equi = (k 0 /3) 1 -(-2) N . For each experiment, we randomly generate ensembles of perturbed initial configurations using the probability distributions (7). The perturbed energy E l 0 (0 + ) ∈ [0, 1] is determined by the initial Table 1. Parameters used for the numerical experiments. Each pair (E l0 (0 + ), k(l 0 )) determines an ensemble of relaxations, for which N s = 500 relaxations are computed. The time of integration for each realization is T f /k(l 0 ). inverse temperature of the perturbed mode: b l 0 (0 + ) = 1/E l 0 (0 + ). The initial averaged helicity of the perturbed mode is then γ l 0 E l 0 . To guarantee the relaxation towards the equipartition ensemble, the initial temperatures of the N -1 other modes need to be chosen so as to compensate the energy and helicity perturbations. A possible choice is to take them as b l (0

SET N k 0 k min k max δt E equi /N log 2 k(l 0 ) E l 0 (0 + ) T f A 23 2 -1 1
+ ) = b B + ζ B (-1)k(l) with b B and ζ B determined by H equi -γ l 0 E l 0 = 1 b B l =l 0 γ l 1 + b -1 B ζ B γ l and 1 -E l 0 = 1 b B l =l 0 1 1 + b -1 B ζ B γ l . ( 25
)
We report the results observed for E l 0 = 10 -1 , 10 -2 and 10 -3 , when perturbing the wavenumbers 0 ≤ k(l 0 ) ≤ 22 and averaging over N s = 500 realizations for each ensemble. For low-mode perturbations, the variation of the bath helicity is then very small compared to the total helicity,

|γ l 0 E l 0 /H equi | < 3 • 2 l 0 -N 1.
This essentially induces ζ B 0, consistent with the approximation b l (t) β that we used in the previous section.

Time integration is performed using a 4 th -order Runge-Kutta scheme, with a constant timestep δt, that guarantees the conservation of energy and helicity up to 10 -4 during the relaxation for each realization. With this numerical protocol, all the high-frequency modes (k ≥ k(l 0 )) are observed to relax towards their equilibrium values (see Figure 2 for an illustration).

Results

The numerics match two main features of the optimal predictions. When the perturbed wavenumbers are low (k N ), both the shapes of the relaxation profiles, and their timescales are in good agreement with the predictions of the optimal closure. On Figure 3, we display the relaxation profiles in terms of the entropy variable σ, which we recall is the natural thermodynamic variable for the optimal closure framework. For each ensemble, we fit the profile (21) tanh σ 4 = tanh σ0 4 e -2t/τ to the averaged DNS profile, by adjusting numerically both σ0 and τ . The numerical data can then be collapsed on the same graph. These data are shown on the left panel of Figure 3, which gathers together all the relaxations observed for the set A for k ≤ 2 11 , regardless of the perturbation size. This collapse of the data validates the main quantitative prediction of the optimal closure theory.

The right panel of Figure 3 shows for both sets A and B the spectral dependency of the time scales inferred from the numerical experiments. For low to intermediate

log 2 k = 0 log 2 k = 1 log 2 k = 2 log 2 k = 3 log 2 k = 4 log 2 k = 5 log 2 k = 6 log 2 k = 7 log 2 k = 8 log 2 k = 9 log 2 k = 10 log 2 k = 11 exp -2t/τ [k]
10 0 10 1 ∝ k -1 k τ [k]/τ [3]
Figure 3. Left : Relaxation of the entropy σ for the set A and log 2 k[l 0 ] < 11. All the relaxations are here superimposed : to each color correspond three datasets, one for each value of the perturbed energy indicated in Table 1 (see text for details). Right : Spectral dependence of the observed relaxation time τ [k] for sets A (in red) and B (in blue). The symbols code the initial perturbed energy : 10 -1 (•), 10 -2 ( ), and 10 -3 (×). The dotted line represent the average taken over all the ensembles. k (say k 12), they indeed scale as k -1 , as anticipated by the optimal formula (24). We note no significant difference between the two sets A and B. Left : Spectral dependence of the weighing constant κ, measured as the ratio between the numerically adjusted time scale and the optimal time obtained with a weighing constant set to 1. Right : Dependence of the parameter σ0 used for the collapse shown on Figure 3. The color code and the symbols are the same as for Figure 3. The symbols code the initial perturbed energy : 10 -1 (•), 10 -2 ( ), and 10 -3 (×). The color codes the set : red for set A and blue for set B (12). From a practical point of view, the value of κ determines the relaxation time scale, obtained as τ κ (k) = 2(N/69) 1/2 (κk(l 0 )) -1 from Equations ( 21) and (24). For each ensemble, we measure κ as the ratio between the time τ [k] previously determined and the optimal time τ opt (k) = τ κ=1 (l 0 ) = 2(N/69) 1/2 (k(l 0 )) -1 . For low and intermediate values of k (k 12), the constant is found to be fairly close to 1 for every ensemble. It does not vary with the resolution N , nor does it vary with the shell number, in good agreement with the closure. However, one observes a slight dependency on the perturbed energy not captured by the optimal framework. Averaging κ over the first 12 shells, we indeed observe κ = κ(E l 0 ), to be a decreasing function of the perturbed energy ranging from from κ 1.25 for E l 0 = 10 -3 to κ 0.5 for E l 0 = 10 -1 .

Comments

The numerical results support the predictions of the optimal closure. This may seem surprising, in view of the crude and simple Ansatz (7) on which the closure is based. It could be argued that the test problem set that we have used is designed to generate only small deviations from Gaussianity. Indeed, we checked that those deviations develop within one relaxation time, before fading away as the system relaxes to Gibbs equilibrium. Moreover, no significant deviations from Gaussianity were observed in the statistics of the bath. Similarly, we monitored the two-mode correlation, which showed only weak correlations between the perturbed modes and the bath. In our test problem, therefore, the Ansatz is not unsound.

While the low k behavior show remarkable agreement with the optimal predictions, strong departures are observed for the high modes. We can think of two main reasons. From a mathematical point of view, perturbation of a mode with large k will result in a large perturbation in the helicity. Hence the approximation b l (t) β may no longer be valid. More crucially though, the relaxation that the optimal closure intends to describe is caused by a thermalized bath, with the underlying assumption that the latter is composed of fluctuating degrees of freedom. This assumption does not hold when the smallest scales are perturbed, as the bath is then only composed of slow degrees of freedom. Those modes l < l 0 are essentially frozen during the relaxation (see the right panel of Figure 2), and hence we can expect that they play no significant role in the thermalization process. That the optimal closure fails to capture the relaxation in this situation is not surprising, given that it is suited to the reduction of the dynamics of the largest scales of the motion only.

Perspectives

This work has exposed a new kind of turbulent closure, which we term "optimal closure." It has been used to predict the relaxation towards a Gibbs thermalized state of an inviscid shell dynamics, which can be thought of as a minimal model for homogeneous turbulence. The optimal closure has two appealing features: it is self-consistent and bears close connections with thermodynamics. Within its framework, the effective dissipation on the large-scales due to unresolved small scales is a consequence of nonvanishing entropy production, which is present even when Gaussians are taken to proxy true ensemble averages. Numerics shows good agreement with the salient predictions for the single-mode test problem: one being the non-trivial shape of the relaxation profile, the other being the correct time scaling for relaxation.

While this investigation has focused on one simplified model of turbulence, we think that the optimal theory can be useful more widely. Most of the calculations made in the derivation of our closure rely on the properties (1)-( 3). This class of dynamics is in fact very general; it includes, in particular, Burgers equation, as well as 2D and 3D Euler dynamics. For example, for 2D Galerkin Euler dynamics on any bounded domain, the "C" coefficients would be C klm = φ k [φ l , φ m ], with [, ] the usual Poisson Bracket and φ k 's an orthornormal family of the Laplacian eigenmodes. For 3D Euler dynamics, the helical decomposition of the 3D Euler dynamics considered by [31] is also closely analogous to (3), and this feature has been exploited in the past to consider more realistic shell models of turbulence ( [32]). In each of these systems, a Gaussian Ansatz like the one exploited in the present work will lead to a corresponding optimal closure that is almost identical to that derived here. In fact, the steps leading from the Liouville equation ( 6) to the Lagrangian density [START_REF] Kraichnan | [END_REF] will proceed in an entirely analogous way.

However, one must bear in mind some drawbacks of the optimal closure framework. In the present paper the optimal closure is analytically tractable, but in more general situations this is not guaranteed. The crux of the issue lies in solving the Euler-Lagrange equations, or the associated Hamilton-Jacobi equations, for the defining optimization problem. In a companion paper ( [START_REF] Turkington | Submitted to Nonlinearity[END_REF]), a closed set of equations for the inhomogeneous statistical dynamics of a set of large-scale 2D Euler modes is derived at the cost of performing a perturbation expansion in the amplitudes of the nonequilibrium disturbances. Whether the optimal closure can be used efficiently in combination with more complex Ansätze describing far-from-equilibrium and strongly non-Gaussian effects is still an open question, as is the treatment of viscous effects. These are challenging matters for future work.

The effective dissipation rate depends on what we choose to be the damping times 1/µ l 0 mn associated to triads of neighboring shells (l 0 , m, n). A standard choice would express the µ lmn s as µ lmn = µ l + µ m + µ n , (A.5)

with the µ l representing the typical frequency associated to the shell l. In our case, it is however not self-evident what those are. On the one hand, we may assume that the typical time is an equilibrium time so that µ(l) ∼ k(l)b(l) -1/2 (up to a nondimensional constant). For small perturbations, this yields µ(l) ∼ k(l)E 1/2 N -1/2 , and hence τ edqnm (l 0 ) ∼ N 1/2 E -1/2 k(l 0 )φ κ=1 (l 0 ) -1 . Up to a non-dimensional constant, the expression is the same as (21), and the scaling is τ edqnm (k) ∼ N 1/2 E -1/2 k -1 . On the other hand, we may as well assume that the transfers between the perturbed mode and its neighboring shells is given by a Kolmogorov phenomenology. The asssumption of a constant rate of energy transfer yields a scaling v(l) ∼ 1/3 k(l) -1/3 and therefore a typical frequency µ(l) ∼ 1/3 k(l) 2/3 . The effective damping time would be τ edqnm (l 0 ) ∼ N E -1 1/3 k(l 0 ) 2/3 φ κ=1 (l 0 ) -1 , with scaling τ edqnm (k) ∼ N E -1 1/3 k -4/3 .

2 22 3

 3 × 10 -8 1/23 0.044 0 → 22 10 -1 , 10 -2 , 10 -3 10 B 31 2 -8 2 -7 2 22 3 × 10 -8 1/31 0.033 3 → 22 10 -1 , 10 -2 , 10 -3 5

Figure 2 .

 2 Figure 2. Inverse temperatures measured at times (left) and final times (right)for the set A with E l0 (0 + ) = 10 -1 . Each set of color represent a different ensemble ( a different perturbed mode). The black dots represent the averages of the inverse temperatures taken over all the ensembles. It can be observed that at final times the small-scale modes (l > l 0 ) have relaxed towards the equipartition equilibrium, represented by the solid black line β = 1.

10 0 10 1

 1 10 2 10 3 10 4 10 5 10 6

4 Figure 4 .

 44 Figure 4. Left : Spectral dependence of the weighing constant κ, measured as the ratio between the numerically adjusted time scale and the optimal time obtained with a weighing constant set to 1. Right : Dependence of the parameter σ0 used for the collapse shown on Figure3. The color code and the symbols are the same as for Figure3. The symbols code the initial perturbed energy : 10 -1 (•), 10 -2 ( ), and 10 -3 (×). The color codes the set : red for set A and blue for set B

Acknowledgments

The work reported in this paper was partially supported by the National Science Foundation under grant DMS-1312576.

Appendix A. EDQNM for the shell dynamics

To obtain the EDQNM prediction for the damping of the perturbed mode l 0 , one first writes the equation for the second and third order cumulant involved in its dynamics :

The EDQNM dynamics is obtained by modifying both sides of the equation for the third moment: the two averages • and • ρ are identified to evaluate the fourth-order moment on the r.h.s and a damping term is added on the l.h.s to compensate for the latter approximation. The equation that we now use is

where the coefficient µ lmn has dimension 1/time. It is meant to capture the damping caused by by the higher order statistics. Identifying • ρ and • for the l.h.s of the equation for the second moment yields

where the Markovian approximation is implemented to make the final approximation. Further assuming b m (t) = β for l = l 0 , we get the evolution of the perturbed energy in terms of the the function f previously defined by (15) :

If the µ l 0 mn do not depend on the perturbed energy, then the evolution of the energy perturbation δ l 0 (t) = β/b l 0 (t) -1 is exponential : δ l 0 (t) = δ l 0 (0 + )e -2t/τ edqnm (l 0 ) with τ edqnm (l 0 ) = β 1≤m,n≤N f [l 0 , m, n] µ l 0 mn -1

.

(A.4)