Pierre Bochard
email: bochard@math.univ-lyon1.fr

Paul Pegon
email: paul.pegon@math.u-psud.fr

A KINETIC SELECTION PRINCIPLE FOR CURL-FREE VECTOR FIELDS OF UNIT NORM

 by Jabin, Otto and Perthame. In their article they proved that planar vector fields taking value into the unit sphere of the euclidean norm and satisfying a given kinetic equation are locally Lipschitz. Here, we study the same question replacing the unit sphere of the euclidean norm by the unit sphere of any norm. Under natural asumptions on the norm, namely smoothness and a qualitative convexity property, that is to be of power type p, we prove that planar vector fields taking value into the unit sphere of such a norm and satisfying a certain kinetic equation are locally 1 p-1 -Hölder continuous. Furthermore we completely describe the behaviour of such a vector field around singular points as a vortex associated to the norm. As our kinetic equation implies for the vector field to be curl-free, this can be seen as a selection principle for curl-free vector fields valued in spheres of general norms which rules out line-like singularities.

Preliminaries and notations

Our notations in this article will be standard. Beware that we will consider in this article maps instead of equivalence classes for the almost everywhere relation; all maps will be taken Borel measurable. More specifically, let Ω, E ⊂ R 2 . We will denote by L 1 loc (Ω, E) the set of Borel maps m which are locally integrable and such that m(x) ∈ B for all x in Ω. Of course we do not lose generality with

Introduction

The goal of this paper is the generalization of regularity results on solutions of a kinetic equation obtained by Jabin, Otto and Perthame in their article [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF]. In order for the reader to understand where we are starting from, we will begin by briefly recalling their results. Let Ω ⊂ R 2 be an open set. Jabin, Otto and Perthame were interested in the study of the following micromagnetic energy on H 1 (Ω, R 2 ):

E ε (m) = ε ˆΩ |Dm| 2 + 1 ε ˆΩ(1 -|m| 2 ) 2 + 1 ε ˆR2 |∇ -1 curl m| 2 , (1.1)
where m is extended by zero outside of Ω. They associate to the measurable vector field m : Ω → R 2 the quantity:

χ m : Ω × S 1 -→ {0, 1} (x, s) -→ 1 {m(x)•s>0} .
The introduction of such a quantity is motivated by the following averaging formula: let u ∈ S 1 , then

u = 1 2 ˆS1 1 {u•s>0} s dH 1 (s) (1.2)
An obvious consequence of (1.2) is that if we suppose that |m| = 1 almost everywhere, then for almost every x in Ω,

m(x) = 1 2 ˆS1 χ m (x, s)s dH 1 (s).
They proved then the following theorem:

Theorem 1.1 (Jabin, Otto, Perthame). Let m be a zero-state energy, that is a limit of a sequence (m ε ) ε>0 ⊂ H 1 (Ω, R 2 ) with E ε (m ε ) ---→ Classical results on the regularity of the average of solutions of kinetic equations tell us (see [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] for example) that equation (1.3) implies for m to be in H 1 2 (Ω, R 2 ). But in the case of this particular kinetic equation, it turns out that combining (1.3) with |m| = 1 implies a much stronger regularity. More precisely, the following holds: Theorem 1.2 (Jabin, Otto, Perthame). Let m be a measurable vector field on Ω satisfying |m| = 1 almost everywhere and equation (1.3). Then m is locally Lipschitz continuous inside Ω except at a locally finite set of singular points. Furthermore, for every singular point p, there is α ∈ {-1, 1} such that in any convex neighborhood of p, m(x) = α xp |x -p| a.e.

Before we state our own results, let us give a bit more insight into the two previous theorems. Looking at Theorem 1.1, we expect a zero-state energy m to be of modulus one and curl-free because of the energy (1.1). As a consequence, the curl-free information about m has to be encompassed into equation (1.3). It is indeed easy to see that if a modulus one vector field checks equation (1.3) then it is curl-free; we will prove this in our more general setting in Proposition 4.1. Keeping this in mind, we can know look at Theorem 1.2 in the following way: equation (1.3) is a selection principle among curl-free vector fields taking values into S 1 which, roughly speaking, does not allow for line-like singularities.

The question we are addressing here is the following: can we find a kinetic equation which will act as a selection principle for curl-free vector field taking value into the unit sphere of any norm instead of S 1 ? The purpose of this article is to prove that if we make simple geometric assumptions on our norm, the answer to this question is yes. In the spirit of Theorem 1.2, we will then prove that this kinetic equation implies some regularity on the vector field itself.

Our paper will be divided into three parts. The first one will be devoted to prerequisites on general norms and their modulus of convexity, the second one to the study of a generalized avering formula in the spirit of (1.2) and the last one to our main result, i.e. the regularity of solutions to the kinetic equation we are going to introduce.

Before stating our results, we need some definitions on norms and on the way to measure their convexity.

Definition 1.1. Let • be a norm and B its closed unit ball. We call modulus of convexity associated to B the quantity ω B : [0, 2] → [0, 1] defined by:

ω B (δ) = inf 1 - x + y 2 : x = y = 1 and x -y ≥ δ . (1.4)
Definition 1.2. Let p ≥ 0. We say that a norm is of power type p if there is a constant K > 0 such that:

(1.5) ∀δ ∈ [0, 2], Kδ p ≤ ω B (δ)
As an example, one should keep in mind that the euclidean norm is of power type 2 and that more generally, for p > 1, l p norms are of power type max{p, 2}: this is a consequence of Clarkson's inequalities and can be found in [START_REF] Lindenstrauss | Classical Banach spaces. II[END_REF]. The main reason for introducing theses two notions is that we need some property on the norm in order to obtain a kinetic equation which acts as a selection principle excluding line-like singularities. It turns out that the power type p property is exactly the good property to look at. We refer the reader to the beginning of our third section for a more detailed discussion.

The first important result of our article is an avering formula in the spirit of (1.2):

Theorem 1.3. Let B ⊂ R 2 be a symmetric convex body, that is the unit ball of some norm, then:

(1.6) ∀x ∈ ∂B, x = 1 2 ˆ∂B ⊥ 1 {x•s>0} n B ⊥ (s) dH 1 (s),
where n B ⊥ (s) stands for the unit normal to ∂B ⊥ at s.

Remark 1.1. Note that formula (4.1) generalizes easily in any dimension in the case of the euclidean sphere. We refer the reader to [START_REF] Bochard | Kinetic formulation of vortex vector fields[END_REF] for such a formula and its use to prove a rigidity result obtained in the study of an analog of equation (1.3) in dimension greater than 2. It would be nice to know if we can obtain an averaging formula like (1.6) in greater dimension for a general symmetric convex body.

The following corollary of Theorem 1.3, which is crucial to prove our main result, is immediate.

Corollary 1.4. Let . be a norm, B its closed unit ball and m : Ω → R 2 a Borel vector field such that m = 1. We associate to m the quantity:

χ m : Ω × ∂B ⊥ -→ {0, 1} (x, s) -→ 1 {m(x)•s>0} .
Then, for all x in Ω,

(1.7) m(x) = 1 2 ˆ∂B ⊥ χ m (x, s)n B ⊥ (s) dH 1 (s).
We need a last definition to state our main result:

Definition 1.3.
Let . be a norm with unit ball B and • ⋆ its dual norm defined as usual by the equality:

x ⋆ = sup y ≤1
x, y .

We call vortex associated to the norm • the function V B : x → ∇ x • ⋆ at the points where this quantity makes sense. The map x → . being convex, V B is well-defined almost everywhere on R 2 according to Rademacher's theorem.

We can now state the main theorem of our article.

Theorem 1.5. Let . be a norm of power type p and B its closed unit ball such that ∂B is a C 1 submanifold in R 2 . Let m : Ω → R 2 be a Borel vector field such that m = 1 and satisfying:

∀s ∈ ∂B ⊥ , n B ⊥ (s) ⊥ • ∇χ m (•, s) = 0 in D ′ (Ω). (1.8)
Then m is locally 1 p-1 -Hölder continuous outside a locally finite set of singular point. Furthermore, for every singular point p, there is α ∈ {-1, 1} such that in any convex neighborhood of p, m(x) = αV B (xp) almost everywhere, where V B is the vortex associated to . defined in Definition 1.3. Remark 1.2. Notice that the two hypotheses on the norm are not redundant: we can find norms which are not of power type p for any p and whose unit sphere is a C 1 submanifold. Take for example the norm associated to the l 1 unit ball whose angles have been smoothed out. On the other hand, an example of a type p norm whose unit ball is not a C 1 submanifold is given by x = x 1 + x p with p > 2. The reader interested in the links between convexity and smoothness of the ball of a norm will consult with profit Section 1.e. of [START_REF] Lindenstrauss | Classical Banach spaces. II[END_REF].

In order to see how Theorem 1.5 could be used, let us note that Theorem 1.2 itself has interesting consequences. For example, together with clever commutator estimates, De Lellis and Ignat proved in [START_REF] De | A regularizing property of the 2D-eikonal equation[END_REF] the following: Theorem 1.6 (De Lellis, Ignat). Let Ω be an open set in R 2 and r ∈ [START_REF] Bochard | Kinetic formulation of vortex vector fields[END_REF][START_REF] De | A regularizing property of the 2D-eikonal equation[END_REF]. Then every m in

W 1 r ,r curl (Ω, S 1 ) := m ∈ W 1 r ,r (Ω, R 2 ) : curl m = 0 in D ′ (Ω) and |m| = 1 a.e.
is locally Lipschitz continuous inside Ω except at a locally finite set of singular points, where W 1 r ,r (Ω, R 2 ) stands for the usual fractional Sobolev space.

A first natural question would be to ask if we can obtain such a theorem when S 1 is changed into the unit sphere of a norm satisfying the hypotheses of Theorem 1.5. Another natural question, in view of Theorem 1.1, would be to ask if we can obtain an analog of Theorem 1.1 if we change the energy (1.1) into:

Ẽε (m) = ε ˆΩ |Dm| 2 + 1 ε ˆΩ(1 -m 2 ) 2 + 1 ε ˆR2 |∇ -1 curl m| 2 ,
where . is now a norm satisfying the hypotheses of Theorem 1.5. Note that the proof of both Theorem 1.6 and Theorem 1.1 rely on the notion of entropy, that is, in this context, smooth functions Φ :

S 1 → R 2 such that for all open Ω in R 2 , ∀m ∈ C ∞ curl (Ω, S 1 ), ∇ • [Φ • m] = 0.
We refer the reader to [START_REF] De | Structure of entropy solutions to the eikonal equation[END_REF] and [START_REF] Ignat | Entropy method for line-energies[END_REF] for more details and applications of this notion. Therefore a first step in order to obtain analogs of Theorems 1.1 and 1.6 would be to develop an adapted notion of entropy. We will adress this question in a paper to come. The reader interested in questions related to the regularity of solutions of the eikonal equation may consult [START_REF] Caffarelli | Distance functions and almost global solutions of eikonal equations[END_REF].

Modulus of convexity and vortices

Our main sources for this section are [START_REF] Lindenstrauss | Classical Banach spaces[END_REF] and [START_REF] Lindenstrauss | Classical Banach spaces. II[END_REF] on the geometry of norms and [START_REF] Ekeland | Convex analysis and variational problems[END_REF] on convex analysis. For the reader's comfort, let us collect some general facts on convex functions and convex bodies:

• A function f : R d → R ∪ {+∞} is said convex if f ((1 -t)x + ty) ≤ (1 -t)f (x) + tf (y) for all x, y ∈ R d , t inf[0, 1]. Its domain is defined as Dom(f ) := {x : f (x) < ∞}. • The subdifferential of f at x is defined as ∂f (x) := {p : ∀y, f (y) ≥ f (x) + p, y -x }. It is nonempty if x ∈ • Dom(f ).
In that case, f is differentiable at x if and only if its subdifferential is a singleton, in which case ∂f (x) = {∇f (x)}.

• The Legendre-Fenchel transform of f is defined as

f ⋆ (p) = sup x x • p -f (x).
If f is a proper ( ≡ +∞) convex lsc function, so is f ⋆ , and f = f ⋆⋆ . • In that case, ∂f ⋆ is the inverse of ∂f in the sense that (2.1) p ∈ ∂f (x) ⇐⇒ x ∈ ∂f ⋆ (p).

• If B is a convex body and • its associated norm, then

χ ⋆ B = • ⋆ . • If x ∈ ∂B, we say that u is a normal vector to B at x if ∀y ∈ B, u • (y -x) ≤ 0.
Notice that the set of normal vectors, denoted by N B (x), is a convex cone which is precisely ∂χ B (x).

• If x ∈ ∂B, the following relation between the subdifferentials of • and χ B holds:

(2.2) ∂ x • = ∂χ B (x) ∩ B ⋆ , or equivalently ∂ x • = N B (x) ∩ B ⋆ .
• There is a unique unit normal vector at x ∈ ∂B if and only if the associated norm is differentiable at x. In general this is true for H 1 -almost every x ∈ ∂B, and it is the case for all x ∈ ∂B for instance if ∂B is a C 1 submanifold.

Before getting to the heart of the matter, let us explain why the norm should satisfy some property for the kinetic equation to act as a selection principle ruling out line-like singularities. For that purpose, let us take another look at the vortex V B associated to a norm • of unit ball B as defined in Definition

1.3. It is clear that V B is curl-free in D ′ (R 2 ). Indeed, taking ϕ ∈ C ∞ 0 (R 2 ), ˆR2 ∂ 1 ϕV B,2 -∂ 2 ϕV B,1 dx = ˆR2 ∇ϕ ⊥ • ∇ x ⋆ dx = ˆR2 ∇ • ∇ϕ ⊥ x ⋆ dx = 0,
where we used the fact that V B is the distributional gradient of

→ P(S 1 ) x -→ N B (x) ∩ S 1 ,
where N B (x) is the normal cone to B at x. Lemma 2.1. Let B be the unit ball of a norm and n B the application defined above. Then, for H 1 -almost every y in S 1 , there is a unique x in ∂B such that y ∈ n B (x).

Proof. Let us consider:

ñB : ∂B -→ P(∂B ⋆ ) x -→ N B (x) ∩ ∂B ⋆ , ñB ⋆ : ∂B ⋆ -→ P(∂B) x -→ N B ⋆ (x) ∩ ∂B.
By (2.1) and (2.2), it is easy to see that p ∈ ñB (x) if and only if x ∈ ñB ⋆ (p). Since ∂B ⋆ admits a normal vector for H 1 -almost every p, we get that there is a unique x satisfying p ∈ ñB (x) for H 1 -almost every p in ∂B ⋆ . We get uniqueness for n B by using the fact that the projection from S 1 to ∂B ⋆ is a bi-Lipschitz bijection.

Remark 2.1. In the rest of the paper, if y ∈ S 1 is such that there is a unique x ∈ ∂B satisfying y ∈ n B (x), we will abusively denote it by x = n -1 B (y). One can deduce from the previous proof that this in in particular the case if

∂B ⋆ is a C 1 submanifold of R 2 . Proposition 2.2. Let • be a norm of unit ball B, then for almost every x, V B (x) = n -1 B x |x| . Proof. We know that • ⋆ is the Legendre-Fenchel transform of χ B . Consequently • ⋆ is differentiable at x if and only if x is in the normal cone of a unique p ∈ ∂B, that is if and only if n -1 B x |x| is well defined, in which case n -1 B x |x| = p = ∇ x • ⋆ . = V B (x)
. By the previous lemma, it is true for almost every x.

We can now prove as promised that V B is always solution to the kinetic equation (1.8). Indeed, let

Ω ⊂ R 2 , ϕ ∈ C ∞ 0 (Ω), and s ∈ ∂B. Using Proposition 2.2, for almost every x ∈ Ω, V B (x) = n -1 B x
|x| , and

ˆΩ ∇ϕ • n B ⊥ (s) ⊥ χ VB (x, s) dx = ˆΩ∩{n -1 B x |x| •s>0} ∇ϕ • n B ⊥ (s) ⊥ dx.
But for almost every s, n -1

B x

|x| • s > 0 is an half-space whose boundary is the line L passing through 0 and directed by n B (s ⊥ ), whose unit normal is ±n B (s ⊥ ) ⊥ = ∓n B ⊥ (s). As a consequence, using Stokes formula and the fact that ϕ is supported in Ω,

ˆΩ∩{n -1 B x |x| •s>0} ∇ϕ • n B ⊥ (s) ⊥ dx = ±n B ⊥ (s) ⊥ • n B ⊥ (s) ˆΩ∩L ϕ dx = 0.
Let us define a last quantity that we will need later. For u, v ∈ R 2 which are not colinear, we define the convex cone C(u, v) generated by u, v as

C(u, v) := λ 1 u + λ 2 v : (λ 1 , λ 2 ) ∈ R + × R + .
Lemma 2.3. Let • be a strictly convex and differentiable (outside 0) norm of closed unit ball B.

Then

n B : ∂B → S 1 is monotonic in the sense that w ∈ C(u, v) ⇐⇒ n B (w) ∈ C(n B (u), n B (v)).
Proof. Notice that C(u, v) is (by Hahn-Banach Theorem) the intersection of the half-spaces which are delimited by vector hyperplanes and which contain u and v. First, let us show that, given any such half-space H, n B is a bijection between ∂B ∩ H and a half-circle of S 1 . Since n B is continuous and ∂B ∩ H is compact and connected, n B (∂B ∩ H) is a compact and connected subset of S 1 , hence it is a closed arc of S 1 . Since ∂B ∩ H contains two symmetric points ±x ∈ ∂B, so does n B (∂B ∩ H), hence it contains a half-circle [θ, θ + π]. It may not be larger than that, otherwise it would contain a nontrivial arc and its symmetric, hence ∂B ∩ H as well since n B is homeomorphic and antisymmetric. This cannot be true. This proves that n B sends half-spaces of ∂B to half-circles and it is clear that it realizes a bijection between the set of half-spaces of ∂B and the set of half-circles of S 1 . Now w ∈ C(u, v) means that w belongs to any half-space (delimited by a vector hyperplane) containing u, v, which means that n B (w) belongs to any half-circle containing n B (u), n B (v) by what we just proved. This exactly means that n B (w

) ∈ C(n B (u), n B (v)).

Modulus of convexity.

In this section we give an alternate definition of the modulus of convexity, which offers some advantages: an intuitive interpretation is given in Proposition 2.4, and a characterization in terms of the regularity of the vortex is given in Proposition 2.5. This will be useful to prove the sharpness of our main theorem. Of course, it is essentially equivalent to Definition 1.1 which is given in the introduction, as showed in Remark 2.3.

Definition 2.1. We define the modulus of convexity of B as the greatest nondecreasing function ρ

B : [0, 2] → R + such that for all x, y ∈ ∂B and all u ∈ N B (x), (2.3) u • y ≤ u • x -u ⋆ ρ B ( y -x ).
We say that B is of power type p ≥ 2 if ρ B (δ) ≥ Cδ p for some C > 0 and that it is elliptic if it is of power type 2.

Remark 2.2. Notice that

∀y ∈ ∂B, u • y ≤ u • x -u ⋆ ρ B ( y -x )
if and only if

∀y ∈ R d , χ B (y) ≥ χ B (x) + u • (y -x) + u ⋆ ρ B ( y -x ).
This last inequality is reminiscent of the inequality f (y) ≥ f (x) + p • (yx) for p ∈ ∂f (x) which holds for arbitrary convex functions. The extra term u ⋆ ρ B ( yx ) measures how much B is convex around its boundary.

Remark 2.3. The two definitions of modulus of convexity are equivalent in the sense that:

ρ B (δ/2) ≤ ω B (δ) ≤ ρ B (δ)/2.
Proof. The proof is quite easy. Take x, y ∈ ∂B and u

∈ N B ((x + y)/2) such that u ⋆ = 1. Notice that u ∈ ∂ x+y 2 • (because u ⋆ = 1
), hence by convexity:

1 = x ≥ x + y 2 + u • x -y 2 .
By definition of ρ B one has:

u • y ≤ u • x + y 2 -ρ B y - x + y 2 ,
or equivalently

u • y -x 2 ≤ -ρ B y -x 2 .
Putting this in the previous inequality yields:

x + y 2 ≤ 1 -u • x -y 2 ≤ 1 -ρ B y -x 2 .
Since ω B is precisely the largest map for which the inequality x+y 2 ≤ 1ω B ( yx ) holds for all x, y ∈ ∂B, it follows that:

ρ B (δ/2) ≤ ω B (δ)
for all δ ∈ [0, 2]. Now let us prove the other inequality. This time we take x, y ∈ ∂B, u ∈ N (x) with u ⋆ = 1. By definition we know that:

x

+ y 2 ≤ 1 -ω B ( y -x ),
but since u ⋆ = 1, one has:

x + y 2 ≥ u • x + y 2 = u • x + u • y -x 2 = 1 + u • y -x 2 .
Putting these two together, one gets:

u • y ≤ u • x -2ω B ( y -x ),
and by homogenity if u has arbitrary norm:

u • y ≤ u • x -u ⋆ • 2ω B ( y -x ).
Since ρ B is the largest function for which such an inequality holds for all x, y and u ∈ N B (x), one has

2ω B (δ) ≤ ρ B (δ)
which concludes the proof. As indicated by its name, the greater ρ B is, the more convex B is. For example, taking a look at Figure 1, it is easy to see that the modulus of convexity associated to the l 1 norm is constant equal to 0 on a neighborhood of 0. On the other side, the euclidean ball is the most convex of all in the sense that if ω 2 is its modulus of convexity, for all symmetric convex body B,

(2.4) ∀δ ∈ [0, 2], ω B (δ) ≤ ω 2 (δ).
We refer the reader to [START_REF] Nordlander | The modulus of convexity in normed linear spaces[END_REF] for a proof of this inequality.

Remark 2.4. Using the symmetry of S 1 , ρ 2 can easily be computed and one finds:

ρ 2 (δ) = 1 -1 -δ 2 ∼ δ→0 δ 2 2 .
As a consequence of this equality, of inequality (2.4) and Remark 2.3, there is no norm of power type p with p < 2.

Throughout the rest of this paper, we will use the modulus of convexity ρ B as defined in this section. The following proposition gives an intuitive characterization of convex bodies of power type p. It states that they are convex bodies whose boundary is locally the graph of a map which is above that of the map t → |t| p around 0.

Proposition 2.4. Let x ∈ ∂B, u a normal vector of unit euclidean norm. Take γ :]ε, +ε[→ ∂B a local parameterization of ∂B around x = γ(0), with direct orientation and unit speed. We denote by τ = γ(0) the unit tangent at x and ν = τ ⊥ the inner normal. In the basis (τ, ν), we write

γ(t) = x + a(t)τ + b(t)ν. Then B is
γ(t) = x + t γ(0) + t 2 /2 • γ(0) + o(t 2 ) = x + tτ + κ(x)t 2 /2 • ν + o(t 2 ).
Consequently a(t) = t + o(t 2 ) and b(t) = κt 2 /2 + o(t 2 ), thus B being elliptic, one has:

C(t + o(t 2 )) 2 ≤ κt 2 /2 + o(t 2 ).
Dividing by t 2 and t → 0 yields κ ≥ 2C and ∂B has positive curvature. The converse is straightforward.

The next proposition establishes a link between the power type p property of a norm and the smoothness of its vortex. Proposition 2.5. Let B be a convex body associated to the norm • . Then B is of power type p if and only if • ⋆ is differentiable everywhere outside 0 and

V B . = ∇ • ⋆ is 1 p-1
-Hölder continuous far from 0, in the sense that for some constant H > 0 one has:

V B (v) -V B (u) ≤ H δ 1 p-1 u -v 1 p-1 ⋆ , for all u , v ≥ δ and all δ > 0. Proof. Take x, y ∈ ∂B and u ∈ N B (x), v ∈ N B (y) with u, v nonzero. One has u • y ≤ u • x -u ⋆ ρ B ( y -x ), v • x ≤ v • y -v ⋆ ρ B ( y -x ),
and by summing these one gets:

2 min( u ⋆ , v ⋆ )ρ B ( y -x ) ≤ (v -u) • (y -x). Now if B is of power type p then C y -x p ≤ ρ B ( y -x ) ≤ (v -u) • (y -x) min( u ⋆ , v ⋆ ) ≤ v -u ⋆ y -x min( u ⋆ , v ⋆ ) , thus y -x ≤ u -v ⋆ C min( u ⋆ , v ⋆ ) 1 p-1 , which yields y -x ≤ H δ 1 p-1 u -v 1 p-1 ⋆ .
provided that u ⋆ , v ⋆ ≥ δ, for some constant H depending only on the norm.

Conversely, take x ∈ ∂B, u ∈ N B (x) with u ⋆ = 1. We want to show that for all y ∈ B:

u • y ≤ u • x -C|y -x| p . Take x 0 a minimizer of g(y) = χ B (y) + (u • (x -y) -C |y-x| p p
). We want to show that for small enough C, g(x 0 ) ≥ 0. The necessary optimality condition reads

u 0 := u + C(x 0 -x) p-1 ∈ ∂χ B (x 0 ). Necessarily, x 0 ∈ B thus u 0 -u ⋆ ≤ δC x 0 -x p-1 ≤ δC2 p-1
where δ

:= sup v =1 v ⋆ . If C is small enough, say Cδ2 p-1 ≤ 1/2 then both u ⋆ , u 0 ⋆ ≥ 1/2.
On the other hand one has by hypothesis:

x 0 -x p-1 ≤ 2H p-1 u 0 -u ⋆ ≤ 2δCH p-1 x 0 -x p-1 .
Now if one takes C even smaller, so that 2δCH p-1 < 1, one must have x = x 0 and u = u 0 , which implies that min R 2 g = g(x 0 ) = g(x) = 0 and the following holds:

∀y ∈ B, u • y ≤ u • x -C|y -x| p /p with C = min(2 -p , 1/(2δH p-1 )).
If u has arbitrary norm, this becomes:

∀y ∈ B, u • y ≤ u • x -C ′ u ⋆ y -x p where C ′ = C/p.
As a corollary, one gets the following regularity theorem.

Theorem 2.6. Let • be a norm with closed unit ball B. It is of power type p if and only if n -1 B : S 1 → ∂B is well-defined and 1 p-1 -Hölder continuous. In particular it is elliptic iff n -1 B is Lipschitz continuous.

Averaging formula

This section is devoted to the proof of Theorem 1.3. A first interesting remark is that if we want an averaging formula in the spirit of formula (1.6) to be true, we need the convex to be symmetric. More precisely, we have the following proposition: Proposition 3.1. Let B ⊂ R 2 be a convex body with 0 in its interior. Assume there is a Borel vector measure µ ∈ M 2 (R 2 \ {0}) such that the following representation formula holds:

∀x ∈ ∂B, x = ˆR2 1 {x•s>0} µ(ds). (3.1)
Then B is symmetric and one may replace µ in the formula by the measure μ defined by μ(A) := µ(A)µ(-A) 2 for all Borel set A, which is an antisymmetric measure in the sense that μ(-A) = -μ(A), and which does not give mass to vector lines.

Proof. Let us set

x + := {s ∈ R 2 : x, s > 0}, x -:= {s ∈ R 2 : x, s < 0}, x ⊥ := {s ∈ R 2 : x, s = 0}. (3.2)
One may define the gauge of B as

j B : R 2 -→ R + x -→ inf{t > 0 : x ∈ tB},
the radial projection onto ∂B and radial symmetry with respect to B respectively as:

p B : R 2 \ {0} -→ ∂B x -→ x jB (x) , s B : R 2 \ {0} -→ R 2 x -→ j B (x)p B (-x).
First, let us assume that µ does not charge lines of the plane, i.e. µ(D) = 0 for all vector line D ⊂ R 2 . Writing R 2 = x + ⊔ x -⊔ x ⊥ and noticing that s B (x) + = x -, we obtain:

µ(x + ) + µ(s B (x) + ) = µ(x + ) + µ(x -) + µ(x ⊥ ) = µ(R 2 ) =: v ∈ R 2 .
If x is in ∂B, according to (3.1) one has x = µ(x + ) and s B (x) = µ(s B (x) + ) since s B (x) ∈ ∂B, which yields

(3.3) ∀x ∈ ∂B, x + s B (x) = v.
But s B (x) is colinear to x, and since

• B = ∅, we can find two non colinear vector x 1 , x 2 ∈ ∂B. Now writing (3.3) with x 1 and x 2 implies v = 0; consequently, for all x ∈ ∂B, x = -s B (x) = -p B (-x), which exactly means that B is symmetric. Now we may get rid of the hypothesis that µ does not charge lines. If µ gives a positive mass to some vector lines, it may only charge countably many of them since µ is finite and these lines only intersect at 0 which is µ-negligible. Let us denote by L the reunion of the perpendiculars to these lines. The previous reasoning shows that x + s B (x) = v for all x ∈ ∂B \ L. But ∂B ∩ L is countable hence ∂B \ L is dense in ∂B and by continuity of s B the identity

x + s B (x) = 0 = v
holds for all x ∈ ∂B, which implies that B is symmetric and µ does not charge vector lines. Thus if x ∈ ∂B then -x ∈ ∂B, which implies:

-x = ˆR2 1 {-x•s>0} µ(ds) = µ(-(x + )) and x = x -(-x) 2 = µ(x + ) -µ(-(x + )) 2 := μ(x + ).
Obviously, μ is antisymmetric and does not charge lines since µ does not either. Now let us pass on to the proof of the representation formula as stated in Theorem 1.3. It turns out that checking such a formula is quite easy once one has found the right candidate for µ, but the following theorem is more precise as it states that µ is essentially unique. Theorem 3.2. Let B ⊂ R 2 be a symmetric convex body, that is the unit ball of some norm, then there is a unique antisymmetric measure µ supported on ∂B ⊥ such that:

(3.4) ∀x ∈ ∂B, x = ˆR2 1 {x•s>0} µ(ds), which is µ = 1/2 • n B ⊥ H 1 ∂B ⊥
where n B ⊥ (s) stands for the unit normal to ∂B ⊥ at s.

Proof. Let B be a symmetric convex body and suppose that there is µ ∈ M 2 (R 2 ) supported in ∂B ⊥ and satisfying the representation formula:

∀x ∈ ∂B, x = ˆ∂B ⊥ 1 {x•s>0} µ(ds). (3.5)
We want to prove that µ =

n B ⊥ 2 H 1 ∂B ⊥ .
Recalling the notations of (3.2), notice that this rewrites more concisely as

(3.6) ∀x ∈ ∂B, x = µ(x + ). 0 v u x y ∂B ∂B ⊥ Figure 2.
If u, v are two points of ∂B ⊥ , we denote by ]u, v] the oriented arc of ∂B ⊥ delimited by u and v (v included and u not included). Let us set x = u ⊥ , y = v ⊥ , which both belong to ∂B since it is symmetric.

Notice that if the oriented angle between u and v is such that 0 < ∢(u, v) < π then ]u, v] = x + \ y + . Finally we set E := ]u, v] and F := x + \ E, so that

x + = E ⊔ F, y + = F ⊔ (-E), thus µ(x + ) = µ(E) + µ(F ), µ(y + ) = µ(F ) + µ(-E).
Since µ is antisymmetric, µ(-E) = -µ(E) hence substracting the last equality to the previous one yields µ(x + )µ(y + ) = 2µ(E). Using (3.6) one gets

x -y = 2µ(]u, v]), i.e. µ(]u, v]) = - 1 2 (v ⊥ -u ⊥ ) = 1 2 -1 (v -u).
In short, setting F = 1 2 R -1 , for all u, v such that 0 < ∢(u, v) < π, we have:

(3.7) µ(]u, v]) = F (v) -F (u).
This relation 1 allows us to assert that µ is the following measure on ∂B ⊥ :

(3.8) µ(ds) = D s F τ (s) ds = F τ (s) ds
where τ (s) is the unit vector tangent to ∂B ⊥ at s (with direct orientation). By definition,

n B ⊥ (s) = R -1 τ (s) hence (3.8) rewrites µ = n B ⊥ 2 H 1 ∂B ⊥ , which is what we shall prove now. Setting ν = n B ⊥ H 1 ∂B ⊥ , it is easy to check that ν(]u, v]) = F (v) -F (u) = µ(]u, v]
) for all u, v such that 0 < ∢(u, v) < π. Indeed, take a curve γ : [0, l] → L ⊥ which describes the oriented arc [u, v] and is 1 Notice that (3.7) would match the definition of the Stieltjes measure associated to a given function F if µ was a measure on the real line. In this case, µ is given by the distributional derivative of F and we are merely transposing this fact to our case, where µ is a measure on a closed curve.

parameterized by arc-length. Then one has

ν(]u, v]) = ˆl 0 n B ⊥ (γ(t)) dt 2 = ˆl 0 R -1 ( γ(t)) dt 2 = 1 2 R -1 (γ(l) -γ(0)) = F (v) -F (u).
Since the Borel sets ]u, v] form a π-system generating the Borel σ-algebra of ∂B ⊥ , µ = ν. Moreover, since ν satisfies relation (3.7), it satisfies the representation formula (3.6) by taking v → -u. This is what we wanted to prove.

Kinetic formulation

4.1. Kinetic formulation and curl-free vector fields of norm one. This subsection is devoted to justifying the heuristic explanation of our introduction, that is that our kinetic equation acts as a selection principle for norm one curl-free vector fields. First, we have the following easy proposition: Proof. Let m be a vector field satisfying our hypothesis. Using the characteristics method, it is easy to see that such a field is constant along lines. Indeed, writing F (x) := x , differentiating the relation F (m(x)) = 1 in Ω and using the fact that m is curl-free, one gets:

Proof. Let ϕ ∈ C ∞ 0 (Ω). Then, ˆΩ ∂ 1 ϕm 2 -∂ 2 ϕm 1 dx = ˆΩ ∇ϕ ⊥ • m dx = ˆΩ ∇ϕ ⊥ • 1 2 ˆ∂B ⊥ χ m (x, s)n B ⊥ (s) dH 1 (s) dx = 1 2 ˆ∂B ⊥ ˆΩ ∇ϕ • n B ⊥ (s) ⊥ χ m (x, s) dx dH 1 (s) = 0,
(4.1) D x m • ∇F (m(x)) = D T x m • ∇F (m(x)) = 0.

Now let t → γ(t) be a smooth curve taking values in Ω and p(t) := m(γ(t))

. Differentiating p, one obtains ṗ(t) = D γ(t) m • γ(t). Therefore, choosing γ to be a solution of the differential equation γ(t) = ∇F (m(γ(t))) and using (4.1) the couple (γ, p) is now solution of the system:

ṗ(t) = 0, γ(t) = ∇F (p(t)).
Thus p is constant equal to m(γ(0)) = m(x), γ satisfies γ = ∇F (m(x)) and m is constant along γ, that is m is constant on L x ∩ Ω where L x is the line passing through x and directed by ∇F (m(x)). Set s ∈ ∂B ⊥ . If m(x 0 ) • s = 0, then m(x 0 ) = ±s ⊥ , say m(x 0 ) = s ⊥ . Since m is constant along the line γ : t → x 0 + t∇F (s ⊥ ) and equal to s ⊥ , then χ m (γ(t), s) = 0 and by differentiation at t = 0 one gets ∇F (s ⊥ ) • ∇χ m (x 0 , s) = 0 (we write ξ • ∇f to designate the partial derivative of f in the direction ξ provided it exists). But ∇F (s ⊥ ) is the normal vector ξ to B at s ⊥ such that ξ ⋆ = 1, in particular it is colinear to n B (s ⊥ ), thus n B (s ⊥ ) • ∇χ m (x 0 , s) = 0. Now if m(x 0 ) • s > 0, then m(x) • s > 0 along γ, otherwise it must vanish somewhere along that line and m ≡ ±s ⊥ on that line, which is a contradiction. Consequently, χ m (•, s) is constant along γ, thus we get again by differentiation along γ at t = 0: n B (s ⊥ ) • ∇χ m (x 0 , s) = 0. The same reasoning works for m(x 0 ) • s < 0. Noticing that n B ⊥ (s) ⊥ = n B (s ⊥ ), this implies that

∀s ∈ ∂B ⊥ , ∀x ∈ Ω, n B ⊥ (s) ⊥ • ∇χ m (x, s) = 0. (1.8)
4.2. Direction conservation and trace theorem. We are now entering the heart of the proof of Theorem 1.5, which is quite similar in spirit to the proof of Theorem 1.2 by Jabin Otto and Perthame. In order to use equation (1.8) to gain regularity, an important point will be to be able to define the trace of m along a line. Remark 4.1. Note that starting from here, we will always place ourselves under the asumptions of Theorem 1.5, that is the unit sphere ∂B associated to a norm will be a C 1 submanifold and the norm itself will be of power type p.

Definition 4.1. Let m ∈ L 1 loc (Ω, R 2 ). We say that x 0 is a Lebesgue point of m if: lim r→0 Br (x0) |m(x) -m(x 0 )| dx = 0.
We denote by Leb(m) ⊂ Ω the set of Lebesgue point of m. It is well known (see [START_REF] Evans | Measure theory and fine properties of functions[END_REF] for example) that

|Ω \ Leb(m)| = 0.
Let us state a useful lemma which relates Lebesgue points of m to those of χ m (•, s). Proof. The proof of (i) is a simple consequence of the averaging formula (1.7). Indeed, if x 0 is a Lebesgue point of χ m (•, s) for H 1 -a.e. s ∈ ∂B ⊥ :

Br (x0) |m(x) -m(x 0 )| dx = 1 2 Br (x0) ˆ∂B ⊥ (χ m (x, s) -χ m (x 0 , s))n B ⊥ (s) dH 1 (s) dx ≤ 1 2 Br (x0) ˆ∂B ⊥ |χ m (x, s) -χ m (x 0 , s)| dH 1 (s) dx = 1 2 ˆ∂B ⊥ Br (x0) |χ m (x, s) -χ m (x 0 , s)| dx dH 1 (s) r→0 ---→ 0,
by the dominated convergence theorem.

Now let us prove (ii)

. Take a Lebesgue point x 0 of m and s ∈ ∂B ⊥ such that m(x 0 ) • s = 0, say for example m(x 0 ) • s > 0. First, let us remark that

m(x 0 ) • s |B r (x 0 ) ∩ {m • s ≤ 0}| |B r (x 0 )| ≤ 1 B r (x 0 ) ˆBr(x0)∩{m•s≤0} (m(x 0 ) • s -m(x) • s) dx ≤ Br(x0) |m(x 0 ) -m(x)| dx.
Dividing by m(x 0 ) • s and taking the limit when r goes to 0, one gets

|B r (x 0 ) ∩ {m • s ≤ 0}| |B r (x 0 )| r→0 ---→ 0, but Br (x0) |χ m (x, s) -χ m (x 0 , s)| dx = Br (x0) |χ m (x, s) -1| dx = |B r (x 0 ) ∩ {m • s ≤ 0}| |B r (x 0 )| r→0 ---→ 0, hence x 0 is a Lebesgue point of χ m (•, s).
The case m(x 0 ) • s < 0 is done in a similar way.

We can now prove the following proposition, which plays a crucial role in the proof of Theorem 1.5.

∈ ∂B ⊥ with z -y ∈ Vect(n B ⊥ (s) ⊥ ). Then m(y) • s > 0 (resp. < 0) =⇒ m(z) • s ≥ 0 (resp. ≤ 0). (4.2)
Proof. Let us suppose that m(y) • s = 0. If m(z) • s = 0, then the proof is over; assume otherwise that m(z) • s = 0. Fix x ∈ Ω and define for ε small enough:

χ ε (x) := Bε(x)
χ m (y, s) dy, which may be written as

χ ε (x) = ρ ε * χ m (•, s) where ρ ε := 1 Bε(0) |B ε (0)| .
χ ε is continuous and for every Lebesgue point x 0 of χ m (•, s), one has χ ε (x 0 ) ε→0 ---→ χ m (x 0 , s). Now let (δ n ) n∈N * be a sequence of smooth mollifiers supported in B 1/n and set χ ε n := δ n * χ ε . We are going to show that χ ε n (y) = χ ε n (z) and for that we introduce g :

[0, 1] -→ R t -→ χ ε n (y + t(z -y))
. The function g is smooth and satisfies

g ′ (t) = (z -y) • ∇χ ε n (y + t(z -y)) = (z -y) • ∇[δ n * ρ ε ] * χ m (•, ξ) (y + t(y -z)) = ˆR2 (z -y) • ∇ [δ n * ρ ε ] (y + t(z -y) -v)χ m (v, s) dv. But (z -y) ∈ Vect(n B ⊥ (s) ⊥ ) and δ n * ρ ε ∈ C ∞ c (Ω)
; therefore because of (1.8), g ′ (t) = 0 and χ ε n (y) = g(0) = g(1) = χ ε n (z). Taking the limit as n goes to +∞, we obtain

χ ε (y) = χ ε (z).
But y, z are Lebesgue points of m and m(y) • s = 0, m(z) • s = 0 hence by (ii) of Lemma 4.3 y, z are Lebesgue points of χ m (•, s), thus taking the limit as ε goes to 0, we have

χ m (y, s) = lim ε→0 χ ε (y) = lim ε→0 χ ε (z) = χ m (z, s),
which concludes the proof.

= {tv : t ∈ [-1, 1]} for some v ∈ S 1 . For r > 0 set P r := {x 1 v ⊥ + x 2 v : (x 1 , x 2 ) ∈ [-r, r] × [-1, 1]}
and assume that P r is included in Ω for small r. Then there exists a measurable function m :

[-1, 1] → R 2 satisfying (i) lim r→0 Pr |m(x) -m(x 2 )| dx = 0, (ii) for almost every x 2 , m(x 2 ) ∈ ∂B, (iii) if x ∈ Leb(m) ∩ L then x 2 ∈ Leb( m) and m(x) = m(x 2 ), (iv) if x 2 ∈ Leb( m), then there exists a sequence x n ∈ Leb(m) such that x n → x and m(x n ) → m(x 2 ).
The vector field m is called the trace of m on the line L. Remark 4.2. We only assume the form of L = {tv : t ∈ [-1, 1]} for commodity but the result obviously holds for any general segment in Ω. This will be used several times in the rest of the article.

Equation (1.8) means that χ m only depends on n B ⊥ (s) • x and s. The next lemma asserts that χ m may be written as a Borel function of these quantities. Lemma 4.6. If m satisfies (1.8), for any convex set V (for instance a convex neighborhood of L) there exists a Borel function χm : R × ∂B ⊥ → {0, 1} such that for a.e. x in V ,

χ m (x, s) = χm (n B ⊥ (s) • x, s). (4.3)
Proof. We know that m and χ m are Borel maps. For s fixed, we denote by

x = u 1 n B ⊥ (s) + u 2 n B ⊥ (s) ⊥ the decomposition of x on n B ⊥ (s), n B ⊥ (s) ⊥ .
Taking arbitrary smooth functions φ, ψ of the variables u 1 , u 2 with (suitable) compact support and testing the kinetic equation against ρ

(x) = φ(u 1 )ψ(u 2 ), one gets ˆφ(u 1 )ψ ′ (u 2 )χ m (x, s) du 2 du 1 = 0,
which implies that for a.e. u 1 , the map

u 2 → χ m (u 1 n B ⊥ (s) + u 2 n B ⊥ (s) ⊥ , s) is a.e. equal to a constant, namely to the mean value of χ m (•, s) over the slice V s,x•n B ⊥ (s) where V s,a = {an B ⊥ (s) + tn B ⊥ (s) ⊥ : t ∈ R} ∩ V . Therefore, setting χm (a, s) = Vs,a χ m (x, s) dx,
one has χ m (x, s) = χm (n B ⊥ (s) • x, s) for all s and almost every x in V . Let us justify briefly that χm is Borel. Setting F (s, a, t) = an B ⊥ (s) + tn B ⊥ (s) ⊥ , which is continuous, and U s,a = {t :

F (s, a, t) ∈ V }, one has χm (a, s) = 1 |U s,a | ˆUs,a χ m (an B ⊥ (s) + tn B ⊥ (s) ⊥ , s) dt = ´R χ m (F (s, a, t), s)1 V (F (s, a, t)) dt ´R 1 V (F (s, a, t)) dt ,
and it is now clear that it is Borel as an integral of a Borel map with parameter.

We can now prove Theorem 4.5.

Proof of Theorem 4.5. We proceed in several steps.

Step 1:

Trace for χ m . Provided that v • n B ⊥ (s) = 0, the function [-1, 1] ∋ x 2 → χm (x 2 v • n B ⊥ (s), s) is the trace of χ m (•, s)
on L in the sense that:

(4.4) lim r→0 Pr |χ m (x, s) -χm (x 2 v • n B ⊥ (s), s)| dx = 0.
Indeed, using Lemma 4.6:

2 Pr |χ m (x, s) -χm (n B ⊥ (s) • vx 2 , s)| dx = ˆ1 -1 r -r χ m (x 1 v ⊥ + x 2 v, s) -χm (x 2 v • n B ⊥ (s), s) dx 1 dx 2 = ˆ1 -1 r -r χm ((x 1 v ⊥ + x 2 v) • n B ⊥ (s), s) -χm (x 2 v • n B ⊥ (s), s) dx 1 dx 2 ≤ 1 |v • n B ⊥ (s)| ˆ1 -1 r -r χm (x 1 v ⊥ • n B ⊥ (s) + y 2 , s) -χm (y 2 , s) dx 1 dy 2 ≤ 1 |v • n B ⊥ (s)| sup |y1|≤r ˆ1 -1 | χm (y 1 + y 2 , s) -χm (y 2 , s)| dy 2 .
This last quantity goes to 0 as r goes to 0 because of the continuity of the translation in L 1 . Note that for a fixed v ∈ S 1 , there are only two vectors s ∈ ∂B ⊥ for which v • n B ⊥ (s) = 0 because B is a convex body which is stricly convex (the norm being of type p). As a consequence equality (4.4) is true for H 1 -a.e. s in ∂B ⊥ .

Step 2: Trace for m. For x 2 ∈ [-1, 1], we define the trace m of m on L by the following equality :

m(x 2 ) = 1 2 ˆ∂B ⊥ χm (x 2 v • n B ⊥ (s), s)n B ⊥ (s) dH 1 (s).
Then, thanks to the averaging formula (1.7),

2 Pr |m(x) -m(x 2 )| dx ≤ Pr ˆ∂B ⊥ |(χ m (x, s) -χm (x 2 v • n B ⊥ (s), s))n B ⊥ (s)| dH 1 (s) dx ≤ ˆ∂B ⊥ Pr |χ m (x, s) -χm (x 2 v • n B ⊥ (s), s)| dx dH 1 (s).
Using (4.4) and the dominated convergence theorem, (i) is proved. Moreover,

ˆ1 -1 |1 -m(x 2 ) | dx 2 ≤ 2 Pr | m(x) -m(x 2 ) | dx ≤ C Pr |m(x) -m(x 2 )| dx,
the last equality resulting from the equivalence of norms on R 2 and the reverse triangle inequality. This last quantity going to 0 as r goes to 0, (ii) is proved.

Step 3: Proof of (iii). Take x a Lebesgue point of m. We know by Lemma 4.3 that x is a Lebesgue point of χ m (•, s) for almost all s. We define the cube

Q s r (x) := x + {an B ⊥ (s) + bn ⊥ B ⊥ (s) : (a, b) ∈ [-r, r] 2 },
and write y = y s 1 n B ⊥ (s) + y s 2 n ⊥ B ⊥ (s) for arbitrary y. Notice that:

Q s r (x) |χ m (y, s) -χ m (x, s)| dy = Q s r (x) | χm (y s 1 , s) -χm (x s 1 , s)| dy = x s 1 +r x s 1 -r | χm (u, s) -χm (x s 1 , s)| du.
The quantity on the left tends to 0 as r goes to 0 hence the quantity on the right as well, which means that x s 1 = x • n B ⊥ (s) ∈ Leb( χm ) for almost all s. Moreover, by definition of the trace, one has for all y ∈ L:

m(y 2 ) = 1 2 ˆ∂B ⊥ χm (y 2 v • n B ⊥ (s), s)n B ⊥ (s) dH 1 (s), thus x2+r x2-r | m(y 2 ) -m(x 2 )| dy 2 ≤ ˆ∂B ⊥ x2+r x2-r | χm (y 2 v • n B ⊥ (s), s) -χm (x 2 v • n B ⊥ (s), s)| dy 2 Js(r)
ds.

Provided that v • n B ⊥ (s) = 0, which is true for a.e. s:

J s (r) = x s 1 +rv•n B ⊥ (s) x s 1 -rv•n B ⊥ (s) | χm (u, s) -χm (x s 1 , s)| du r→0 ---→ 0,
hence by the dominated convergence theorem, J s (r) being bounded by 2, one gets:

L s r (x) = x + {tv + un B ⊥ (s) : (t, v) ∈ [-r, r]}. Since x is a Lebesgue point of m, it is a Lebesgue point of χ m (•, s) for almost all s, hence L s r (x)
χ m (y, s) dy → χ m (x, s).

Passing to the limit as r goes to 0 in the integral:

m(x 2 ) = 1 2 ˆ∂B ⊥ χ m (x, s)n B ⊥ (s) ds = m(x),
using again the representation formula.

Step 4: Proof of (iv). Fix ε > 0, and take δ > 0 to be suitably chosen later in terms of ε.

= x + {u 1 v ⊥ + u 2 v : (u 1 , u 2 ) ∈ [-r, r] × [-a, a]}.
Since x 2 is a Lebesgue point of m, one may find a small enough such that

x2+a x2-a | m(y 2 ) -m(x 2 )| dy 2 ≤ δ.
Then for this fixed a, by (i) the quantity ffl Pr,a |m(y)m(y 2 )| dy goes to 0 as r goes to 0, thus one may find r small such that it is less then δ, yielding E(a, r) ≤ 2δ. Now E(a, r) may be bounded from below

m(y) • s > 0 (resp. < 0) =⇒ m(z 2 ) • s ≥ 0 (resp. ≤ 0), . (4.5)
Proof. The key point lies in (iv) of Theorem 4.5. With the notations of this theorem, if z is a point of L such that z 2 is a Lebesgue point of m, one may find a sequence z n ∈ Leb(m) such that z n → z and m(z n ) → m(z 2 ). We know that (zy) • n B ⊥ (s) = 0. The map n B ⊥ : ∂B ⊥ → S 1 being a homeomorphism, one may find a sequence s n ∈ ∂B ⊥ such that (z ny) • n B ⊥ (s n ) = 0 and s n → s (set

s n = ±n -1 B ⊥ (z n -y) ⊥ |z n -y|
). Applying Proposition 4.4, one gets m(z n ) • s n ≥ 0, and passing to the limit n → ∞ yields m(z 2 ) • s ≥ 0.

4.3. The regularity theorem. Now that we have defined a trace of our vector field along lines, our goal is to prove a form of invariance along lines just like what would happen in the smooth case thanks to the characteristic method. Proof. By translation of the domain and rotation of the target space, we may assume without loss of generality that x 0 = 0 and p = (λ, 0) with λ > 0. Let h > 0, r 0 > 0 be fixed and set

E ± r0 := x ∈ Ω ∩ Leb(m) : x = hv ± tv ⊥ 0 < t ≤ r 0 . For x ± ∈ E ± r0 , we will note s ± = (s ± 1 , s ± 2 ) := n -1 B ⊥ (x ± ) ⊥ |x ± | ∈ ∂B ⊥ and α ± = (α ± 1 , α ± 2 ) := n -1 B ⊥ hv ⊥ ∓r0v |hv ⊥ ∓r0v| . Remark that s ± is chosen in a way such that x ± ∈ Vect(n B ⊥ (s ± ) ⊥ ). We claim O p h L r0 E + r 0 E - r 0 Figure 3. The set E ± r0 that for r 0 small enough, s ± 2 ≥ λ 2 > 0 and there is C > 0 such that: (0 > s + 1 ≥ Cα + 1 and 0 < s - 1 ≤ Cα - 1 ) or (0 < s + 1 ≤ Cα + 1 and 0 > s - 1 ≥ Cα - 1 ), Let us prove this claim. First note that n B ⊥ (s ± ) is in the cone C(v ⊥ , n B ⊥ (α ± )) as defined above Lemma 2.3. Indeed, n B ⊥ (s ± ) = (x ± ) ⊥ |(x ± ) ⊥ | = hv ⊥ ∓ tv |hv ⊥ ∓ tv| , and 1 |hv ⊥ ∓ tv| hv ⊥ ∓ tv = 1 |hv ⊥ ∓ tv| t r 0 (hv ⊥ ∓ r 0 v) + h 1 - t r 0 v ⊥ , = 1 |hv ⊥ ∓ tv| t|hv ⊥ ∓ r 0 v| r 0 n B ⊥ (α ± ) + h 1 - t r 0 v ⊥ . Using Lemma 2.3, this implies s ± ∈ C(n -1 B ⊥ (v ⊥ ), α ± ). But n -1 B ⊥ (v ⊥ ) = n -1 B (v) ⊥ = p ⊥ so that: s ± ∈ C(p ⊥ , α ), meaning that there are t 1 , t 2 in R + * such that s ± 1 s ± 2 = t 1 0 λ + t 2 α ± .
Because of the continuity of n -1 B ⊥ proved in Theorem 2.6, α ± r0→0 ---→ p ⊥ = (0, λ), which implies that for r 0 small enough s ± 2 ≥ λ 2 > 0. To conlude the proof of our claim, we just have to remark that there are only two possibilities for the position of α ± with respect to p ⊥ since p ⊥ ∈ C(α + , α -) by Lemma 2.3.

In the first configuration, because s

± , α ± ∈ ∂B ⊥ , s + ∈ C(p ⊥ , α + ) implies 0 > s + 1 ≥ Cα + 1 and s -∈ C(p ⊥ , α -) implies 0 < s - 1 ≤ Cα - 1 .
In the second configuration, using the same trick ones gets

s + ∈ C(p ⊥ , α + ) implies 0 < s + 1 ≤ Cα + 1 and s -∈ C(p ⊥ , α -) implies 0 > s - 1 ≥ Cα - 1 .
This concludes the proof of our claim.

p ⊥ α + α - p ⊥ α - α + Figure 4. Two differents configurations
Without loss of generality, we will place ourselves in the second case that is we will chose r 0 small enough such that s

± 2 ≥ λ 2 > 0, 0 < s + 1 ≤ Cα + 1 and 0 > s - 1 ≥ Cα - 1 .
Then, using Proposition 4.4, for

x + ∈ E + r0 , m(0) • s + = λs + 1 > 0 =⇒ m(x + ) • s + = m 1 (x + )s + 1 + m 2 (x + )s + 2 ≥ 0 =⇒ m 2 (x + ) ≥ - m 1 (x + )s + 1 s + 2 ≥ -Cα + 1 .
Identically, we get for

x -∈ E - r0 : m(0) • s -= λs - 1 < 0 =⇒ m(x -) • s -= m 1 (x -)s - 1 + m 2 (x -)s - 2 ≤ 0 =⇒ m 2 (x -) ≤ - m 1 (x -)s - 1 s - 2 ≤ -Cα - 1 .
Notice that Theorem 4.5 implies that

ˆ1 -1 r 0 |m(x) -m(x 2 )| dx 1 dx 2 = 2 Pr |m(x) -m(x 2 )| dx r→0 ---→ 0 and ˆ1 -1 0 -r |m(x) -m(x 2 )| dx 1 dx 2 = 2 Pr |m(x) -m(x 2 )| dx r→0 ---→ 0.
Consequently there is a sequence r n → 0 such that:

rn 0 |m(x) -m(x 2 )| dx 1 n→+∞ -----→ 0 and 0 -rn |m(x) -m(x 2 )| dx 1 n→+∞ -----→ 0
for almost every x 2 . Moreover we know that for almost every

x 2 , x 1 v ⊥ + x 2 v is a Lebesgue point of m for almost every x 1 , so that m 2 (x 1 v ⊥ + x 2 v) ≥ -Cα + 1 if x 1 ≥ 0 and m 2 (x 1 v ⊥ + x 2 v) ≤ -Cα - 1 if x 1 ≤ 0. Thus it follows that for almost every x 2 : m2 (x 2 ) = lim n→∞ rn 0 m 2 (x 1 v ⊥ + x 2 v) dx 1 ≥ -Cα + 1 r→0 ---→ 0, and m2 (x 2 ) = lim n→∞ 0 -rn m 2 (x 1 v ⊥ + x 2 v) dx 1 ≤ -Cα - 1 r→0 ---→ 0.
This shows that m2 (x 2 ) = 0 for a.e. x 2 , i.e. m(x 2 ) ∈ {±p} which is what we wanted to prove.

We can now finish the proof of Theorem 1.5.

Proof of Theorem 1.5. Consider any open convex ω ⊂ Ω with d = d(ω, ∂Ω) > 0. It is enough to prove Theorem 1.5 in any such ω. Let x, y in ω ∩ Leb(m), L x be the line passing through x and directed by n B (m(x)) and L y be the line passing through y and directed by n B (m(y)). If L x and L y are parallel and distinct, we claim that m(x) = m(y). Indeed, if it is not the case, m(x) = -m(y). We choose s ∈ ∂B ⊥ such that yx ∈ Vect(n B ⊥ (s) ⊥ ). Note that m(y) • s = 0, otherwise s and m(y) ⊥ are colinear and yx ∈ Vect(n B (m(y))) and L x = L y . Now using Proposition 4. |x| , the trace of m along L x and L y would be determined in the same way, but with the opposite sign. We will use this in the end of our proof.

We are now going to distinguish two cases: Case 1: dist(O, ω) ≤ d. We claim that m is a vortex in ω. Let us take z ∈ ω ∩ Leb(m) ∩ (L x ∪ L y ) c and let L z be the line directed by m(z) and passing through z. If we prove that L z intersects L x and L y in O, we are done. Suppose now that this is not the case. Using the same argument than in the beginning, L z cannot be parallel neither to L x nor to L y . Let P 1 = L z ∩ L x and P 2 = L z ∩ L y . By convexity, there is a small portion of L z in Ω. But using the remark in our first part to the lines L x and L z , one gets: for a.e. But n -1 B (-x) = -n -1 B (x) and the two traces obtained for mLz are opposed on the segment [P 1 , P 2 ], wich leads to a contradiction (see Figure 5). Remark 4.3. Notice that this result is sharp, in the sense that we cannot hope to get a better regularity for vector fields satisfying the kinetic equation and valued in the sphere of a power type p norm. Indeed, as shown in Section 2 the vortex V B associated to the ball B of this norm is always solution, and Proposition 2.5 shows that it is 1 p-1 -Hölder continuous far from 0 if and only if the norm is of power type p.

ε→0 0 .

 0 Then |m| = 1 almost everywhere and m satisfies the kinetic equation: ∀s ∈ S 1 , s ⊥ • ∇χ m (•, s) = 0 in D ′ (Ω). (1.3)

Figure 1 .

 1 Figure 1. Unit ball of . 1 and vortex associated to . ∞ , singular along the axes.

Proposition 4 . 1 .

 41 Let Ω be an open set in R 2 , B the closed unit ball associated to the norm • and m a Borel vector field such that m = 1. If m satisfies the kinetic equation (1.8), then m is curl-free in the distributional sense.

Proposition 4 . 2 .

 42 where we use the Corollary 1.4 in the first line and the kinetic equation(1.8) in the second, which concludes the proof.The next proposition gives kind of a reciprocal if the field m is smooth. Let Ω be an open convex set in R 2 , x → x a norm in C 2 (R 2 \ {0}), and m ∈ C 1 (Ω, R 2 ) a curl-free vector field such that m = 1. Then m satisfies the kinetic equation(1.8).

Lemma 4 . 3 .

 43 Let Ω ⊂ R 2 be an open set and m ∈ L 1 loc (Ω, ∂B).(i) If x 0 is a Lebesgue point of χ m (•, s) for H 1 -almost every s ∈ ∂B ⊥ then it is a Lebesgue point of m. (ii) If x 0 is a Lebesgue point of m then x 0 is a Lebesgue point of χ m (•, s) for all s ∈ ∂B such that m(x 0 ) • s = 0 . In particulier x 0 is a Lebesgue point of m ifand only if it is a Lebesgue point of χ m (•, s) for almost every s.

Proposition 4 . 4 .

 44 Let Ω ⊂ R 2 be a convex open set and m ∈ L 1 loc (Ω, ∂B) satisfying the kinetic equation(1.8). Assume that y, z ∈ Leb(m) and that s

Theorem 4 . 5 .

 45 Let Ω ⊂ R 2 be a convex open set and m ∈ L 1 loc (Ω, ∂B) satisfying the kinetic equation(1.8). Let L be a segment of the form L

| m(y 2 )lim r→0 1 2

 21 m(x 2 )| dy 2 → 0, and x 2 is a Lebesgue point of m. Moreover:• n B ⊥ (s), s) dt n B ⊥ (stv + un B ⊥ (s) ⊥ ) • n B ⊥ (s), s) du dt n B ⊥ (sχ m (tv + un B ⊥ (s) ⊥ , s) du dt n B ⊥ (s) ds = ˆ∂B ⊥ L s r (x) χ m (y, s) dy n B ⊥ (s) dswhere we have defined the parallelogram (which is non-flat for a.e. s):

  For a, r positive and smaller than ε, we look at the quantity E(a, r) = ffl x2+a x2-a ffl r -r |m(y)m(x 2 )| dy. This quantity is bounded from above as follows: y 2 )m(x 2 )| dy ≤ Pr,a(x) |m(y)m(y 2 )| dy + x2+a x2-a | m(y 2 )m(x 2 )| dy 2 , where P r,a :

4 . 4 . 7 .

 447 |m(y)m(x 2 )| ≥ ε |y ∈ Leb(m) ∩ P r,a : |m(y)m(x 2 )| > ε| |P r,a | , which implies that |y ∈ Leb(m) ∩ P r,a : |m(y)m(x 2 )| > ε| |P r,a | ≤ 2δ ε .Choosing δ = ε/4 this quantity is strictly less than 1 hence the set{y ∈ Leb(m) ∩ P r,a : |m(y)m(x 2 )| ≤ ε} is not empty. Any point x ε in this set is a Lebesgue point such that |x ε -x| ≤ √ 2ε (because x ε ∈ P ε,ε (x)), and sastisfying |m(x ε )m(x 2 )| ≤ ε. Taking ε = 1/n gives the desired conclusion.With this definition of the trace, we obtain as a corollary an extension of Proposition 4.Corollary Assume that Ω ⊂ R 2 is a convex open set and m ∈ L 1loc (Ω, ∂B) satisfies the kinetic equation(1.8). If L is a given line segment in Ω and z ∈ L is such that z 2 is a Lebesgue point of the trace m on L, then for all y ∈ Leb(m) such that zy ∈ Vect(n B ⊥ (s) ⊥ ) one has:

Proposition 4 . 8 .

 48 Let m ∈ L 1 (Ω, ∂B) satisfying the kinetic equation(1.8). Suppose that x 0 is a Lebesgue point of m and denote by L be the line passing trough x 0 and directed by v := n B (p) ∈ S 1 where m(x 0 ) = p. Then for almost every x 2 , m(x 2 ) ∈ {±p} where m is the trace of m on L in the sense ofTheorem 4.5.

.=

  4 sign(m(y) • s) = sign(m(x) • s) hence m(x) = m(y).If L x and L y intersect each other, let O be their intersection point. Up to a change of variable in the target space, we can suppose that O is the origin, L x = Vect( x |x| ) and L y = Vect( y |y| ). Now, up to a change of sign for m, we can suppose that m(x) = n -1 B ( x |x| ) and m(y) = ±n -1 B ( y |y| ). We are going to prove that the trace of m along L x and L y in the sense of Theorem 4.5 is completely determined by m(x) and m(y). More precisley, for a.e.t ∈ R such that t x |x| ∈ L x ∩ ω, mLx (t) = n -1 B x |x| , for a.e. t ∈ R such that t y |y| ∈ L y ∩ ω, mLy (t) = n -1 B y |y| ,where mLx and mLy stand for the trace of m along L x (resp L y ) in the sense of Theorem 4.5. Let z = t y |y| ∈ L y ∩ ω chosen such that t is a Lebesgue point of mLy and choose s∈ ∂B ⊥ such that zx ∈ Vect(n B ⊥ (s) ⊥ ). Because of Proposition 4.8, m(t) = ±m(y) = ±n -1 B y |y| . But m(x) • s = n -1Because z and x are not colinear,(z-x) |z-x| ∈ Vect(x) and n -1 B being bijective, m(x) • s = 0. Because of Proposition 4.7, this implies sign( mLy (t) • s) = sign(m(x) • s). But y |y| = z |z| ∈ C x |x| , there is α, β in R + such that: αm(x) • s, so that sign( mLy (t) • s) = sign(n -1 B y |y| • s). This forces mLy (t) = n -1 B y |y| and we proved that mLy is determined along L y . We show in the same way that mLx = n -1 B x |x| along L x . Note that if we started with m(x) = -n -1

  First note that up to a change of sign for m, we can suppose that m(x) = n -1 B x |x| and m(y) = n -1 B y |y| .

Figure 5 . 1 p- 1 ,

 511 Figure 5. An impossible configuration Case 2: dist(O, ω) > d. Using the first part, we know that up to a change of sign: m(x) = n -1 B x |x| = V B (x) and m(y) = n -1 B

Properties of the normal field. We

  • by Rademacher's theorem. Furthermore, V B takes its values into ∂B. Indeed by (2.1), for almost every x (those where • ⋆ is differentiable, i.e. where V B is defined), one has p = V B (x) ⇐⇒ x ∈ ∂ p • , so that p is a point where the subdifferential ∂ p • contains a nonzero vector, i.e. a point on the boundary of the ball. Consequently V B (x) = p ∈ ∂B. We are going to prove that V B is always a solution to the kinetic equation(1.8). As a consequence, if we want this kinetic equation to rule out line-like singularities then V B should not exhibit such singularities, and the regularity of V B is related to the convexity of B, as we will see in Proposition 2.5. This is the fundamental reason why we need some convexity property on our norm in order to obtain Theorem 1.5. start off by stating some useful properties of the normal field n

	2.1.

B : ∂B -

  t ∈ R such that t x -P 1 |x -P 1 | ∈ L x ∩ ω, mLx (t) = -n -1 B x -P 1 |x -P 1 | , for a.e. t ∈ R such that t z -P 1 |z -P 1 | ∈ L y ∩ ω, mLz (t) = -n -1 B z -P 1 |z -P 1 |, and using it to the lines L y and L z , one gets: for a.e. t ∈ R such that t y -P 2 |y -P 2 | ∈ L x ∩ ω, mLx (t) = -n -1 B y -P 2 |y -P 2 | , for a.e. t ∈ R such that t z -P 2 |z -P 2 | ∈ L y ∩ ω, mLz (t) = -n -1

B z -P 2 |z -P 2 |

.

Acknowledgments:

The first author would like to thank warmly Felix Otto for helpful discussions and comments and the Max Planck Insitute of Leipzig for the six months he spent there. This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Pierre Philippe

Pablo Cuéllar

Florian Brunier-Coulin

Li-Hua H Luu

Nadia Benahmed

Stéphane Bonelli

Jean-Yves Delenne

Physics of soil erosion at the microscale

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Erosion and transport of cohesive soil sediments by fluid flows are of primary concern in many processes, either in natural or manmade environments, as fluvial morphological changes, spreading of pollutant or radionucleides in river sediments, impact of agricultural soil erosion on the global carbon cycle, local scour at bridge piers, external and internal erosion of earthen hydraulic structures [START_REF]Erosion in Geomechanics Applied to Dams and Levees[END_REF].

The present contribution focuses on two related fundamental issues: (i) How to properly predict a critical condition, if any, for occurrence of erosion? (ii) How to accurately quantify the amount of eroded soil particles? It is first presented here a brief synthesis of the most usual methods used to tackle these issues with particular attention to what are the major remaining problems or contradictions which could limit the general scope, or even the validity, of these previous approaches. As a consequence, an alternative methodology is proposed, based on systematic investigations, either experimental or by means of numerical models, with artificial cohesive materials and aims to combine erosion tests in some specific fluid flow configurations with better targeted, or even specially developed, mechanical characterisations of our model systems. As illustrated by several preliminary tests following the previous strategy, a specific scope is put on the local processes of matter entrainment by a fluid flow inducing what is more globally called soil's erosion at a more wider scale, both in space and time.

⋆ e-mail: pierre.philippe@irstea.fr

Quantification of soil erosion and related issues

Being able to quantify erosion of a soil by a neighboring fluid flow requires first to select both space and time scales over which the amount of eroded matter and the hydrodynamical load exerted by the fluid flow should be considered and quantified. Owing to the large variety of practical issues, the space scale can span from kilometer squares to microns, including for instance watersheds, riverbanks, soil specimens in the lab, or only few sediment layers, while the duration under consideration can go from seconds to months, or even more in some geological phenomena. This first choice prescribes the type of erosion models needed to achieve a consistent prediction, in particular regarding the quantities that can be used to account for the action of the eroding fluid flow. In a soil mechanics perspective as will be considered in the following, the size of a soil sample under study is typically equal to some centimeters, or tens of centimeters, and the rates of erosion are estimated over time periods of seconds to minutes. Regarding the flow regime which is turbulent in most applications, an hydrodynamical indicator, able to account for the hydrodynamic force exerted by the flow on the soil surface, is not easy to quantify since the fluid flow highly fluctuates in time and also, most often, in space. The simplest and most commonly followed solution consists in a crude averaging of the chosen quantity. Alternative approaches can be developed, mostly based on statistical modeling, but might be hard to apply successfully in practice since a realistic probability density function is needed. The same uncertainty about the use of average quantities arises considering the mechanical resistance of a soil to erosion. Here again, it is likely that the sensitivity to erosion varies from location to location within a soil, reflecting somehow the inherent heterogeneous nature of the material. Note also that in the particular case of a sandy soil, following the pioneer studies by Shields on fluvial sediment transport, the resistance to erosion can be expressed through a critical dimensionless shear-stress whose value can be approximated from simplified mechanistic models describing the equilibrium of a single particle and accounting for drag force, buoyant weight and friction. The more usual situation of a cohesive soil is far much complicated and knowledge is lacking on local processes of erosion at a natural soil surface to consider developing similar mechanistic models.

In practice, the very large majority of previous and actual studies seeking to assess the sensitivity of natural soils to fluid flow erosion have based their analysis assuming the following expression of a local erosion law that accounts for a threshold and relates ṁ, the rate of mass removal by units of time and surface, to τ, the shear-stress exerted by the fluid flow at the soil's surface [START_REF] Partheniades | [END_REF]4]:

ṁ = k er (τ -τ c ) if τ > τ c , (1) ṁ
= 0 otherwise. (2)
In this purely empirical modeling, τ c (in Pa) is a critical shear-stress and k er (in s.m -1 ) is a kinetic erosion coefficient. Both parameters are sometimes called soil's erodibility. Several erosion tests have been developed to provide direct measurements or (sometimes crude) estimates of eroded mass and hydrodynamical shear-stress. Based on the local erosion law in Eq. 1 (or a slightly modified version), a fitting procedure ultimately provides the erodibility parameters. Correlations between those parameters and more common soil's properties are still being explored. If rather accurate trends are observed for a same soil, very different and hardly predictable values of erodibility can be found from one soil to another and also possibly from one erosion device to another. The latter result raises in particular questions regarding the postulated intrinsic nature of erodibility. The many different, and sometimes contradictory, empirical relationships proposed in the literature are far from providing a definitive answer. Another general issue concerns the use of average quantities, especially when describing the fluid flow, which appears in serious contradiction with a process as erosion which assumes the existence of an inherent soil's threshold. Indeed, the local hydrodynamic stress values are naturally distributed (fluctuations, turbulence) and the highest portion of this distribution might exceed the soil's critical threshold τ c while the average stress remains below it, generating intermittent particle extraction and consequently sustaining over time a non-zero rate of erosion.

Methodology

Regarding the related literature, very little is known about the real processes at work at particle scale and which commonly hide behind the general term of erosion. Furthermore, as discussed previously, no clear picture has emerged from the most popular route based on direct empirical correlations between resistance to erosion and many different usual material properties obtained for a large variety of natural soils. This approach possibly reaches its limits, facing a rather complex phenomenon and the present contribution proposes to reverse the methodology by using purely artificial materials which can advantageously allow for both a much more accurate control of some key properties (as cohesion, particle size, etc) and, whenever possible, for a direct visualization of the basic local processes of solid matter removal by a fluid flow. Our goal is first to gain insights from the fundamental scale in order to reach a better understanding of the microstructural scenarios of soil's erosion. Second, to identify and rationalize these local mechanisms would constitute an important step towards clearer relationships between some pertinent usual properties of a material, whose characterization might requires the development of specific mechanical tests, and some more application-oriented empirical characterizations as the previous erodibility parameters k er and τ c , or possibly some new ones related to alternative local erosion laws to be proposed. Such a breakthrough could considerably supplement the usual approach for erosion risk assessment and accordingly motivates this work.

Model materials

For the present contribution, three different types of cohesive soils have been considered. The first type of materials that can be rather simply prepared and studied is made of large granular particles with adhesive bonds at inter-particle contacts. Such contact bonds can be either liquid or solid as illustrated in Figure 1. It is to be noticed the main differences between liquid and solid bonds: solid bridges are usually far much resistant than liquid ones but the latter can be broken and then created back with strong hysteresis while the rupture of a solid bond is permanent and irreversible. Regarding the grains, they can come from either a natural sandy material or an artificial one as the previous glass beads in Fig. 1 which can advantageously enable the additional implementation of two coupled optical techniques, Refractive Index Matching and Planar Laser Induced Fluorescence, to access to a direct visual- ization of the local erosion process provided that the eroding fluid is an index-matched mixture [5]. The behaviour of such a cohesive granular material with solid bonds can also be investigated by means of the coupling between the Discrete Element Method (DEM) and the Lattice Boltzmann Method (LBM), which is certainly today one of the numerical methods being the most increasingly used for the modeling of fluid-grain systems. An attractive bond at each grain contact is implemented thanks to an additional cohesion model between the solid particles which is based on a relatively straightforward rheology (i.e. viscoelasticperfectly-plastic) and on the de-bonding limit (yield) conditions introduced by Delenne and coworkers [3] which permit the consideration of bond ruptures in tension, shear and bending conditions.

The second type of model cohesive soils that can be easily studied through systematic investigations are chemically-treated sands, as magic sand and kinetic sand. The first one is a building toy made of hydrophobic sand grains and can consequently be viewed as a granular material with an effective "skin" cohesion arising from the Laplace over-pressure at the interface between air and water. Kinetic sand , made of 98% sand and 2% polydimethylsiloxane (a type of silicone oil), is another toy material specially developed to mimic the physical behaviour of wet sand with reversible adhesion between grains.

The third kind of artificial cohesive soils corresponds to homogeneous materials, modeled in a continuous medium perspective and able to mimic loamy soils made of very small particles with strong attractive interactions. A synthetic clay as Laponite appears as a sound choice since its yield-stress rheology can be systematically varied and because it also has the great advantage to be transparent and thus to allow here again for direct visualization of solid matter retrieval during an erosion test.

Coupling hydrodynamic and mechanical tests

As an integral part of our methodology, erosion and mechanical tests must be implemented in parallel. The choice of the hydrodynamical configuration inducing erosion determines the ability to measure and/or model accurately some key quantities as shear-stress or velocity and to reach sufficient intensity level for erosion to occur. In the present paper, some already existing configurations, as jet erosion test and hole erosion test, were used but sometimes with specific adaptations (fluid, geometry, optical techniques) enabling direct visualization of the local process. By way of example, Figure 2 shows a jet erosion test on a magic sand where the geometry was restricted to 2D for observation purpose. Some rather usual mechanical tests were carried out with our artificial materials, as fall-cone test, while several other ones were specifically developed for instance to measure tensile strength both at contact and sample scales [START_REF] Brunier-Coulin | [END_REF].

Some preliminary results

Quite an extensive study of our first model material (i.e. glass beads bonded by solid bridges), subjected to erosion by impinging of an immersed jet, has already been carried out by part of the present authors and will be presented elsewhere [START_REF] Brunier-Coulin | [END_REF]. However, several recent and original results can be presented, even if they are still preliminary at this stage. To start with, our first type of model material has also been studied in a jet erosion configuration but this time by means of a numerical DEM-LBM modeling as previously mentioned. More precisely, the influence of the cohesion model at particle contacts was tested through deliberately exaggerated soil's behaviours by imposing a very weak value of the solid bond yield alternatively under tension, shear and bending loading. As can be seen on Figure 3, widely different erosion processes are observed in the three distinct cases: (i) almost usual grainby-grain erosion, (ii) remote debonding to a rather long extent within the sample, (iii) in-depth fracturing creating large aggregates.

Another example deals with the same kind of jet erosion test but performed this time on one of our second model materials, namely kinetic sand, and with pressurised air as eroding fluid. Changing the initial density of the kinetic sand allows to control quite accurately the undrained cohesion Cu estimated via a cone-fall test (by calibration from penetration depth of a falling cone, released from a given height, within a soil's specimen). During a jet test, a classical fitting procedure of the scour crater depth versus time using Eq. 1 provides the erodibility parameters τ c and k er which, as underlined in Figure 4, are nicely and almost linearly correlated to Cu.

Finally, a last preliminary study has focused on hole erosion test performed on a Laponite synthetic clay, our third model material of cohesive soils, using the classical Hole Erosion test (HET) device [START_REF]Erosion in Geomechanics Applied to Dams and Levees[END_REF] where a controlled flow is generated along a preformed axial hole within the soil sample to determine a critical value of hydraulic shear stress. Figure 5 gives a general overview of the pipe enlargement versus time while Figure 6 is a close-up where crack initiation and propagation can be observed till a large aggregate is ultimately washed away.

Conclusion

The starting point of the present contribution was the identified limitations and remaining issues regarding theoretical and experimental state-of-the-art in quantifying erosion of natural soils. It was then proposed to reverse the classical methodology, substituting real soils by artificial materials, in both experimental and numerical perspectives, with the purpose to better control key properties of the materials so as to implement systematic analysis of sensitivity to erosion, including whenever possible direct visualization of the basic local processes of solid matter removal by a fluid flow as well as specifically selected, or developed, mechanical tests. Several examples of such model materials were first described before some encouraging preliminary results were presented and discussed. Much more work is now required to achieve comprehensive results continuing this strategy.

Figure 1 .

 1 Figure 1. (Left) Capillary bridge of Ucon Oil between 3mm glass beads. (Right) Solid bridge of paraffin between 7mm glass beads after solidification of the original capillary bridge of liquid paraffin.

Figure 2 .

 2 Figure 2. Typical sequence (from left to right and top to bottom) of scour erosion of a magic sand during a 2D water jet test. The arrow indicates the location of the impinging water jet.

Figure 3 .

 3 Figure 3. Jet erosion tests on a soil where the cohesion model at particle contacts presents a small yield value under tension (top), under shear (middle), and under bending (bottom), respectively.

Figure 4 .

 4 Figure 4. Linear correlations between the erodibility parameters τ c (black squares) and k er (blue circles) of a kinetic sand, deduced from the classical interpretation model with a 2D pressurised-air jet test (differential pressure ∆P = 200Pa), and the undrained cohesion Cu, obtained from a fall-cone test.

Figure 5 .

 5 Figure 5. Time evolution of an initially cylindrical pipe in a transparent synthetic clay sample during a Hole Erosion Test with water flow from left to right. The flow rate is progressively increased and the pictures correspond (from left to right and top to bottom) to successive times t = 94s, t = 200s, t = 350s, and t = 800s. The dimension of the pictures is 70mm × 58mm.

Figure 6 .

 6 Figure 6. Close-up from the previous test (Fig. 5) where a crack propagation is clearly identified above the pipe and induces the final pull-out of a large aggregate.

in : Powders & Grains 2017, 8th International Conference on Micromechanics of Granular Media, Montpellier, France, 3-7 July 2017

Acknowledgements

The authors acknowledge a financial support from the region PACA (France) through the APEX-ESCAPE project.