Gilles Carbou
email: gilles.carbou@univ-pau.fr

Pierre Fabrie
email: pierre.fabrie@math.u-bordeaux1.fr

Kévin Santugini
email: kevin.santugini@math.u-bordeaux1.fr

G Métivier

J Rauch

Weak solutions to the Landau-Lifshitz-Maxwell system with nonlinear Neumann boundary conditions arising from surface energies

We study the Landau-Lifshitz system associated with Maxwell equations in a bilayered ferromagnetic body when super-exchange and surface anisotropy interactions are present in the spacer in-between the layers. In the presence of these surface energies, the Neumann boundary condition becomes nonlinear. We prove, in three dimensions, the existence of global weak solutions to the Landau-Lifshitz-Maxwell system with nonlinear Neumann boundary conditions.

Introduction

Ferromagnetic materials are widely used in the industrial world. Their four main applications are data storage (hard drives), furtivity, communications (wave circulator), and energy (tranformers). For an introduction to ferromagnetism, see Aharoni [START_REF] Aharoni | Introduction to the theory of ferromagnetism[END_REF] or Brown [START_REF] William | Micromagnetics[END_REF].

The state of a ferromagnetic body is characterized by its magnetization m, a vector field whose norm is equal to 1 inside the ferromagnetic body and null outside. The evolution of m can be modeled by the Landau-Lifshitz equation ∂m ∂t = -m∧h tot -αm∧(m∧h tot ),

where h tot depends on m and contains various contributions. In particular, in this paper, h tot includes various volumic and surfacic energies, among [START_REF] Alouges | On global weak solutions for Landau-Lifshitz equations : existence and nonuniqueness[END_REF] The micromagnetic model

One possible model of ferromagnetism is the micromagnetic model introduced by W.F Brown [START_REF] William | Micromagnetics[END_REF]. In the micromagnetic model, the magnetization M is the mean at the mesoscopic scale of the microscopic magnetization and has constant norm M s in the ferromagnetic material and is null outside. In this paper, we only work with the dimensionless magnetization m = M /M s .

To each interaction p present in the ferromagnetic material is associated an energy E p (m) and an operator H p linked by

DE p (m) • v = - Ω H p (m)(x) • v(x)dx
The vector field h p = H p (m) is the magnetic effective field associated to interaction p. The total energy is the sum of all the energies associated with every interaction. These energies completely characterize the stationary problem: the steady states of the magnetization are the minimizers of the total energy under the constraint m = 1.

To have an evolution problem, a phenomenological partial differential equation was introduced in Landau-Lifshitz [START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF], the Landau-Lifshitz equation: ∂m ∂t = -m∧h tot -αm∧(m∧h tot ).

where h tot contains all the contributions to the magnetic effective field. These contributions can either be volumic or surfacic in nature.

Volume energies

Exchange

Exchange is essential in the micromagnetic theory. Without exchange, there would be no ferromagnetic materials. This interaction aligns the magnetization over short distances. In the isotrope and homogenous case, the exchange energy may be modeled by the following energy

E e (m) = A 2 Ω ∇m 2 dx.
The associated exchange operator is H e (m) = -A △ m.

Anisotropy

Many ferromagnetic materials have a crystalline structure. This crystalline structure can penalize some directions of magnetization and favor others.

Anisotropy can be modeled by

E a (m) = 1 2 Ω (K(x)m(x)) • m(x)dx.
where K is a positive symmetric matrix field. The associated anisotropy operator is H a (m) = -Km.

Maxwell

This is the magnetic interaction that comes from Maxwell equations. The constitutive relations in the ferromagnetic medium are given by:

B = µ 0 (h + m), D = ε 0 e,
where m is the extension of m by zero outside Ω.

Starting from the Maxwell equations, the magnetic excitation h and the electric field e are solutions to the following system:

µ 0 ∂(h + m) ∂t + curl e = 0,
µ 0 ∂e ∂t + σ(e + f )1 Ω -curl h = 0.
As these are evolution equations, initial conditions are needed to complete the system. The energy associated with the Maxwell interaction is

E maxw (h, e) = 1 2 h 2 L 2 (R 3 ) + ε 0 2µ 0 e 2 L 2 (R 3 ) .
We recall the Law of Faraday: div B = 0. Here, the constitutive relation reads B = µ 0 (h + m). Therefore, in order to satisfy the law of Faraday, we must assume that it is satisfied at initial time. For positive times, by taking the divergence of the first Maxwell's equation, we remark that the divergence free condition is propagated by the system.

Surface energies

When a spacer is present inside a ferromagnetic material, new physical phenomena may appear in the spacer. These phenomena are modeled by surface energies, see M. Labrune and J. Miltat [START_REF] Labrune | Wall structure in ferro / antiferromagnetic exchange-coupled bilayers : a numerical micromagnetic approach[END_REF].

Super-exchange

This surface energy penalizes the jump of the magnetization across the spacer. It is modeled by a quadratic and a biquadratic term:

E se (m) = J 1 2 Γ γ + m -γ -m 2 dS( x) + J 2 Γ γ + m∧γ -m 2 dS( x). (3.1)
The magnetic excitation associated with super-exchange is:

H se (m) = J 1 (γ * m-γm)+2J 2 (γm•γ * m)γ * m-γ * m 2 γm dS(Γ + ∪Γ -),
where γ * is defined in §3. Integration over dS(Γ + ∪Γ -) should be understood as integrating over both faces of the surface Γ.

Surface anisotropy

Surface anisotropy penalizes magnetization that is orthogonal on the boundary. In the micromagnetic model, it is modeled by a surface energy:

E sa (m) = K s 2 Γ + γm∧ν 2 dS( x) + K s 2 Γ - γm∧ν 2 dS( x) = K s 2 Γ ± γm∧ν 2 dS( x). (3.2)
The magnetic excitation associated with surface anisotropy is:

H sa (m) = K s (γm • ν)ν -γm dS(Γ + ∪ Γ -).

New boundary conditions

Without surface energies, the standard boundary condition is the homogenous Neumann condition. When surface energies are present, the boundary conditions are the ones arising from the stationarity conditions on the total magnetic energy:

Aγm∧ ∂m ∂ν = K s (ν • γm)γm∧ν + J 1 γm∧γ * m + 2J 2 (γm • γ * m)γm∧γ * m
on the interface Γ ± . A more convincing justification for these boundary conditions is that they are the ones needed to recover formally the energy inequality. These boundary conditions are nonlinear.

The Landau-Lifshitz system

We consider the following Landau-Lifshitz-Maxwell system:

∂m ∂t = -m∧h vol tot -αm∧(m∧h vol tot ) in R + × Ω, (4.1a)
m(0, •) = m 0 on Ω, (4.1b) m = 1 in R + × Ω, (4.1c) ∂m ∂ν = 0 on ∂Ω \ Γ ± , (4.1d
)

∂m ∂ν = Ks A (ν • γm)(ν -(ν • γm)γm) + J 1 A (γ * m -(γm • γ * m)γm) + 2 J 2 A (γm • γ * m)(γ * m -(γm • γ * m)γm) on R × Γ ± , (4.1e)
where h vol tot = h -Km + A △ m and (e, h) is solution to Maxwell equations:

µ 0 ∂(m + h) ∂t + curl e = 0 in R + × R 3 , (4.2a)
ε 0 ∂e ∂t + σ(e + f )1 Ω -curl h = 0 in R + × R 3 , (4.2b) e(0, •) = e 0 in R 3 , (4.2c) h(0, •) = h 0 in R 3 . (4.2d)
We first begin by defining the concept of weak solution to the Landau-Lifshitz-Maxwell system with surface energies. This concept of weak solutions is present in [START_REF] Alouges | On global weak solutions for Landau-Lifshitz equations : existence and nonuniqueness[END_REF][START_REF] Carbou | Time average in micromagnetism[END_REF][START_REF] Labbé | Simulation numérique du comportement hyperfréquence des matériaux ferromagnétiques[END_REF][START_REF] Santugini-Repiquet | Solutions to the Landau-Lifshitz system with nonhomogenous Neumann boundary conditions arising from surface anisotropy and super-exchange interactions in a ferromagnetic media[END_REF]. The key point is that the Landau-Lifschitz equation (4.1a) is formally equivalent to the following Landau-Lifschitz-Gilberg equation:

∂m ∂t -αm∧ ∂m ∂t = -(1 + α 2 )m∧h vol tot ,
which is more convenient to obtain the weak formulation defined by: Definition 1 (Weak solutions to Landau-Lifshitz-Maxwell with surface energies). Functions m in L ∞ (0, +∞;

H 1 (Ω)) and in H 1 loc ([0, +∞[; L 2 (Ω)) with ∂m ∂t in L 2 (R + ×Ω), e in L ∞ (R + ; L 2 (R 3 )), and h in L ∞ (R + ; L 2 (R 3
)) are said to be weak solutions to the Landau-Lifshitz Maxwell system with surface energies if

1. m = 1 almost everywhere in ]0, T [×Ω.

2. For all T > 0 and φ in H 1 (]0, T [×Ω),

Q T ∂m ∂t • φdxdt -α Q T m(t, x)∧ ∂m ∂t (t, x) • φ(t, x)dxdt = (1 + α 2 )A Q T 3 i=1 m(t, x)∧ ∂m ∂x i (t, x) • ∂φ ∂x i (t, x)dxdt + (1 + α 2 ) Q T (m(t, x)∧K(x)m(t, x)) • φ(t, x)dxdt -(1 + α 2 ) Q T (m(t, x)∧h(t, x)) • φ(t, x)dxdt -(1 + α 2 )K s ]0,T [×Γ ± (ν • γm)(γm∧ν) • γφdS( x)dt -(1 + α 2 )J 1 ]0,T [×Γ ± (γm∧γ * m) • γφdS( x)dt -2(1 + α 2 )J 2 ]0,T [×Γ ± (γm • γ * m)(γm∧γ * m) • γφdS( x)dt.
(4.3a)

3. In the sense of traces, m(0,

•) = m 0 . 4. For all ψ in C ∞ c ([0, +∞[, R 3 ): -µ 0 R + ×R 3 (h + m) • ∂ψ ∂t dxdt + R + ×R 3 e • curl ψdxdt = = µ 0 R 3 (h 0 + m 0 ) • ψ 0 dx (4.3b) 5. For all Θ in C ∞ c ([0, +∞[×R 3 ): -ε 0 R + ×R 3 e• ∂Θ ∂t dxdt- R + ×R 3 h•curl Θdxdt+σ R + ×Ω (e+f )•Θdxdt = = ε 0 R 3 e 0 • Θ 0 dx. (4.3c) 6.
The following energy inequality holds

E(m(T ), h(T ), e(T )) + α 1 + α 2 Q T ∂m ∂t 2 dxdt + σ µ 0 T 0 e 2 L 2 (Ω) dt + σ µ 0 Q T e • f dxdt ≤ E(m 0 , h 0 , e 0 ), (4.3d)
where

E(m, h, e) = A 2 Ω ∇m 2 dx + 1 2 Ω (K(x)m(x)) • m(x)dx + ε 0 2µ 0 R 3 e(x) 2 + 1 2 R 3 h(x) 2 + K s 2 Γ + ∪Γ - γ + m∧ν 2 dS(x) + J 1 2 Γ γ + m -γ -m 2 dx + J 2 Γ γ + m∧γ -m 2 dx.
Our first result states the existence of a global in time weak solution to the Laudau-Lifschitz-Maxwell system . Theorem 2. Let m 0 be in H 1 (Ω) such that m 0 = 1 almost everywhere in Ω. Let h 0 and e 0 be in L 2 (Ω). Let f be in L 2 (R + ×Ω) Suppose div(h 0 + m 0 ) = 0 in R 3 , where m 0 is the extension of m 0 by 0 outside Ω. Then, there exists at least one weak solution to the Landau-Lifshitz-Maxwell system in the sense of Definition 1.

Uniqueness is unlikely as the solution isn't unique when only the exchange energy is present, see [START_REF] Alouges | On global weak solutions for Landau-Lifshitz equations : existence and nonuniqueness[END_REF].

In our second result we characterize the ω-limit set of a trajectory. The definition is the following: Definition 3. Let (m, h, e) be a weak solution of the Landau-Lifschitz-Maxwell system given by Theorem 2. We call ω-limit set of this trajectory the set:

ω(m) = v ∈ H 1 (Ω), ∃(t n ) n , lim n→+∞ t n = +∞, m(t n , .) ⇀ v weakly in H 1 (Ω) .
We remark that m ∈ L ∞ (0, +∞; H 1 (Ω)) so that ω(m) is non empty. Theorem 4. Let (m, e, h) be a weak solution of the Landau-Lifschitz-Maxwell system given by Theorem 2. Let u ∈ ω(m). Then u satisfies:

1. u ∈ H 1 (Ω), |u| = 1 almost everywhere, 2. for all ϕ ∈ H 1 (Ω), 0 = A Ω 3 i=1 u(x)∧ ∂u ∂x i (x) • ∂ϕ ∂x i (t, x)dx + Ω (u(x)∧K(x)u(x)) • ϕ(x)dx - Ω (u(x)∧H(x)) • ϕ(x)dx -K s (Γ ± ) (ν • γu)(γu∧ν) • γϕdS( x) -J 1 (Γ ± ) (γu∧γ * m) • γϕdS( x) -2J 2 Γ ± (γu • γ * u)(γu∧γ * u) • γϕdS( x).
(4.4)

3.

H is deduced from u by the relations: div (H + u) = 0 and curl H = 0 in D ′ (R 3 ).

Technical prerequisite results on Sobolev Spaces

In this section, we remind the reader about some useful previously known results on Sobolev Spaces that we use in this paper. In the whole section O is any bounded open set of R 3 , regular enough for the usual embeddings result to hold. For example, it is enough that O satisfy the cone property, see [1, §4.3].

We start with Aubin's lemma [START_REF] Aubin | Un théorème de compacité[END_REF], as extended in [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 4].

Lemma 5 (Aubin's lemma). Let X ⊂⊂ B ⊂ Y be Banach spaces. Let F be bounded in L p (0, T ; X). Suppose {∂ t u, u ∈ F } is bounded in L r (0, T ; Y ).
Suppose for all t in .

• If r ≥ 1 and 1 ≤ p < +∞, then F is a compact subset of L p (0, T ; X) .

• If r > 1 and p = +∞, then F is a compact subset of C(0, T ; B). Lemma 6. For all T > 0, the imbedding from

H 1 (]0, T [×O) to C([0, T ], L 2 (O)) is compact.
Proof. Use the Aubin's lemma, see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 4], extended to the case p = +∞, with

X = H 1 (O) and B = Y = L 2 (Ω). Lemma 7. Let u belong to H 1 (]0, T [×O) ∩ L ∞ (]0, T [; H 1 (O)), then u be- longs to C([0, T ]; H 1 ω (O)) where H 1 ω (O) is the space H 1 (O) but with the weak topology. Proof. The function u, belongs to C([0, T ], L 2 (O)). Let now (t n ) n be a se- quence in [0, T ] converging to t. Then, u(t n , •) converges to u(t, •) in L 2 (O). Also, the sequence (u(t n , •)) n∈N is bounded in H 1 (O)
, therefore from any subsequence of (u(t n , •)) n∈N , one can extract a subsequence that converges weakly in H 1 (O). The only possible limit is u(t, •) therefore the whole sequence converges weakly in H 1 (O).

Lemma 8. Let (u n ) n∈N be bounded in H 1 (]0, T [×O) and in L ∞ (]0, T [; H 1 (O)).
Let (u n k ) k∈N be a subsequence which converges weakly to some u in H 1 (]0, T [×O).

Then, for all t in [0, T ], the same subsequence

u n k (t, •) converges weakly to u(t, •) in H 1 (O). Proof. For all t in [0, T ], u n k (t, •) converges strongly to u(t, •) in L 2 (O). Therefore, any subsequence u n k j (t, •) that converges weakly in H 1 (O) has u(t, •) for limit. Since u n k (t, •) is bounded in H 1 (O), from any subsequence of u n k (t,
•), one can extract a further subsequence that converges weakly in H 1 (O), therefore, for all t in [0, T ], the whole subsequence

u n k (t, •) converges weakly to u(t, •) in H 1 (O).
6 Proof of Theorem 2

Idea of the proof

We proceed as in [START_REF] Carbou | Time average in micromagnetism[END_REF] and [START_REF] Santugini-Repiquet | Solutions to the Landau-Lifshitz system with nonhomogenous Neumann boundary conditions arising from surface anisotropy and super-exchange interactions in a ferromagnetic media[END_REF] and combine the ideas of both papers. We start by extending the surface energies to a thin layer of thickness 2η > 0.

As in [START_REF] Santugini-Repiquet | Solutions to the Landau-Lifshitz system with nonhomogenous Neumann boundary conditions arising from surface anisotropy and super-exchange interactions in a ferromagnetic media[END_REF], we consider the operator

H η s : H 1 (Ω) ∩ L ∞ (Ω) → H 1 (Ω) ∩ L ∞ (Ω) m → 1 2η    0 in R 3 \ ( B×(I \ I η ) ), 2K s ((m • ν)ν -m) + 2J 1 (m * -m) +4J 2 (m • m * )m * -m * 2 m in B×(I \ I η ), (6.1
) where m * is the reflection of m, i.e. m * (x, y, z, t) = m(x, y, -z, t), see Figure 1. The associated energy is:

η Figure 1: Artificial boundary layer E η s (m) = K s 2η B×(I \I η ) m 2 -(m • ν) 2 dx + J 1 2η B×(I \I η ) m 2 + m * 2 2 -(m • m * ) dx + J 2 2η B×I \I η m * 2 m 2 -(m • m * ) 2 dx. (6.2)
This energy will replace the surfacic ones (3.1) and (3.2). The idea is to consider the Landau-Lifshitz-Maxwell system with homogenous Neumann boundary conditions with the excitation containing this new component then have η tend to 0. We consider the doubly penalized problem:

α ∂m k,η ∂t + m k,η ∧ ∂m k,η ∂t = (1 + α 2 )(A △ m -Km + h k,η + H η s (m k,η )) -k(1 + α 2 )(( m k,η 2 -1)m k,η ), (6.3a) ∂m k,η ∂ν = 0 on ∂Ω, (6.3b) m k,η (0, •) = m 0 , (6.3c)
with Maxwell equations:

ε 0 ∂e k,η ∂t + σ(e k,η + f )1 Ω -curl h k,η = 0, (6.4a) µ 0 ∂(m k,η + h k,η ) ∂t + curl e k,η = 0, (6.4b) e k,η (0, •) = e 0 , (6.4c)
h k,η (0, •) = h 0 . (6.4d)
The basic idea is to prove the existence of weak solutions to the penalized problem via Galerkin, then have k tend to +∞ to satisfy the local norm constraint on the magnetization, then have η tend to 0 to transform the homogenous Neumann boundary condition into the nonlinear condition above.

First Step of Galerkin's method

As in [START_REF] Alouges | On global weak solutions for Landau-Lifshitz equations : existence and nonuniqueness[END_REF] we consider the eigenvectors (v j ) j≥1 of the Laplace operator with Neumann homogenous conditions. This basis is, up to a renormalisation, an hilbertian basis for the spaces L 2 (Ω), H 1 (Ω), and {u ∈ H 2 (Ω), ∂u ∂ν = 0}. The eigenvectors v k all belong to C ∞ (Ω; R 3 ). We call V n the space spanned by (v j ) 1≤j≤n . As in [START_REF] Carbou | Time average in micromagnetism[END_REF], we consider an hilbertian basis (ω j ) j≥1 of L 2 (R 3 ; R 3 ) such that every ω j belongs to C ∞ c (R 3 ; R 3 ). We call W n the space spanned by (ω j ) 0≤j≤n .

Set n ≥ 1, η > 0 and k > 0. We search for m n,k,η in

H 1 (R + ; (V n ) 3 ), h n,k,η in H 1 (R + ; W n ), and e n,k,η in H 1 (R + ; W n ) such that α dm n,k,η dt = -P Vn (m n,k,η ∧ dm n,k,η dt ) + (1 + α 2 )P Vn (A △ m n,k,η -Km n,k,η ) + (1 + α 2 )P Vn (h n,k,η + H η s (m n,k,η )) -(1 + α 2 )kP Vn (( m n,k,η 2 -1)m n,k,η ), (6.5a)
ε 0 de n,k,η dt = -P Wn (curl h n,k,η ) -P Wn (1 Ω (e n,k,η + f )), (6.5c)
with the inital conditions:

m n,k,η (0, •) = P Vn (m 0 ), (6.6a) h n,k,η (0, •) = P Wn (h 0 ), (6.6b) e n,k,η (0, •) = P Wn (e 0 ), (6.6c)
where P Vn is the orthogonal projection on V n in L 2 (Ω) and P Wn is the orthogonal projection on

W n in L 2 (Ω; R 3 )). Let a(t) = (a i (t)) 1≤i≤n , b = (b i ) 1≤i≤n and c(t) = (c i (t)) 1≤i≤n be the coefficients of m n,k,η (t, •), h n,k,η (t, •)
and e n,k,η (t, •) in the decomposition

m n,k,η (t, •) = n i=1 a i (t)v i , h n,k,η (t, •) = n i=1 b i (t)ω i , e n,k,η (t, •) = n i=1 c i (t)ω i .
Then, System (6.5) is equivalent to

da dt + φ(a, da dt ) = F m (a, b), (6.7a) d(b + La) dt = F h (c), (6.7b) dc dt = F e (h n,k,η , e n,k,η ) + f * , (6.7c)
where L is linear, F m , F h and F e are polynomial thus of class C ∞ , and f * is in L 2 (R + ; R n ). These are supplemented by initial conditions

a(0, •) = a 0 , b(0, •) = b 0 , c(0, •) = c 0 , (6.8)
where a 0 , b 0 , and c 0 are obtained by orthogonal projection of m 0 , h 0 , e 0 over the v i or the ω i . As φ(•, •) is bilinear continuous and φ(a, •) is antisymmetric, the linear application Id -φ(a, •) is invertible. Finally f * is L 2 . Therefore, by the Carathéorody theorem, System (6.7) has local solutions. Therefore, there exists

T * > 0 and m n,k,η in H 1 (]0, T * [; (V n ) 3 ), h n,k,η in H 1 (]0, T * [; W n
) and e n,k,η in H 1 (]0, T * [; W n ) that satisfy (6.5) and (6.6).

Multiplying (6.5) by test functions and integrating by part yields:

α Q T ∂m n,k,η ∂t • φdxdt + Q T m n,k,η ∧ ∂m n,k,η ∂t • φdxdt = -(1 + α 2 )A Q T 3 i=1 ∂m n,k,η ∂x i • ∂φ ∂x i dxdt -(1 + α 2 ) Q T (K(x)m n,k,η (x)) • φdxdt + (1 + α 2 ) Q T h n,k,η • φdxdt -(1 + α 2 )k Q T ( m n,k,η 2 -1)m n,k,η • φdxdt + (1 + α 2 ) K s η ]0,T [×(B×]-η,η[) ((ν • m n,k,η )ν -m n,k,η ) • φdxdt + (1 + α 2 ) J 1 η ]0,T [×(B×]-η,η[) (m * n,k,η -m n,k,η ) • φdxdt + 2(1 + α 2 ) J 2 η ]0,T [×(B×]-η,η[) (m n,k,η • m * n,k,η )m * n,k,η -m * n,k,η 2 m n,k,η • φdxdt, (6.9a)
for all φ in C ∞ ([0, T * ], V 3 n ). And

µ 0 ]0,T [×R 3 ∂h n,k,η ∂t + ∂m n,k,η ∂t • ψdxdt + ]0,T [×R 3 curl e n,k,η • ψdxdt = 0, (6.9b) for all ψ in C ∞ ([0, T * ], W n ). And ε 0 ]0,T [×R 3 ∂e n,k,η ∂t • Θdxdt - ]0,T [×R 3 curl h n,k,η • Θdxdt +σ Q T (e n,k,η + f ) • Θdxdt = 0, (6.9c) for all Θ in C ∞ c ([0, T * ], W n ). By density, (6.9) also holds if φ belongs to L 2 (]0, T * [; V 3 n ), ψ belongs to L 2 (]0, T * [, W n ), and Θ belongs to L 2 (]0, T * [, W n ).
As in [START_REF] Carbou | Time average in micromagnetism[END_REF], set φ = ∂m n,k,η ∂t in (6.9a), we obtain

A 2 Ω ∇m n,k,η (T, x) 2 dx + 1 2 Ω (K(x)m n,k,η (T, x)) • m(T, x)dx + k 4 Ω ( m n,k,η (T, x)) 2 -1) 2 dx - Q T h n,k,η • ∂m n,k,η ∂t dxdt + E η s (m n,k,η (T, •)) + α 1 + α 2 Q T ∂m n,k,η ∂t 2 dxdt ≤ A 2 Ω ∇P n (m 0 ) 2 dx + 1 2 Ω (K(x)P Vn (m 0 )) • P Vn (m 0 )dx + k 4 Ω ( P Vn (m 0 )) 2 -1) 2 dx + E η s (P Vn (m 0 )).
Set ψ = h n,k,η in (6.9b), we obtain

µ 0 2 R 3 h n,k,η (T, x) 2 dxdt + µ 0 Q T ∂m n,k,η ∂t • h n,k,η dxdt + ]0,T [×R 3 h n,k,η • curl e n,k,η dxdt ≤ µ 0 2 R 3 P Wn (h 0 ) 2 dx,
Set Θ = e n,k,η in (6.9c), we obtain

ε 0 2 R 3 e n,k,η (T, •) 2 - ]0,T [×R 3 e n,k,η • curl h n,k,η dxdt + σ ]0,T [×R 3 e n,k,η 2 dxdt + σ ]0,T [×R 3 f • e n,k,η dxdt ≤ ε 0 2 R 3 P W N (e 0 ) 2 dx.
Combining these three inequalities, we get an energy inequality

A 2 Ω ∇m n,k,η (T, •) 2 dx + 1 2 Ω (K(x)m n,k,η (T, x)) • m n,k,η (T, x)dx + k 4 Ω ( m n,k,η (T, x)) 2 -1) 2 dx + ε 0 2µ 0 R 3 e n,k,η (T, x) 2 dx + 1 2 R 3 h n,k,η (T, x) 2 dx + E η s (m n,k,η (T, •)) + α 1 + α 2 Q T ∂m n,k,η ∂t 2 dxdt + σ µ 0 ]0,T [×R 3 e n,k,η 2 dxdt + σ µ 0 ]0,T [×R 3 f • e n,k,η dxdt ≤ A 2 Ω ∇P Vn (m 0 ) 2 dx + 1 2 Ω (K(x)P Vn (m 0 )) • P Vn (m 0 )dx + k 4 Ω ( P Vn (m 0 ) 2 -1) 2 dx + E η s (P Vn (m 0 )) + ε 0 2µ 0 R 3 P W N (e 0 ) 2 dx + 1 2 R 3 P W N (h 0 ) 2 dx (6.10)
The projection P n (m 0 ) converges to m 0 in H 1 (Ω) and in L 6 (Ω) by Sobolev imbedding. The terms on the right hand-side remain bounded independently of n. The last term on the left hand-side may be dealt with by Young inequality. Thus, m n,k,η , h n,k,η and e n,k,η cannot explode in finite time and exist globally.

Final step of Galerkin's method

We now have n tend to +∞ By (6.10) and using Young inequality to deal with the term containing f :

• m n,k,η is bounded in L ∞ (R + ; L 4 (Ω)) independently of n. • ∇m n,k,η is bounded in L ∞ (R + ; L 2 (Ω)) independently of n. • ∂m n,k,η ∂t is bounded in L 2 (R + ; L 2 (Ω)) independently of n. • h n,k,η is bounded in L ∞ (R + ; L 2 (Ω)) independently of n. • e n,k,η is bounded in L ∞ (R + ; L 2 (Ω)) independently of n.
Thus, there exist

m k,η in H 1 loc ([0, +∞[; L 2 (Ω)) ∩ L ∞ (0, +∞; H 1 (Ω)), h k,η in L ∞ (R + ; L 2 (Ω)), e k,η in L ∞ (R + ; L 2 (Ω))
, such that up to a subsequence:

• m n,k,η converges weakly to m k,η in H 1 (]0, T [×Ω). • m n,k,η converges strongly to m k,η in L 2 (]0, T [×Ω). • m n,k,η converges strongly to m k,η in C([0, T ]; L 2 (Ω)) and thus in C([0, T ]; L p (Ω))
for all 1 ≤ p < 6.

• ∇m n,k,η converges weakly to ∇m k,η in L 2 (]0, T [×Ω).

• For all time T , ∇m n,k,η (T, •) converges weakly to ∇m k,η (T, •) in L 2 (Ω).

The same subsequence can be used for all time T ≥ 0, see Lemma 8.

• ∂m n,k,η ∂t converges star weakly to

∂m k,η ∂t in L ∞ (R + ; L 2 (Ω)).
• h n,k,η converges star weakly to h k,η in L ∞ (R + ; L 2 (Ω)).

• e n,k,η converges star weakly to e k,η in L ∞ (R + ; L 2 (Ω)).

Moreover, by Aubin's lemma, see [START_REF] Aubin | Un théorème de compacité[END_REF], m n,k,η converges strongly to m k,η in L p (R + ; L q (Ω)) for 1 ≤ p < +∞ and 1 ≤ q < 6.

Taking the limit in the energy inequality (6.10) as n tend to +∞ is tricky: the terms involving the L 2 (Ω) norm of e n,k,η (T, •) and h n,k,η (T, •) are tricky. For all T > 0, we can extract a subsequence of e n,k,η (T, •) that converges weakly to e T k,η in L 2 (Ω) as n tends to +∞. The tricky part is that it is unproven that e T k,η is equal to e k,η (T, •). If we had strong convergence of e n,k,η as a function defined on R + ×Ω or if we had the existence of a subsequence along which e n,k,η (T, •) converged weakly in L 2 (Ω) for almost all time T , then we could conclude directly. Unfortunately, while we have for all T > 0, the existence of a subsequence of e n,k,η (T, •) that converges weakly in L 2 (Ω), the subsequence depends on T . We have the same problem for h n,k,η . There's no such problem with m(T, •), see Lemma 8. To solve the problem, we first integrate (6.10) over ]T 1 , T 2 [ where 0 ≤ T 1 < T 2 < +∞ then we can take the limit as n tend to +∞:

A 2 T 2 T 1 Ω ∇m k,η (T, •) 2 dxdT + 1 2 T 2 T 1 Ω (K(x)m k,η (T, x)) • m k,η (T, x)dxdT + k 4 T 2 T 1 Ω ( m k,η (T, x)) 2 -1) 2 dx + ε 0 2µ 0 T 2 T 1 R 3 e k,η (T, x) 2 dx + 1 2 T 2 T 1 R 3 h k,η (T, x) 2 dx + T 2 T 1 E η s (m k,η (T, •)) + α 1 + α 2 T 2 T 1 Q T ∂m k,η ∂t 2 dxdt + σ µ 0 T 2 T 1 ]0,T [×R 3 e k,η 2 dxdt + σ µ 0 T 2 T 1 ]0,T [×R 3 f • e k,η dxdt ≤ A 2 T 2 T 1 Ω ∇m 0 2 dx + 1 2 T 2 T 1 Ω (K(x)m 0 ) • m 0 dx + T 2 T 1 E η s (m 0 ) + ε 0 2µ 0 T 2 T 1 R 3 e 0 2 dx + 1 2 T 2 T 1 R 3 h 0 2 dx,
for all 0 ≤ T 1 < T 2 < +∞. Since the equality holds for all T 1 and T 2 , we have for almost all T > 0

A 2 Ω ∇m k,η (T, •) 2 dxdT + 1 2 Ω (K(x)m k,η (T, x)) • m k,η (T, x)dxdT + k 4 Ω ( m k,η (T, x)) 2 -1) 2 dx + ε 0 2µ 0 R 3 e k,η (T, x) 2 dx + 1 2 R 3 h k,η (T, x) 2 dx + E η s (m k,η (T, •)) + α 1 + α 2 Q T ∂m k,η ∂t 2 dxdt + σ µ 0 ]0,T [×R 3 e k,η 2 dxdt + σ µ 0 ]0,T [×R 3 f • e k,η dxdt ≤ A 2 Ω ∇m 0 2 dx + 1 2 Ω (K(x)m 0 ) • m 0 dx + E η s (m 0 ) + ε 0 2µ 0 R 3 e 0 2 dx + 1 2 R 3 h 0 2 dx, (6.11)
We take the limit in (6.9a) as n tends to +∞:

Q T α ∂m k,η ∂t • φdxdt + Q T m k,η ∧ ∂m k,η ∂t • φdxdt = -(1 + α 2 )A Q T 3 i=1 ∂m k,η ∂x i • ∂φ ∂x i dxdt -(1 + α 2 ) Q T (K(x)m k,η (t, x)) • φ(t, x)dxdt + (1 + α 2 ) Q T h k,η • φdxdt + (1 + α 2 ) K s η ]0,T [×(B×]-η,η[) ((ν • m k,η )ν -m k,η ) • φdxdt + (1 + α 2 ) J 1 η ]0,T [×(B×]-η,η[) (m * k,η -m k,η ) • φdxdt + 2(1 + α 2 ) J 2 η ]0,T [×(B×]-η,η[) (m k,η • m * k,η )m * n,k,η -m * k,η 2 m k,η • φdxdt, (6.12a)
for all φ in n C ∞ ([0, T [; V 3 n ).
By density, it also holds for all φ in H 1 (]0, T [×Ω). We integrate (6.9b) by parts then take the limit as n tends to +∞.

-µ 0 R + ×R 3 (h k,η + m k,η )) ∂ψ ∂t dxdt + R + ×R 3 e k,η • curl ψdxdt = µ 0 R 3 (h 0 + m 0 )) • ψ(0, •)dx, (6.12b)
for all ψ in n C ∞ c ([0, +∞[; W n ). By density, it also holds for all ψ in L 1 (R + ; H 1 (Ω)) such that ∂ψ ∂t belongs to L 1 (R + ; H 1 (Ω)). We integrate (6.9c) by parts then take the limit as n tends to +∞.

-ε 0 R + ×R 3 e k,η • ∂Θ ∂t dxdt - R + ×R 3 h k,η • curl Θdxdt + σ R + ×Ω (e k,η + f ) • Θdxdt = ε 0 R 3 e 0 • Θ(0, •)dx, (6.12c) for all Θ in n C ∞ c ([0, +∞[; W n ).
By density, it also holds for all Θ in L 1 (R + ; H 1 (Ω)) such that ∂Θ ∂t belongs to L 1 (R + ; H 1 (Ω)).

Limit as k tends to +∞

By (6.11) and using Young inequality to deal with the term containing f :

• m k,η is bounded in in L ∞ (R + ; L 4 (Ω)) independently of n. • ∇m k,η is bounded in L ∞ (R + ; L 2 (Ω)) independently of n. • ∂m k,η ∂t is bounded in L 2 (R + ; L 2 (Ω)) independently of n. • h k,η is bounded in in L ∞ (R + ; L 2 (Ω)) independently of n. • e k,η is bounded in in L ∞ (R + ; L 2 (Ω)) independently of n. • k( m k,η 2 -1) is bounded in in L ∞ (R + ; L 2 (Ω)) independently of n.
Thus, there exist m η , h η , e η , such that up to a subsequence:

• m k,η converges weakly to m η in H 1 (]0, T [×Ω).
• m k,η converges strongly to m η in L 2 (]0, T [×Ω).

• m k,η converges strongly to m η in C([0, T ]; L 2 (Ω)) and thus in C([0, T ]; L p (Ω)) for all 1 ≤ p < 6.

• ∇m k,η converges weakly to ∇m η in L 2 (]0, T [×Ω).

• For all time T , ∇m k,η (T, •) converges weakly to ∇m η (t, •) in L 2 (Ω).

• ∂m k,η ∂t converges star weakly to ∂mη ∂t in L ∞ (R + ; L 2 (Ω)). • h k,η converges star weakly to h η in L ∞ (R + ; L 2 (Ω)).
• e k,η converges star weakly to e η in L ∞ (R + ; L 2 (Ω)).

Moreover, by Aubin's lemma m η converges strongly to m η in L p (R + ; L q (Ω)) for 1 ≤ q < +∞ and 1 ≤ q < 6. Since m k,η 2 -1 converges to 0, therefore m η = 1 almost everywhere on R + ×Ω.

For the reasons explained in §6.3, we integrate (6.11) over [T 1 , T 2 ], drop the term k m η 2 -1 2 L 2 /4, and compute the limit as k tends to +∞. After the limit is taken, we drop the integral over [T 1 , T 2 ] and obtain that for almost all T > 0:

A 2 Ω ∇m η (T, •) 2 dx + 1 2 Ω (K(x)m η (T, x)) • m η (T, x)dx + ε 0 2µ 0 R 3 e η (T, x) 2 dx + 1 2 R 3 h η (T, x) 2 dx + E η s (m η (T, •)) + α 1 + α 2 Q T ∂m η ∂t 2 dxdt + σ µ 0 ]0,T [×R 3 e η 2 dxdt + σ µ 0 ]0,T [×R 3 f • e η dxdt ≤ A 2 Ω ∇m 0 2 dx + 1 2 Ω (K(x)m 0 ) • m 0 dx + E η s (m 0 ) + ε 0 2µ 0 R 3 e 0 2 dx + 1 2 R 3 h 0 2 dx. (6.13)
We replace φ in (6.12a) with m k,η ∧ϕ where ϕ is C ∞ c (R + ×Ω):

-α

Q T m k,η ∧ ∂m k,η ∂t • ϕdxdt + Q T m k,η 2 ∂m k,η ∂t • ϕdxdt = Q T m k,η • ∂m k,η ∂t (m k,η • ϕ)dxdt + (1 + α 2 )A Q T 3 i=1 m k,η ∧ ∂m k,η ∂x i • ∂ϕ ∂x i dxdt + (1 + α 2 ) Q T (m k,η (t, x)∧K(x)m k,η (t, x)) • ϕ(t, x)dxdt -(1 + α 2 ) Q T (m k,η ∧h k,η ) • ϕdxdt -(1 + α 2 ) K s η ]0,T [×(B×]-η,η[) (ν • m k,η )(m k,η ∧ν) • ϕdxdt -(1 + α 2 ) J 1 η ]0,T [×(B×]-η,η[) (m k,η ∧m * k,η ) • ϕdxdt -2(1 + α 2 ) J 2 η ]0,T [×(B×]-η,η[) (m k,η • m * k,η )(m k,η ∧m * k,η ) • ϕdxdt,
We then take the limit as k tends to +∞:

-α Q T m η ∧ ∂m η ∂t • ϕdxdt + Q T ∂m η ∂t • ϕdxdt = +(1 + α 2 )A Q T 3 i=1 m η ∧ ∂m η ∂x i • ∂ϕ ∂x i dxdt + (1 + α 2 ) Q T (m η (t, x)∧K(x)m η (t, x)) • ϕ(t, x)dxdt -(1 + α 2 ) Q T (m η ∧h η ) • ϕdxdt -(1 + α 2 ) K s η ]0,T [×(B×]-η,η[) (ν • m η )(m η ∧ν) • ϕdxdt -(1 + α 2 ) J 1 η ]0,T [×(B×]-η,η[) (m η ∧m * η ) • ϕdxdt -2(1 + α 2 ) J 2 η ]0,T [×(B×]-η,η[) (m η • m * η )(m η ∧m * η ) • ϕdxdt, (6.14a)
We take the limit in (6.12b) as k tends to +∞:

-µ 0 R + ×R 3 (h η + m η )) ∂ψ ∂t dxdt + R + ×R 3 e η curl ψdxdt = µ 0 R 3 (h 0 + m 0 )) • ψ(0, •)dx (6.14b)
for all ψ in L 1 (R + ; H 1 (Ω)) such that ∂ψ ∂t belongs to L 1 (R + ; L 2 (Ω)). We take the limit in (6.12c) as k tends to +∞.

-ε 0 R + ×R 3 e η • ∂Θ ∂t dxdt - R + ×R 3 h η • curl Θdxdt + σ R + ×Ω (e η + f ) • Θdxdt = ε 0 R 3 e 0 • Θ(0, •)dx, (6.14c)
for all Θ in in L 1 (R + ; H 1 (Ω)) such that ∂Θ ∂t belongs to L 1 (R + ; L 2 (Ω)).

Limit as η tends to 0

Since H 1 (Ω) is continuously imbedded in C 0 ]-L -, L + [\{0}; L 4 (B) , E η s (m 0 ) remains bounded independently of η and converges to E s (m 0 ) . Thus, using (6.13) and the constraint m η = 1 almost everywhere:

• m η is bounded in L ∞ (R + ×Ω) by 1. • ∇m η is bounded in L ∞ (R + ; L 2 (Ω)) independently of η. • ∂m k,η ∂t is bounded in L 2 (R + ; L 2 (Ω)) independently of η. • h k,η is bounded in in L ∞ (R + ; L 2 (Ω)) independently of η. • e k,η is bounded in in L ∞ (R + ; L 2 (Ω)) independently of η.
Thus, there exists m in L ∞ (R + ; H 1 (Ω)) and in

H 1 loc ([0, +∞[; L 2 (Ω)), h in L ∞ (R + ; L 2 (Ω)) and e in L ∞ (R + ; L 2 (Ω)) such that up to a subsequence • m η converges weakly to m in H 1 (]0, T [×Ω).
• m η converges strongly to m in L 2 (]0, T [×Ω).

• m η converges strongly to m in C([0, T ]; L 2 (Ω)) and thus in C([0, T ]; L p (Ω)) for all 1 ≤ p < +∞.

• ∇m η converges weakly to ∇m in L 2 (]0, T [×Ω).

• For all time T , ∇m η (t, •) converges weakly to ∇m(t, •) in L 2 (Ω).

• ∂mη ∂t converges star weakly to ∂m ∂t in L ∞ (R + ; L 2 (Ω)). • h η converges star weakly to h in L ∞ (R + ; L 2 (Ω)).
• e η converges star weakly to e in L ∞ (R + ; L 2 (Ω)).

As m η = 1 almost everywhere, m = 1 almost everywhere. Moreover, as m η (0, •) = m 0 , we have m(0, •) = m 0 .

For the reasons explained in §6.3, we integrate (6.13) over [T 1 , T 2 ], and compute the limit as k tends to +∞. All the volume terms converge to their intuitive limit. After the limit is taken, we drop the integral over [T 1 , T 2 ] and obtain that for almost all T > 0: Taking the limit in the surfacic terms requires more work. For easier understanding, First, the space

H 1 (]0, T [×Ω) is compactly imbedded into C 0 ([-L -, 0]; L 2 (]0, T [×B) ⊗ C 0 ([0, L + ]; L 2 (]0, T [×B)).
This is a direct application of Lemma 6 with O =]0, T [×B and, thus a direct consequence of the extended Aubin's lemma 5. Therefore, m η converges strongly to m in

C 0 ([-L -, 0]; L 2 (]0, T [×B) ⊗ C 0 ([0, L + ]; L 2 (]0, T [×B)).
Since m η = 1, the convergence is strong in

C 0 ([-L -, 0]; L p (]0, T [×B) ⊗ C 0 ([0, L + ]; L p (]0, T [×B)), for all p < +∞. lim sup η→0 T 2 T 1 E η s (m η (t, •)) -E η s (m(t, •)) ≤ lim sup η→0 1 2η η -η T 2 T 1 B P (m η (t), m * η (t)) -P (m(t), m * (t)) dxdydzdt ≤ lim sup η→0 sup z∈[-η,η] T 2 T 1 B P (m η (t), m * η (t)) -P (m(t), m * (t)) dxdy ≤ 0.
where P is some polynomial. Moreover, m(•, •) belongs to:

C 0 [-L -, 0]; L p (]0, T [×B) ⊗ C 0 [0, L + ]; L p (]0, T [×B) .
Therefore, we have

lim η→0 T 2 T 1 E η s (m(t, •)) -E s (m(t, •)) ≤ lim η→0 1 2η η -η B P (m(t), m * (t)) -P (m(x, y, 0 + , t), m(x, y, 0 -, t)) dxdydt ≤ lim η→0 sup z <η T 2 T 1 B P (m(z, T ), m * (z, t)) -P (m(x, y, 0 + , T ), m(x, y, 0 -, t)) dxdydt ≤ 0.
Hence, the integral over [T 1 , T 2 ] of inequality (4.3d) hold for all 0 < T 1 < T 2 , therefore inequality (4.3d) is satisfied for almost all t > 0.

We take the limit in (6.14a) as η tends to 0. All the volume terms converges to their intuitive limit. Moreover, because of the strong convergence, along a subsequence, of m η to m in

C 0 ([-L -, 0]; L p (]0, T [×B) ⊗ C 0 ([0, L + ]; L p (]0, T [×B)),
for all p < +∞, we have lim sup

η→0 1 η ]0,T [×(B×]-η,η[) (ν • m η )(m η ∧ν) • ϕ(t, x)dxdt - ]0,T [×(B×]-η,η[) (ν • m)(m∧ν) • ϕ(t, x)dxdt = 0, lim sup η→0 1 η ]0,T [×(B×]-η,η[) (m η ∧m * η ) • ϕ(t, x)dxdt - ]0,T [×(B×]-η,η[) (m∧m * ) • ϕ(t, x)dxdt = 0, lim sup η→0 1 η 1 η ]0,T [×(B×]-η,η[) (m η • m * η )(m η ∧m * k,η ) • ϕ(t, x)dxdt - ]0,T [×(B×]-η,η[) (m • m * )(m∧m * ) • ϕ(t, x)dxdt = 0.
Since m belongs to

C 0 ([-L -, 0]; L p (]0, T [×B) ⊗ C 0 ([0, L + ]; L p (]0, T [×B)),
each surface term also converges to its surface intuitive limits. Therefore, the weak formulation (4.3a) is also satisfied. We take the limits as η tends to 0 in (6.14b) and (6.14b). All the volume terms converges to their intuitive limit. Hence, relations (4.3b) and (4.3c) are satisfied. This finishes our proof of Theorem 2.

Characterization of the ω-limit set

We consider (m, h, e) a weak solution to the Landau-Lifschitz-Maxwell system given by Theorem 2.

We consider u ∈ ω(m). There exists a non decreasing sequence (t n ) n such that t n -→ +∞, and m(t n , .) ⇀ u in H 1 (Ω) weak. Since Ω is a smooth bounded domain, then m(t n , .) tends to u in L p (Ω) strongly for p ∈ [1, 6[, and extracting a subsequence, we assume that m(t n , .) tends to u almost everywhere, so that the saturation constraint |u| = 1 is satisfied almost everywhere.

In addition, we remark that for all n, |m(t n , .)| = 1 almost everywhere, so that m(t n , .) L ∞ (Ω) = 1. By interpolation inequalities in the L p spaces, we obtain that for all p < +∞, m(t n , .) tends to u in L p (Ω) strongly.

First

Step. we fix a a non negative real number. for s ∈] -a, a[ and x ∈ Ω, for n large enough, we set

U n (s, x) = m(t n + s, x).
We have the following estimate: Since m(t n , .) tends strongly to u in L 2 (Ω), then

1 2a a -a Ω |U n (s, x) -m(t n , x)| 2 dxds = 1 2a a -a Ω
U n tends strongly to u in L 2 (-a, a; L 2 (Ω)). (7.1) We remark now that the sequence (∇U n ) n is bounded in L ∞ (-a, a; L 2 (Ω)). In addition, ( ∂Un ∂t ) n is bounded in L 2 (-a, a; L 2 (Ω)). So, by applying Aubin's Lemma with X = H 1 (Ω), B = H 3 4 (Ω), Y = L 2 (Ω), r = 2 and p = +∞, we obtain that (U n ) n is compact in C 0 ([-a, a]; H 3 4 (Ω)), so that

U n tends strongly to u in C 0 ([-a, a]; H 3 4 (Ω)). (7.2)
By continuity of the trace operator, since

H 1 4 (Γ) ⊂ L 2 (Γ), we obtain that γ(U n ) -→ γ(u) strongly in C 0 ([-a, a]; L 2 (Γ)).
In addition, by classical properties of the trace operator, for all n, U n L ∞ ([-a,a]×Ω) = 1, so γ(U n ) L ∞ ([-a,a]×Γ) ≤ 1. We obtain then in particular that γ(U n ) -→ γ(u) strongly in L p ([-a, a] × ∂Ω), p < +∞ Second step. We consider a smooth positive function ρ a compactly supported in [-a, a] such that

ρ a (τ ) = 1 for τ ∈ [-a + 1, a -1], 0 ≤ ρ a ≤ 1, |ρ ′ a | ≤ 2.
For n great enough, we set By construction of (m, h, e), we know that h and e are in L ∞ (R + ; L 2 (R 3 )).

h n a (x) = 1 2a
We have the following estimate:

h n a 2 L 2 (R 3 ) = x∈R 3 1 2a a -a h(t n + s, x)ρ a (s)ds 2 ≤ 1 2a a -a ρ 2 a (s)ds 1 2a R 3 a -a |h(t n + s, x)| 2 dsdx ≤ 2a + 2 2a h L ∞ (R + ;L 2 (R 3 )) .
Therefore,

∀a ≥ 1, ∀n, h n a L 2 (R 3 ) ≤ 2 h L ∞ (R + ;L 2 (R 3 )) . (7.3)
In the same way, we prove that

∀a ≥ 1, ∀n, e n a L 2 (R 3 ) ≤ 2 e L ∞ (R + ;L 2 (R 3 )) . (7.4)
So for a fixed value of a we can assume by extracting a subsequence that h n a and e n a converge weakly in L 2 (R 3 ) when n tends to +∞:

h n a ⇀ h a and e n a ⇀ e a weakly in L 2 (R 3 ) when n → +∞.

In the weak formulation (4.3a), we take φ(t, x) = 1 2a ρ a (t-t n )ψ(x) where ψ ∈ D(Ω). We obtain after the change of variables s = t -t n :

1 2a a -a Ω ∂U n ∂t -αU n ∧ ∂U n ∂t ψ(x)ρ a (s)dxds = T 1 + . . . + T 6
with

T 1 = (1 + α 2 )A 1 2a a -a Ω 3 i=1 U n (s, x)∧ ∂U n ∂x i (t, x) • ∂ψ ∂x i (x)dxds, T 2 = (1 + α 2 ) 1 2a a -a Ω (U n (s, x)∧K(x)U n (s, x)) • ψ(x)ρ a (s)dxds, T 3 = -(1 + α 2 ) 1 2a a -a Ω (U n (s, x)∧h(t n + s, x)) • ψ(x)ρ a (s)dxds, T 4 = -(1 + α 2 )K s 1 2a a -a (Γ ± ) (ν • γU n )(γU n ∧ν) • γψ( x)ρ a (s)dS( x)ds, T 5 = -(1 + α 2 )J 1 1 2a a -a (Γ ± ) (γU n ∧γ * U n ) • γψ( x)ρ a (s)dS( x)ds, T 6 = -2(1+α 2 )J 2 1 2a a -a (Γ ± ) (γU n •γ * U n )(γU n ∧γ * U n )•γψ( x)ρ a (s)dS( x)ds.
Now for a fixed value of the parameter a, we take the limit of the previous equation when n tends to +∞.

Left hand side term: we have the following estimates.

1 2a a -a Ω ∂U n ∂t -αU n ∧ ∂U n ∂t ψ(x)ρ a (s)dxds ≤ (1 + α) 1 2a a -a ρ a (s) ∂U n ∂t (s, .) L 2 (Ω) ψ L 2 (Ω) ≤ 1 √ 2a ψ L 2 (Ω) (1 + α) a -a Ω ∂U n ∂t 2 dxds 1 2 ≤ 1 √ 2a ψ L 2 (Ω) (1 + α) +∞ tn-a Ω ∂m ∂t 2 dxds 1 2 Since ∂ ∂m t ∈ L 2 (R + ; L 2 (Ω)
), the last right hand side term tends to zero when n (and so t n ) tends to +∞. Therefore

1 2a a -a Ω ∂U n ∂t -αU n ∧ ∂U n ∂t ψ(x)ρ a (s)dxds -→ 0 when n -→ +∞. Limit for T 1 : since U n -→ u strongly in L 2 ([-a, a] × Ω), since ∂U n ∂x i ⇀ ∂u ∂x i in L 2 (] -a, a[×Ω) weak, we obtain that T 1 -→ (1 + α 2 )A 1 2a a -a ρ a (s)ds Ω 3 i=1 u(x∧ ∂u ∂x i (x) • ∂ψ ∂x i (x)dx. Limit for T 2 : since U n tends to u strongly in L 2 ([-a, a] × Ω), T 2 -→ (1 + α 2 )A 1 2a a -a ρ a (s)ds Ω (u(x)∧K(x)u(x)) • ψ(x)dx.
Limit for T 3 : we write

T 3 = -(1+α 2 ) Ω u∧h n a ψdx+(1+α 2 ) 1 2a a -a Ω (u-U n )h(t n +s, x)ψ(x)ρ a (s)dxds.
We estimate the right hand side term as follows:

1 2a 1 2a a -a Ω (u -U n )h(t n + s, x)ψ(x)ρ a (s)dxds ≤ ψ L ∞ (Ω) u -U n L 2 (-a,a×Ω) h L 2 ([tn-a,tn+a]×Ω) .
So since U n tends to u in L 2 (-a, a × Ω), we obtain that

T 3 -→ -(1 + α 2 ) Ω u∧h a ψdx.
Limit for T 4 , T 5 and T 6 : since γ(U n ) -→ γ(u) strongly in L p ([-a, a] × Γ ± ) for p < +∞, the same occurs for γ * (U n ) so that we obtain:

T 4 -→ -(1 + α 2 )K s 1 2a a -a ρ a (s)ds (Γ ± ) (ν • γu)(γu∧ν) • γψ( x)dS( x), T 5 -→ -(1 + α 2 )J 1 1 2a a -a ρ a (s)ds (Γ ± ) (γu∧γ * u) • γψ( x))dS( x), T 6 -→ -2(1+ α 2 )J 2 1 2a a -a ρ a (s)ds (Γ ± ) (γu•γ * u)(γu∧γ * u)•γψ( x))dS( x).
So we obtain that u satisfies for all ψ ∈ D ′ (Ω):

A Ω 3 i=1 u(x∧ ∂u ∂x i (x) • ∂ψ ∂x i (x)dx + A Ω (u(x)∧K(x)u(x)) • ψ(x)dx - 2a a -a ρ a (s)ds (1 + α 2 ) Ω u∧h a ψdx -K s (Γ ± ) (ν • γu)(γu∧ν) • γψ( x)dS( x) -J 1 (Γ ± ) (γu∧γ * u) • γψ( x))dS( x) -2J 2 (Γ ± ) (γu • γ * u)(γu∧γ * u) • γψ( x))dS( x) = 0.
We remark that by density, we can extend this equality for all ψ ∈ H 1 (Ω).

We take now the limit when a tends to +∞. By definition of ρ a we obtain that 2a a -a ρ a (s)ds -→ 1.

Concerning h a , by taking the weak limit in Estimate (7.3), we obtain that:

∀a ≥ 1, h a L 2 (R 3 ) ≤ 2 h L ∞ (R + ;L 2 (R 3 )) . (7.5)
So by extracting a subsequence, we can assume that

h a -⇀ H in L 2 (R 3
) weak when a -→ +∞.

In (4.3b), we take ψ(t, x) = θ a (t -t n )∇ξ(x) where ξ ∈ D ′ (R 3 ) and where θ a (t) = U n (s, x)ρ a (s)ds) • ∇ξ(x)dx = 0.

We take the limit of this equality when n tends to +∞ for a fixed a:

-µ 0 R 3 (h a (x) + 1 2a
a -a ρ a (s)dsu(x)) • ∇ξ(x)dx = 0, and taking the limit when a tends to +∞, we get:

-µ 0 R 3
(H(x) + u(x)) • ∇ξ(x)dx = 0, that is div (H + u) = 0 in D ′ (R 3 ).

In (4.3c), we take Θ(t, x) = 1 2a ρ a (t -t n )ξ(x), where ξ ∈ D ′ (R 3 ). We obtain: For n large enough, the right hand side term vanishes. We denote by γ n a the term:

-ε 0 1 2a a -a
γ n a = -ε 0 1 2a a -a R 3
e(t n + s, x) • ρ ′ a (s)ξ(x)dxds.

We have:

|γ n a | ≤ ε 0 a ξ L 2 (R 3 ) e L ∞ (R + ;L 2 (R 3 )) .
So for a fixed a, we can extract a subsequence till denoted γ n a which converges to a limit γ a such that

|γ a | ≤ ε 0 a ξ L 2 (R 3 ) e L ∞ (R + ;L 2 (R 3 )) .
Moreover, thus for a fixed a, since f ∈ L 2 (R + × Ω), this term tends to zero as n tends to +∞.

Therefore taking the limit when n tends to +∞ in (7.6) we obtain:

γ a - R 3
h a • curl ξdx + σ Ω e a • ξ(x)dx = 0.

Taking now the limit when a tends to +∞ yields

- R 3 H • curl ξdx + σ Ω E • ξ(x)dx = 0, (7.7)
where E is a weak limit of a subsequence of (e a ) a .

In the same way, in (4.3b), we take ψ(t, x) = ρ a (t -t n )ξ(x). By the same arguments, we obtain that

R 3 E curl ξ = 0, that is curl E = 0 in D ′ (R 3 ).
So we remark the E is in H curl (R 3 ) and by density of D(R 3 ) in this space, we can take ξ = E in (7.7). We obtain then that This concludes the proof of Theorem 4.

Conclusion

In this paper, we have proven the existence of solutions to the Landau-Lifshitz-Maxwell system with nonlinear Neumann boundary conditions arising from surface energies. We have also characterized the ω-limit set of those weak solutions. Further improvements should be possible. On the one hand, we expect that extending these results to curved spacers should be possible. No fundamental new idea should be necessary to carry out such an extension of our results as long as the spacer fully separates the domain in two. However, even in that case, the technicalities would lengthen the proof and the statement of the theorem as it would be necessary to write down geometric conditions on the spacers (the spacer cannot share a tangent plane with the domain boundary as it would create cusps).

On the other hand, the construction of more regular solutions for this model remains open.

  L 2 (R + × Ω), we obtain that a -a Ω |U n (s, x)m(t n , x)| 2 dxds -→ 0 as n tends to + ∞.

  n + s, x)ρ a (s)ds and e n a (n + s, x)ρ a (s)ds.

3 ( 0 R 3 (h 0

 3030 h(t n + s, x) + U n (s, x)) • ∇ξ(x)ρ a (s)dxds = µ + m 0 ) • ∇ξ(x)θ a (0)dx = 0 since div (h 0 + m 0 ) = 0So for all ξ ∈ D ′ (R 3 ), for all a ≥ 1 and all n great enough,

  from (7.7) that ∀ξ ∈ D(R 3 ),

R 3 H

 3 • curl ξdx = 0, that is curl H = 0 in D ′ (R 3 ).So H satisfies: div (H + u) = 0, curl H = 0.

  R 3 e(t n + s, x) • ρ ′ a (s)ξ(x)dxds -

	R 3	h n a • curl ξdx
	+ σ

Ω e n a • ξ(x)dx + σ Ω 1 2a a -a f (t n + s, x)ρ a (s)ξ(x)dxds = = ε 0 R 3 e 0 • ξ(x)ρ a (-t n )dx.

(7.6)


Nicolas Houy
email: houy@gate.cnrs.fr

Keywords: JEL Classification: D23, E42 Bitcoin, transaction fee, mining, crypto-currency

We study the economics of Bitcoin transaction fees in a simple static partial equilibrium model with the specificity that the system security is directly linked to the total computational power of miners. We show that any situation with a fixed fee is equivalent to another situation with a limited block size. In both cases, we give the optimal value of the transaction fee or of the block size. We also show that making the block size a non binding constraint and, in the same time, letting the fee be fixed as the outcome of a decentralized competitive market cannot guarantee the very existence of Bitcoin in the long-term.

Introduction

Bitcoin 1 has been invented in 2008 ( [START_REF] Nakamoto | [END_REF]) but it really became popular and left the circle of strict early adopters in 20132 . It is usually described by laymen as an electronic or internet money even though this definition is much criticized by the computer science community that rather talks about a disruptive and revolutionary protocol. As a protocol, Bitcoin is still in its early stages of development and its specifications are still often modified. In order to reach the stage of implementation, a proposed modification goes through a whole process of validation. Some questions regarding some specifications are still more or less open and left undecided. Among those questions is the value and nature of the transaction fees in the Bitcoin network. In this paper, we intend to give an economist's point of view on this question.

In order to do that, we need to describe, even superficially, how Bitcoin actually works. When an individual sends some bitcoins to another individual, this information is broadcast to the peer-to-peer Bitcoin network. However, for technical purposes we won't address here, this transaction needs to be included, together with other transactions forming a block, in the blockchain in order to be confirmed and secured. The blockchain is a public ledger that contains the full history of all the transactions in bitcoins ever processed. It is the role of miners to do this work of confirming and securing transactions. Practically, this mining process consists in solving a mathematical problem and the first miner to do so can include a block of transactions in the blockchain. As it requires computational resources, the successful miner is rewarded in bitcoins for his useful work. This reward comes both from an ex-nihilo creation of some new bitcoins and from some fees that Bitcoin users can add to their transactions. In order to control the monetary base, mining is made more complex than it could be. And since the probability for each miner to solve the mining problem depends on his relative computational power, the complexity of mining is made dependent on the total computational power of all miners. Precisely, the complexity is dynamically adjusted so that a block solving and hence a creation of bitcoins occurs every ten minutes in expectation. Moreover, the amount of bitcoins newly created for each block solved is halved every 210,000 blocks (or about 4 years). Hence, the number of bitcoins ever created will asymptotically reach 21 millions3 . The first amount created was 50 bitcoins, on January 3 rd , 2009. This whole process is described as brilliant by some but it has been criticized for the inefficiency due to the loss of resources it induces (see [Krugman, 2013] for instance). Indeed, Bitcoin miners have engaged in an arm race to computational power and in the end, much hardware, engineering and energy are used to solve mathematical problems that are artificially made extremely complex. What is often forgotten or not known by people criticizing this aspect of Bitcoin is that Bitcoin security is directly depending on the total computational power of the miners. Then, the "lost" computational resource is the only determinant of Bitcoin security.

From the point of view of the existence of Bitcoin, if we need much and powerful mining for security reasons and miners have the incentive to mine through some rewards, all the better. In current days, this incentive is almost only due to new bitcoins creation4 . As we said above, this process of bitcoin creation will slowly become insignificant. Ceteris paribus, when there is no new bitcoins created, the incentive to mine will decrease dramatically and the security of Bitcoin will not be achieved any longer. In fact, mining has strong positive externalities. It allows the very existence of Bitcoin. And transaction fees are, in the longterm, the only way Bitcoin users can make sure miners keep mining. Hence, the problem of how to deal with the transaction fees in the Bitcoin protocol becomes extremely important and is certainly one of the most determinant when looking at Bitcoin's future.

In this article, we study the economics of Bitcoin's transaction fees in a very simple partial equilibrium setting. We show that a fixed and imposed transaction fee can keep Bitcoin secure enough if the transaction fee is high enough. We also show that any situation with a fixed transaction fee can be obtained equivalently by setting a maximum block size instead. However, if we let the transaction fee be the result of a decentralized market and have no constraint on the maximum block size, the transaction fee will eventually go to 0 and miners will not have the necessary incentives to keep mining, hence to keep Bitcoin alive.

As main recommendations, we support the view that the Bitcoin protocol should integrate a block size constraint effectively binding or equivalently a corresponding built-in transaction fee. Put the other way around, we warn for the risk of considering that the block size should be adapted so that Bitcoin can scale up for accepting all transactions and, in the same time, trust a decentralized and free market for space inside blocks that would set the transaction fee. Bitcoin mining is an activity with a strong positive externality (securing the network for other users) and economists know that, in this case, free markets are not efficient. Imposing a transaction fee implies tax revenues for miners and imposing a maximum block size creates a monopoly rent for miners. In both cases, it boils down to correcting a market inefficiency by introducing another one.

Model

The model we use is the simplest static model we can imagine of a partial equilibrium setting. It should be understood as modeling the situation that occurs between two inclusions of two following blocks of transactions in the blockchain. We consider an economy with two goods, a physical one and bitcoin to be used as means of exchange.

Without frictions due to the payment system

First, let us assume as is usually done in partial equilibrium that there is no friction due to the payment system. This case does not represent the situation with cash, debit/credit cards or checks since it assumes that there is no opportunity cost due to unearned interests, time wasting entering PIN codes or writing checks or even printing bills or minting coins ( [Baumol, 1952, Tobin, 1956, Baumol and Tobin, 1989, Bounie and François, 2008]). It is rather written as a benchmark case.

The demand for physical good as a function of price is

Q d (p) = d 0 -αp.
The physical good is produced with a cost function depending on the quantity produced C(q) = q 2 /(2β) so that supply as a function of price is given by Q s (p) = βp. Obviously, in a frictionless economy, the equilibrium price is

p * = d 0 α + β ,
the quantity exchanged at equilibrium is

q * = β α + β d 0
and the profit of the supplier is

Π * = β 2(α + β) 2 d 2 0 .
We make the additional assumption that consumers have utility quasi-linear in bitcoin so that the consumer surplus is

CS * = β 2 2α(α + β) 2 d 2 0 ,
and the total surplus is

T S * = Π * + CS * = β 2α(α + β) d 2 0 .
Those results are standard. We will not interpret or comment them.

With a fixed transaction fee

Now, we consider that for each transaction, a fee c is required and is included in its payment by the buyer and is earned by a miner. In this case, the demand function becomes

Q d (p, c) = d 0 -α.(p+c).
We will make the following assumption: 0 ≤ c ≤ d 0 /α. Otherwise, the fee is so large that there is no physical good exchanged. In this case, the equilibrium price is

p f (c) = d 0 -αc α + β ,
the quantity exchanged at equilibrium is

q f (c) = β α + β (d 0 -αc),
the profit of the supplier is

Π f (c) = β 2(α + β) 2 (d 0 -αc) 2 , the consumer surplus is CS f (c) = β 2 2α(α + β) 2 (d 0 -αc) 2 ,
and the total transaction fees transferred to the miner is

M f (c) = βc (α + β) (d 0 -αc).
Now, let us see what happens on the market for mining equipment. Any individual can buy some hashing power at a constant unit price k. If an individual has a hashing power h and the total network hashing power is H, he will earn, with a probability h/H, a reward R plus the transaction fees given above. Considering risk-neutral miners, each of them makes his decision regarding his hashing power demand as a solution to

max h h H (M f (c) + R) -kh.
There is only one possible equilibrium with

H = M f (c) + R k .
Moreover, the profit of the miners is null. Hence, the total surplus in this economy with a fixed transaction fee c is

T S f (c) = Π f (c) + CS f (c) = β(d 0 -αc) 2 2α(α + β) .
Bitcoin, as a payment network, requires a sufficiently high degree of security. The level of security is directly linked to the hashing power of the network ( [START_REF] Eyal | Majority is not enough: Bitcoin mining is vulnerable[END_REF], Kroll et al., 2013, Houy, 2014]). Hence, we require for the exchanges to be possible that H ≥ H > 0. In this case, for Bitcoin to be sustainable, we need:

M f (c) + R k ≥ H.
Then, for normative reasons, the choice of c should be made as a solution to

max c T S f (c) st: M f (c)+R k ≥ H c ≥ 0
If kH -R ≤ 0, obviously, we should have c = 0. Indeed, in this case, the reward is enough to give incentives for the miners to mine. Then, the distortive transaction fee can be set to 0. Otherwise, there exist no solution to this maximization problem if

kH -R > d 0 2 2 β α(α + β) .
Finally, when there is a non null solution, it is given by

c = d 0 -d 2 0 -4(kH -R) α(α+β) β 2α . (1)
A transaction fee higher than c implies a higher loss of efficiency that is not justified by an insufficient security level. A lower transaction fee implies that the minimal requirement in terms of security is not met, the miners having insufficient incentives.

Notice that this result suggests that the transaction cost should be depending on the reward R. When R is large enough, transaction fees can be insignificant. The optimal transaction is also increasing in the minimum level of security required. Indeed, the higher the level of security to be reached (H), the higher the incentive to mine should be.

With a maximum block size

Assume now that there is no fixed transaction fee but instead the number of transactions that can be included in a block is limited.5 Each individual can add to its transaction a fee for the miner. The amount of this fee can have any value. Then, this fee is fixed as a price on a market for space in blocks. Then, we need to consider the demand and supply for this space market. Supply is the inelastic fixed maximum number of transactions that can be included in a block and is denoted B. For this block size constraint to be weakly binding, we will assume that 0 ≤ B ≤ q * = β α+β d 0 . For a price p m for block space and a price of the physical good p, the demand for block space is given by the number of individuals who are willing to pay more than p + p m for the good, i.e.

Q d (p, p m ) = d 0 -α(p + p m ).
Equilibrium on the market for block space is reached when Q d (p, p m ) = B, hence, for space price

p m = d 0 -B α -p.
Then, when we combine both the market for block space and for physical good, we get a physical good equilibrium price

p b (B) = B β ,
and a physical good quantity exchanged at equilibrium

q b (B) = B,
the profit of the supplier is

Π b (B) = B 2 2β , the consumer surplus is CS b (B) = B 2 2α ,
and the total transaction fees transferred to the miner is

M b (B) = B α d 0 -B α + β β .
Notice that we have full equivalence between a fixed transaction fee c and a maximum block size B(c) whenever

B(c) = β α + β (d 0 -αc).
Indeed, it is straightforward to check that

p f (c) = p b (B(c)), q f (c) = q b (B(c)), CS f (c) = CS b (B(c)), Π f (c) = Π b (B(c)), M f (c) = M b (B(c)).
Hence, having a fixed transaction fee in the Bitcoin protocol and having a binding fixed maximum block size are equivalent. In the first case, the fee should be imposed in the Bitcoin protocol. In the second case, the fee will be set as an equilibrium price on a market for block space. Then, it is the maximum block size that is imposed by the protocol. And by making the maximum block size smaller than it could be, the artificially imposed scarcity of the total block space creates the reward for the miner.

Notice that if B = β α+β d 0 , i.e. the maximum block size is not constraining, we have p m = 0 which leads to a situation equivalent to a case with a fixed transaction fee c = 0. Then, the revenue of the miners is only the creation of new bitcoins. The security of the system is guaranteed only as long as R/k > H which cannot hold in the long-term, when R tends to 0, whenever H > 0. Then, making sure that the maximum block size is large enough to handle all transactions and in the same time, letting the transaction fee floating, will lead to a situation where transaction fees will go to 0 and hence, the security of Bitcoin cannot be guaranteed. Now, the optimal block size is given as B = B(c) where c is given by Equation 1:

B = β α + β   d 0 + d 2 0 -4(kH -R) α(α+β) β 2   .

The cost of Bitcoin

Let us finish by an estimation of the cost of Bitcoin as a payment system. We only consider the sum of private costs here. In particular, externalities due to the environmental damage caused by mining activities are not estimated. Assume that Bitcoin is run at its maximum efficiency point, i.e. with transaction fee, c, given in Equation 1. Then, the total surplus cost of Bitcoin is

T S * -T S f (c) = β 2(α + β) c(2d 0 -αc).
This evaluation is to be compared with the estimations of costs for other payment systems ([Schmiedel et al., 2012, Hayashi and[START_REF] Hayashi | [END_REF]) and is left for future studies.

Conclusion

In this article, we have led a simple study on the economics of Bitcoin transaction fees. In order to do that, we have used a very simple partial equilibrium setting with a market for a physical good with bitcoins used as means of exchange. As a specificity of our study, we introduced the market for mining equipment in order to check for the security of the system. Indeed, the security of Bitcoin depends on the computational power of miners. Finally, when we studied the possibility to have a limit on the block size, we introduced the market for block space.

As a result, we showed that having a mandatory transaction fee or limiting the block size and letting the price for block space be determined on a decentralized market are equivalent. However, letting the transaction fee be the outcome of a market and make the block size irrelevant or not binding would lead to a too low level of security for Bitcoin. The reason why the market is not an answer in this case is that mining has a private value different from its social one. Privately, it represents rewards for the miners. Socially, it allows the system to be secure. This is a perfect case of externality that implies the inefficiency of a free market. The transaction fee is a way to bring closer the private and social benefits of mining. So is the limited size of blocks that allows the miner to sell a scarce resource.

In the latter case a question arises: why don't miners limit the size of what they include in the blocks themselves? The reason is the following. When a miner tries to solve a block, the transactions are already in the network. In this case, he is a Stackelberg follower and he has an incentive to include all transactions in the limitless block he tries to solve. He could announce before solving a block that if he indeed solves a block, he will not include all transactions. But this announcement would not be credible. Indeed, since miners are in competition with distributed hashing power, the transactions a miner would not include would certainly be included just ten minutes later by another miner. Including all transactions whatever the fee attached is the only subgame perfect Nash equilibrium of the game we described.

Let us finish our study by mentioning its weaknesses. First, with a static setting we could not consider the dynamic aspects of mining. In addition to what we said above about the competition between miners, it is obvious that mining power purchase decisions are ones that are tightly linked with dynamic aspects. Indeed, buying hashing power is mainly buying a very expensive ASIC machine to be used at almost no marginal cost for a long time with total hashing power highly variable. Second, we did not consider the price volatility of Bitcoin that is huge. Third and related, we did not consider the inflation that should be linked with new bitcoins creation as rewards. Since we did not, it would be optimal in our setting to always have R sufficiently high in order to have a level of security high enough. However, this would create inflation and should have social costs that we cannot capture in our model. Fourth, we did not consider variable or unknown levels of demands and supply for the physical good. This could be problematic especially if we had to set a maximum block size. Finally, our model cannot capture ideas almost philosophical but that are very important for the community of people currently using

At the time this article is written, bitcoins can be bought and sold at about $630 apiece on some exchange markets. The monetary base is about $8,000,000,000.

At the time this article is written, there are about 12,500,000 bitcoins in circulation.

At the time this article is written, the transaction fees represent about 0.4% of the miners' rewards. The remainder is due to bitcoins creation.

In reality, the maximum size of a block is given in bytes. Transactions can have different sizes in bytes. The size of a transaction depends on many parameters (number of inputs and outputs mainly) but not directly on the amount paid in the transaction. We will make the simplifying assumption that each transaction has size 1.

v.0.1 Bitcoin like "Bitcoin neutrality": our study is limited to the economic aspects of Bitcoin transaction fees.


Mohammad Al Haj

Łukasz Paszkowski
email: lukasz.paszkowski@math.uni.wroc.pl

M Al Haj
email: al-hajm@cermics.enpc.fr

L Paszkowski

Convergence to walls of dislocations in the periodic case

Keywords: Dynamical system, Cauchy Lipschitz theorem, comparison principle, periodic solution, viscosity solutions

come

Convergence to walls of dislocations in the periodic case

Introduction

It is well known, in real materials with dislocations, that we can observe several accumulation of dislocations in walls of dislocations or more general in cells with several walls. In this paper our aim is to investigate the dynamics of dislocations that interact together and converge to such walls of dislocations.

Presenting the problem

Let us consider a model describing horizontal motion of dislocation lines parallel to the z-axis. Considering the cross section of these lines, we can reduce the problem to its two-dimensional counterpart where each dislocation line is represented by its position (x i (t), i) ∈ R × Z. Finally, such horizontal evolution can be characterized as follows

x ′ i = j =i f (x j -x i , j -i) for i ∈ Z. (1.1)
Here f : R × Z\{0} → R is an anisotropic force of two-body interactions. An example of such a force, according to [START_REF] Hajj | Derivation and study of dynamical models of dislocation densities[END_REF], is

f (x, y) = x(y 2 -x 2 ) (y 2 + x 2 ) 2 . (1.2)
An important aspect of interatomic interactions is that atoms can attract each other at longer distances and repel at short distances aggregating into various bulk forms. Such behaviour, of course, depends on the form of the considered potentials.

One of the forces describing both long-range attraction and short-range repulsion between atoms is the interaction force given by (1.2). In such an example two particles attract each other if the vertical angle between them is less then π 4 and, on the other hand, repel each other if the angle is greater then π 4 , see Figure 1 and Figure 2.

In the literature, however, there is a convention to express force in terms of energy potentials or commonly called interatomic potentials. Thus a general force acting on an atom can be seen as the negative derivative of some potential function with respect to its position: f (r) = -φ ′ (r).

The system of all particles acting together under the force defined in (1.2) can be rewritten in the following way

   d dt X(t) = F (X(t)), t > 0, X(0) = X 0 , (1.3)
where X(t) = (x i (t)) i∈Z , F (X) = (F i (X)) i∈Z and X 0 is some given initial position of dislocations. Moreover, F i (X) describes a resultant force acting on an i-th particle, i.e.

F i (X) def = j =i
f (x jx i , ji) for each i ∈ Z. Since our aim is to study a long time behaviour of the dynamics of particles which converges to walls of dislocations, the property of the force f described in (1.2) forces us to consider the problem (1.3) with a special condition for the initial data. Namely, we assume

f (x, y) = x(y 2 -x 2 ) (y 2 + x 2 ) 2 , (1.4a) X 0 ∈ Ω ∩ ℓ ∞ , (1.4b)
where

Ω = X : |x i -x j | 3 -2 √ 2 |i -j| . (1.5)
and ℓ ∞ = ℓ ∞ (R) is the Banach space of all bounded sequences over R supplemented with the norm

• ∞ = sup n∈Z |x n |. Remark 1.1 (Sign of f when X ∈ Ω). If X(t) = (x i (t)) i∈Z ∈ Ω, then, in particular, we have (j -i) 2 ≥ (x j -x i ) 2 . This implies that if X ∈ Ω, then (because of (1.2)) f (x j -x i , j -i) has the sign as x j -x i .
Notice here that arctan 3 -2 √ 2 = π 8 guarantees that the force f restricted to Ω is not only attractive but also non-decreasing with respect to the first variable. Therefore, we are able to prove a comparison principle, which helps us to conclude e.g. existence of global-in-time solutions, which stays in Ω.

Figure 2: A fixed particle x i attracts all other particles if they are placed in a region marked in blue and pink. However, the force f is non-decreasing only if the particles are located in the region marked in pink. Such domain we call Ω i and thus we can present Ω, defined in (1.5), as Ω = ∩ i∈Z Ω i .

Main results

Our first result deals with the existence of solutions to the considered problem. More precisely, it reads as follows Theorem 1.2 (Existence of a unique solution). Let (1.4) hold. Then there exists a unique solution X ∈ C 1 ([0, +∞), Ω ∩ ℓ ∞ ) of the Cauchy problem (1.3). Moreover, if the initial data X 0 is N -periodic (i.e. x 0 i = x 0 i+N , for every i ∈ Z), then the solution remains N -periodic for every time t > 0.

The proof of the theorem consists in the application of the classical Cauchy-Lipschitz theorem and the comparison principle result. Notice that in general the locally Lipschitz condition with respect to the first variable of the function f is sufficient to obtain a unique local-in-time solution. In order to extend it to the global-in-time one we need to provide an apriori estimate, e.g. the comparison principle that ensures us that ℓ ∞ -norm of the solution does not blow up.

However, if the function f satisfies the Lipschitz condition globally, which happens when f is defined by (1.4a), we immediately obtain a unique globalin-time solution by extending it with the universal step T > 0, see [START_REF] Brézis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Thm 7.3,p. 184]. Thus, in that case the comparison principle is needed only to ensure that the solution belongs to Ω for all times t > 0.

To prove the comparison principle for the problem (1.3), the monotonicity of a function f is a necessary assumption. Hence, the reason why we consider the initial condition in the special domain Ω is that the function f defined by (1.4a) is indeed monotone over that set.

Our second result is the long time behaviour of the dynamics of particles where we prove that dislocations accumulate creating so-called walls of dislocations. This result can be stated in the following way For the proof of the above theorem we refer to Section 4, and Section 6 for numerical experiments which show the convergence and more information.

We have also proved the following ℓ p contraction for periodic solutions:

Proposition 1.4 (l p contraction). Let X and Y be two N -periodic solutions of the problem (1.3)-(1.4) with N -periodic initial data X 0 and Y 0 respectively. Then the following estimate

X(t) -Y (t) p X 0 -Y 0 p
, for all t > 0 holds true provided p 2.

Related results

Another possible model, first proposed in 1924 and repeatedly improved in subsequent years, involves the Lennard-Jones potential [START_REF] Lennard-Jones | On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature[END_REF]

φ(r) = 4ε r σ 0 -12 - r σ 0 -6
, where r is a distance between two atoms, ε is the depth (minimum) of the energy and σ 0 is the finite distance at which the interparticle potential is zero. Due to its computational simplicity and relatively good approximations, the Lennard-Jones potential is extensively used to describe the properties of gases and in computer simulations [START_REF] Lennard-Jones | On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature[END_REF][START_REF] Lennard-Jones | On the Determination of Molecular Fields. II. From the Equation of State of a Gas[END_REF].

There is no necessity to deal only with two-body potentials. One approach to represent the many-body potentials energy is to consider it as a sum of two-body, three-body, . . ., N -body terms. An example of such constructed energy potential is the Stillinger-Weber potential [START_REF] Stillinger | Computer simulation of local order in condensed phases of silicon[END_REF] for semiconductor silicon containing only two-and three-body terms.

More facts about dislocations, examples of potentials used in various models, and numerical simulations performed on these models can be found in the book of Bulatov and Cai [START_REF] Bulatov | Computer simulations of dislocations, Oxford Series on Materials Modelling[END_REF].

A similar model to ours, where a finite number of dislocations of different types occur (for instance positive and negative ones), was considered by El Hajj, Ibrahim and Monneau [START_REF] Hajj | Derivation and study of dynamical models of dislocation densities[END_REF]. The authors studied horizontal motion of dislocation lines and they derived formally a two-dimensional mean field model called Groma-Balogh model. In the same paper, they also investigated a model with additional boundary conditions. They observed that positive dislocations move to the right, whereas the negative ones move to the left. In particular, numerical simulations of deformations of a slab under an external shear stress have been performed.

Related models called individual cell-based models occur not only in the theory of dislocations but also in the study of e.g. chemotherapy where x i denotes a center of a tumour cell [START_REF] Drasdo | Individual-based approaches to birth and death in avascular tumors[END_REF], chemotaxis [START_REF] Stevens | A stochastic cellular automaton modeling gliding and aggregation of myxobacteria[END_REF] and many others. Moreover, particles may also evolve according to stochastic differential equations, see [START_REF] Bodnar | Derivation of macroscopic equations for individual cell-based models: a formal approach[END_REF] and references therein for numerical simulations.

Comparison principle

In this section, we prove a comparison principle result for a general system of equations:

d dt X(t) = G(X(t)), t > 0, (2.1)
where

X(t) = (x i (t)) i∈Z , G(X) = (G i (X)) i∈Z with G i (X) def = j =i g(x j - x i , j -i)
for each i ∈ Z, and g : R × Z\{0} → R is C 1 and globally nondecreasing with respect to first variable. We will apply this later in Section 3.

Lemma 2.1 (Comparison principle). Let T > 0 and assume that

X, Y ∈ C 1 ([0, T ), ℓ ∞ ) be two solutions of (2.1) with X(0) = X 0 and Y (0) = Y 0 . Assume that X 0 Y 0 , then X(t) Y (t) for every t ∈ [0, T ).
Proof. Notice that the assumption X 0 = X(0) Y (0) = Y 0 reads as x 0 n := x n (0) y n (0) =: y 0 n for every n ∈ Z, and we shall prove that x n (t) y n (t) for every n ∈ Z and t ∈ [0, T ).

Define then new functions Z(t) = (z n (t)) n∈Z and M (t) as

z n (t) = x n (t) -y n (t), M (t) = sup n∈Z z n (t). (2.2) Since X(t) , Y (t) ∈ ℓ ∞ for all t ∈ [0, T ), then from the definition of M, we have ∀ t * ∈[0,T ) ∃ n * (t * ) M (t * ) = z n * (t * ) (t * ), (2.3)
where n * (t * ) may not be necessarily finite. Our goal is to show that

M (t) 0 for all times t ∈ [0, T ). (2.4)
The way to prove (2.4) is to show that for all t ∈ [0, T ), we have M ′ (t) ≤ 0 in the viscosity sense. Then by a comparison principle or the Gronwall inequality we deduce (2.4).

Step 1: n * (t * ) ∈ Z. Let t * ∈ [0, T ) and consider a test function φ such that

M (t) φ(t), M (t * ) = φ(t * ).
Then M ′ (t * ) ≤ 0 in the viscosity sense if φ ′ (t * ) ≤ 0, see [4, Definition 2.2] for a definition of viscosity solutions.

From (2.2) and (2.3), we have

z n * (t) (t) M (t) φ(t), z n * (t * ) (t * ) = M (t * ) = φ(t * ), (2.5) thus φ ′ (t) = d dt z n * (t) (t) at t = t * , since z n * (t) , φ are sufficiently smooth (X, Y ∈ C 1 ([0, T ), ℓ ∞ )). If φ ′ (t) = d dt z n * (t) (t) ≤ 0 at t = t *
, then we have M ′ (t * ) ≤ 0 in the viscosity sense. Thus using the Gronwall inequality (which in the viscosity solutions framework is nothing else but the comparison principle), we deduce

M (t) ≤ M (0). But z n (0) = x n (0) -y n (0) ≤ 0 for all n ∈ Z, thus M (t) ≤ M (0) ≤ 0.
Therefore our goal now is to show that indeed d dt z n * (t) (t) 0 at t = t * . Set n * = n * (t * ), using the Taylor expansion of the function G, we have

dz n * (t) dt = dx n * (t) dt - dy n * (t) dt = G n * (X(t)) -G n * (Y (t)) = m∈Z ∂ m G n * (Θ(t))(x m (t) -y m (t)),
where

Θ(t) = αX(t) + (1 -α)Y (t) for some α ∈ (0, 1). Here ∂ m G n (X) is to be understood as ∂ m G n (X) := dG n (X) dx m .
In particular for t = t * , we obtain

dz n * (t * ) dt = ∂ n * G n * (Θ(t * ))z n * (t * ) + m∈Z m =n * ∂ m G n * (Θ(t * ))(x m (t * ) -y m (t * )) ∂ n * G n * (Θ(t * ))z n * (t * ) + z n * (t * ) m∈Z m =n * ∂ m G n * (Θ(t * )) = z n * (t * ) m∈Z ∂ m G n * (Θ(t * )) = 0.
(2.6)

The inequality in the middle line and the last equality in the above computations can be justified as follow. First, we notice that for every m = n * and by the assumption on monotonicity of the function g, we have

∂ m G n * (Θ(t * )) = g x (Θ m (t) -Θ n * (t), m -n * )α 0.
Here g x denotes the partial derivative of g = g(x, y) with respect to the first variable x.

Second, we can calculate explicitly

∂ n * G n * (Θ(t * ))
. Namely, by the structure of the function G, we get

∂ n * G n * (Θ(t * )) = - m =n * g x (Θ m (t) -Θ n * (t), m -n * )α.
Summing up all the derivatives of G, we arrive at the last equality of (2.6).

Step 2: n * (t * ) = +∞. Then there exists a subsequence n k such that

M (t * ) = sup n∈Z z n (t * ) = lim k→+∞ z n k (t * ).
(2.7)

Let us redefine the sequences up to shift the indices, we have

     x k n (t) = x n+n k (t) → x ∞ n y k n (t) = y n+n k (t) → y ∞ n z k n (t) = z n+n k (t) → z ∞ n = x ∞ n -y ∞ n as k → +∞.
The convergence of the sequences takes place up to subsequence of k, since x k n , y k n and z k n are bounded. Moreover, we have from (2.7) that

M (t * ) = lim k→+∞ z n k (t * ) = lim k→+∞ z k 0 (t * ) = z ∞ 0 (t * ) ≤ sup n∈Z z ∞ n (t * )
and

M (t * ) = sup n∈Z z n (t * ) ≥ z n+n k (t * ) = z k n (t * ) for all n ∈ Z, i.e. M (t * ) ≥ z ∞ n (t * ), and hence M (t * ) ≥ sup n∈Z z ∞ n (t * ). Therefore, M (t * ) = z ∞ 0 (t * ) = sup n∈Z z ∞ n (t * ),
and hence n * = 0. In addition, we have

z ∞ n (0) = x ∞ n (0) -y ∞ n (0) ≤ 0.
Thus, applying the result of Step 1 for z ∞ n (t), we prove the desired result.

Existence and uniqueness of solution

We give, in this section, the proof of Theorem (1.2) which combines the classical Cauchy-Lipschitz theorem [2, Thm 7.3, p. 184] and the comparison principle, Lemma 2.1.

Proof of Theorem 1.2. In the proof we argue in several steps.

Step 0: Properties of the function f . Consider the function f be defined in (1.4a). Clearly, we have f (•, y) ∈ C ∞ (R) and f (±∞, y) = ∓0 for every y ∈ Z\{0} fixed. Moreover, f (•, y) is antisymmetric and there exists

x y = 3 - √ 2|y| such that f (x y , y) = max x∈R f (x, y), f (-x y , y) = min x∈R f (x, y)
and f (•, y) is non-decreasing over [-x y , x y ], see for instance Figure 1. Moreover, we see that for fixed y ∈ Z\{0}

d dx f (x, y) d dx f (0, y) = 1 y 2 . (3.1)
Hence, f is globally Lipschitz continuous over R with 1 y 2 Lipschitz constant depending on fixed y.

Step 1: Existence of a unique global solution for

(1.3). Let X = (x i ) i∈Z , Y = (y i ) i∈Z ∈ ℓ ∞ . Using (3.1), we have F (X) -F (Y ) ℓ ∞ = max i∈Z |F i (X) -F i (Y )| = max i∈Z j =i f (x j -x i , j -i) -f (y j -y i , j -i) max i∈Z j =i |f (x j -x i , j -i) -f (y j -y i , j -i)| max i∈Z j =i 1 (j -i) 2 |x j -y j | + |x i -y i | 4 X -Y ℓ ∞ +∞ k=1 1 k 2 . Thus F (X) -F (Y ) ℓ ∞ ≤ 2 3 π 2 X -Y ℓ ∞ . (3.2)
Therefore, using the classical Cauchy-Lipschitz theorem [2, Thm 7.3, p. 184], there exists a unique solution X ∈ C 1 ([0, +∞), ℓ ∞ ) of (1.3).

Step 2: Invariance: X(t) ∈ Ω for every t ≥ 0. In this step we show that if X(0) = X 0 ∈ Ω, then the solution X(t) given in

Step 1 satisfies X(t) ∈ Ω for every t ≥ 0.

Step 2.1: Variant system of (1.3). For y ∈ Z\{0} fixed, define a new function f = f (x, y) as follows

     f (x, y) = f (x, y) for x ∈ [-x y , x y ], f (x, y) = f (x y , y)
for all x x y , f (x, y) = f (-x y , y) for all x -x y .

(3.3)
Clearly, f is Lipschitz and non-decreasing with respect to the first variable over the whole space. Moreover, since d dx f (±x y , y) = 0 for fixed y = 0, then f is C 1 with respect to the first variable on R.

Then we consider the following system

   d dt X(t) = F ( X(t)) t 0, X(0) = X 0 ∈ Ω ∩ ℓ ∞ , (3.4)
where again X(t) = (x i (t)) i∈Z and F ( X) = ( Fi ( X)) i∈Z , with Fi ( X(t)) :=

j =i f (x j -xi , j -i). (3.5)
Similarly to Step 1 we show for every X = (

x i ) i∈Z , Ỹ = (ỹ i ) i∈Z , that F ( X) -F ( Ỹ ) ℓ ∞ ≤ 2 3 π 2 X -Ỹ ℓ ∞ .
Therefore, using the classical Cauchy-Lipschitz theorem [2, Thm 7.3, p. 184], there exists a unique solution X ∈ C 1 ([0, +∞), ℓ ∞ ) of the variant problem (3.4).

Step 2.2: X(t) ∈ Ω for every t ≥ 0. We have X(0) = X 0 ∈ Ω, i.e.

-3 -2 √ 2|i -j| x 0 i -x 0 j 3 -2 √ 2|i -j|, ∀i, j ∈ Z.
Setting m = ij, we obtain

x m i (0) := x 0 i-m -3 -2 √ 2|m| x 0 i x 0 i-m + 3 -2 √ 2|m| =: x m i (0), for every i, m ∈ Z.
Moreover, from the definition of the function F , see (3.5), it is clear that the problem (3.4) is invariant by translations. Hence,

X m = xi-m (t) -3 -2 √ 2|m| =: x m i i∈Z
and

X m = xi-m (t) + 3 -2 √ 2|m| =: x m i i∈Z
are two solutions of (3.4) for each m ∈ Z. Now, since f is non-decreasing, we can apply the comparison principle, Lemma 2.1 (with T = +∞), and deduce that for every i, m ∈ Z

x m i (t) xi (t) x m i (t) for t > 0.
Hence, X(t) ∈ Ω for all t > 0.

Step 2.3: X solves (1.3).

We have that X(t) ∈ Ω for all t > 0. Thus

-x y = -3 -2 √ 2|i -j| xi -xj 3 -2 √ 2|i -j| = x y , ∀i, j ∈ Z. However, f (•, i -j) | [-xy ,xy ] = f (•, i -j) | [-xy ,xy ] , hence F ( X(t)) = F ( X(t)) over t > 0.
Therefore, X solves (1.3).

Step 2.4: Conclusion: X(t) ∈ Ω for every t ≥ 0. Since X(0) = X 0 = X(0), then by the uniqueness of the solution of system (1.3) (See Step 1), we get X(t) = X(t) ∈ Ω for every t ≥ 0.

Thereupon, we have proved that X(t) is the unique global solution of the problem (1.3)-(1.4).

Step 3: X is periodic. Assume that X(0) = X 0 is N -periodic; i.e. x 0 i = x 0 i+N for every i ∈ Z. Define Y = (y i ) i∈Z = (x i+N ) i∈Z , where we recall that X = (x i ) i∈Z . Then X and Y are two solutions of (1.3) with X(t) , Y (t) ∈ Ω for all t ≥ 0. Moreover, we have Y (0) = (x 0 i+N ) i∈Z = X 0 (Y (0) ≤ X 0 and X 0 ≤ Y (0)). Since f is non-decreasing over Ω, then using the comparison principle (Lemma 2.1 with T = +∞), we deduce that Y (t) = X(t) for every t ≥ 0, i.e x i = x i+N for every t ≥ 0.

Convergence to flat walls

The aim of this section is to prove that under the periodicity assumption imposed on the initial data, the solution constructed in Theorem 1.2 converges to a special stationary solution of the problem (1.3). Precisely, in this section, we prove Theorem 1.3.

Proof of Theorem 1.3.

Step 0: preliminary (reformulation of

(1.3)). Let X = (x i ) i∈Z ∈ C 1 ([0, ∞), Ω ∩ ℓ ∞ ) be a N -periodic (i.e. x i+N = x i ) solution of (1.3). For i = 1, ..., N we have d dt x i (t) = j =i f (x j -x i , j -i) = N j=1 j =i k∈Z f (x j+kN -x i , j -i + kN ).
Using the periodicity of X (x j+kN = x j ), we get that

d dt x i (t) = N j=1 j =i k∈Z f (x j -x i , j -i + kN ).
Hence, we can transform (1.1) into the following equation

d dt x i (t) = N j=1 j =i g(x j -x i , j -i), i = 1, . . . , N, (4.1)
where g(x, y) = k∈Z f (x, y + kN ). Moreover, since for every y = 0, the map

x → f (x, y) is Lipschitz with 1 y 2 Lipschitz constant (see Step 0, in the proof of Theorem 1.2), g(0, y) = f (0, y) = 0 and x ∈ ℓ ∞ , then |g(x, y)| = k∈Z f (x, y + kN ) -f (0, y + kN ) ≤ k∈Z 1 (y + kN ) 2 |x| ≤ M (4.
2) for some M > 0. Hence, g is uniformly bounded in x.

In order to prove the convergence of the solution, we set

M (t) = 1 2 N i=1 x 2 i (t) (4.3)
and we argue by steps.

Step 1: M is non-increasing. Indeed, we have

1 2 d dt N i=1 x 2 i (t) = N i=1 N j=1 j =i x i (t) g(x j (t) -x i (t), j -i) = N i=1 N j=i+1 x i (t) g(x j (t) -x i (t), j -i) + N i=1 i-1 j=1 x i (t) g(x j (t) -x i (t), j -i) = N i=1 N -i k=1 x i (t) g(x i+k (t) -x i (t), k) - N j=1 N i=j+1 x i (t) g(x i (t) -x j (t), i -j) = N i=1 N -i k=1 x i (t) g(x i+k (t) -x i (t), k) - N j=1 N -j k=1 x j+k (t) g(x j+k (t) -x j (t), k) = N i=1 N -i k=1 (x i (t) -x i+k (t)) g(x i+k (t) -x i (t), k) 0.
First, let us mention that due to the fact that the function f = f (x, y), defined by (1.4a) is symmetric in y and antisymmetric in x, the function g = g(x, y) possesses such property as well. Moreover, as a result of the boundedness of the function g(•, y) (which comes form the Lipschitz condition of f (•, y)) and the fact that only finite sums are considered, we are allowed to use Fubini's theorem and change the order of summation. These facts justify the third equality. The inequality is obtained by the fact that each single expression under the sums is nonpositive due to the definitions of the functions g, f and the fact that X(t) ∈ Ω ∩ ℓ ∞ for t 0 (see Remark 1.1).

Finally, we conclude that M (t) → M 0 as t → ∞ since M (t) is nonnegative and non-increasing.

Step 2: limit of X(t) as t → +∞. Let us define X n (t) := X(t+n). Then X n is a solution of (1.3). Since f (x, y) is Lipschitz with 1 y 2 Lipschitz constant (independent if x), hence d dt X n (t) ∈ ℓ ∞ uniformly in n. Using Ascoli's theorem, up to some subsequence, X n (t) → X ∞ (t) as n → ∞ for every t > 0. Thus, we can write

M 0 = lim n→∞ M (t + n) = lim n→∞ 1 2 N i=1 x n i (t) 2 = 1 2 N i=1 x ∞ i (t) 2 . (4.4)
Since X n is a solution of (4.1) and X n (t) ∈ Ω ∩ ℓ ∞ , then the limit X ∞ is a classical solution and X ∞ (t) ∈ Ω ∩ ℓ ∞ . Therefore, repeating all the computations performed in Step 1 for X ∞ , we arrive at

0 = d dt M 0 = d dt 1 2 N i=1 x ∞ i (t) 2 = N i=1 N -i k=1 (x ∞ i (t) -x ∞ i+k (t)) g(x ∞ i+k (t) -x ∞ i (t), k).
Since the solution lives in Ω,

g(x ∞ i+k (t) -x ∞ i (t), k) and x ∞ i+k (t) -x ∞ i (t) have the same sign (see Remark 1.1), then (x ∞ i (t) -x ∞ i+k (t)) g(x ∞ i+k (t) -x ∞ i (t), k) ≤ 0
for all i ∈ {1, .., N -1} and k ∈ {1, ..., N -i}. Thus, either

x ∞ i (t) = x ∞ j (t) for all i = 1, . . . , N -1 and j = i + 1, . . . , N, (4.5)
or we have g = 0. However, since X ∞ (t) ∈ Ω ∩ ℓ ∞ , then g = 0 immediately implies (4.5) (see (1.4a) and the definition of g). Therefore, we get from (4.5) that

x ∞ 1 (t) = x ∞ 2 (t) = • • • = x ∞ N (t).
Next, we plug X ∞ into the equation (4.1) to see that indeed d dt x ∞ i (t) = 0 (since g(0, y) = 0); thus, x ∞ i (t) = x ∞ i (0) = c, for all i = 1, . . . , N , and for some c ∈ R. Moreover, we can write the explicit value of M 0 , i.e.

M 0 = 1 2 N c 2 .
Now take another convergent subsequence, X m (t) of X(t) such that X m (t) → X ∞ (t) as m → ∞. Repeating all the calculations performed for the sequence X n , we may show that x ∞ i (t) = x ∞ i (0) = b for all i = 1, . . . , N , t 0 and some b ∈ R. As before we conclude that

M 0 = 1 2 N b 2 .
Thus, b = c, because we may assume, without loss of generality, that b, c 0, since the problem (4.1) is invariant by translations and the initial data can be shifted to be positive. This implies that the accumulation point of X is unique. Hence, x i (t) → c as t → ∞ for all i = 1, . . . , N .

Step 3: identification of the limit. In this step we prove that the barycenter is preserved in time, i.e. x i (0) for all t > 0, which allows us to determine the value of the constant c.

We have d dt

N i=1 x i (t) = N i=1 N j=1 j =i g(x j -x i , j -i). But N i=1 N j=1 j =i g(x j -x i , j -i) = N j=1 N i=1 i =j g(x j -x i , j -i) = N i=1 N j=1 j =i g(x i -x j , i -j),
where we have changed the order of summation in the first equality (this is possible because g is a bounded function), and we replaced i and j in the second equality. Moreover, since g(x, y) is antisymmetric w.r.t. x and symmetric w.r.t. y (because f antisymmetric and symmetric w.r.t. x and y respectively), then we get

N i=1 N j=1 j =i g(x j -x i , j -i) = N i=1 N j=1 j =i g(x i -x j , i -j) = - N i=1 N j=1 j =i g(x j -x i , j -i).
Therefore, we deduce that

d dt N i=1 x i (t) = N i=1 N j=1 j =i g(x j -x i , j -i) = 0, and hence N i=1 x i (t) = N i=1 x i (0). Finally, since x(t) → c as t → ∞, we conclude that N i=1 x i (0) = lim t→∞ N i=1 x i (t) = N c, i.e. c = 1 N N i=1 x i (0) (4.6)
thus, we have proved the desired result. . condition on x. Suppose now that I 2 0. This immediately implies, in view of the above information, that x i+k (t)y i+k (t) x i (t)y i (t); hence, I 1 0. On the contrary, if I 2 0, then I 1 0. Hence, we conclude

d dt 1 p X(t) -Y (t) p p 0,
which completes the proof.

Corollary 5.1 (l p contraction for a rescaling of x i ). Let p 2. Fix ε > 0 and let us define new variables in the following way

x i (t) = 1 ε u ε (εi, εt), ∀i ∈ Z.
Then the above theorem reads as

u ε (•, τ ) -v ε (•, τ ) p u ε 0 -v ε 0 p .

Numerical experiments

Here we present results of some numerical experiments to confirm the results obtained in Theorem 1.3. We construct an adaptive scheme as follows. Let N > 0 denote the total number of interacting particles. Let ∆t denote a time-step and let us define an approximate solution of (1.3) by a solution X n = (X n 1 , . . . , X n N ) of the following forward Euler scheme

X n+1 = X n + ∆tF (X n ) def = S(X n ). ( 6.1)
Lemma 6.1 (Monotonicity of the scheme). The scheme derived in (6.1) is monotone if and only if the time-step satisfies ∆t 3 π 2 and the initial data X 0 ∈ Ω defined in (1.5).

Proof. To prove the monotonicity it is enough to show that ∂ j S i (X n ) 0 for all i, j = 1, . . . , N . First, we notice that due to Lemma 2.1 we get X n ∈ Ω for all n ∈ N.

Step 1:

j = i. ∂ j F i (X n ) = ∆t f x (X n j -X n i , j -i) 0, (6.2)
since considered function f is nondecreasing with respect to the first variable.

Step 2: j = i.

∂ i F i (X n ) = 1 -∆t N j=1 j =i f x (X n j -X n i , j -i) 1 -∆t N j=1 j =i f x (0, j -i) 1 -∆t j∈Z j =i 1 (j -i) 2 1 -∆t π 2 3 0.
To justify the first inequality we use the properties of the function f described in the proof of Theorem 1.2 Step 1. Moreover, we extend the finite sum by its infinite version and we conclude its nonnegativity. In our numerical experiments we assume the initial data X 0 ∈ Ω ∩ ℓ ∞ which is denoted by "x" on the left-upper plot in Figure 3. Furthermore, in every picture, by "*" we emphasised what the limit solution (by Theorem 1.3 the limit solution is at the barycenter of initial data) is. In Figure 3 we observe the evolution of dislocations which eventually converge.

However, we may also consider the initial data X 0 ∈ Ω where

Ω = X : 3 -2 √ 2 |i -j| < |x i -x j | < |i -j| , (6.3)
see the blue region in Figure 2. It is worth noticing that for such initial data, the force acting on dislocations is still attractive; however, we do not have a comparison principle and we cannot guarantee that solution stays in Ω, but we can perform numerical experiments to see what happen with dislocations. In the above pictures we can see that even small perturbation of initial data produces completely different solutions. The only one (right-lower plot in Figure 4) converged to a flat wall, while the remainder does not.

Figure 1 :

 1 Figure 1: Interaction force f (x, y) as a function of the distance between two atoms for some fixed y ∈ Z\{0} with the property f (-x, y) = -f (x, y). A vertical angle between two particles corresponds to arctan( x y ). Thus π 4

Theorem 1 . 3 (

 13 Convergence to flat walls). Let X be the N-periodic solution of the problem (1.3)-(1.4). Then it converges to a constant stationary solution of the problem (1.3)-(1.4) i.e. for every i ∈ Z, we have lim t→∞ x i (t) = c, where c = 1 N N i=1 x 0 i is the barycenter of the initial data.

Figure 3 :

 3 Figure 3: Evolution of dislocations of (1.3) with initial data X 0 ∈ Ω.

Figure 4 :

 4 Figure 4: Evolution of dislocations of (1.3) with initial data X 0 ∈ Ω. Each simulation starts with different initial data.

Acknowledgments. The authors would like to thank Professor R. Monneau for fruitful discussions in the preparation of this paper. The first author would like to thank the Lebanese National Council for Scientific Research (CNRS-L) and the Campus France (EGIDE earlier) for supporting him. The first author was partially supported by ANR HJNet (ANR-12-BS01-0008-01) and by ANR-12-BLAN-WKBHJ: Weak KAM beyond Hamilton-Jacobi. The second author was supported by the Polish Ministry of Science (MNSzW) grant N201 418839, and the Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007-2013 funded by European Regional Development Fund (Ph.D. Programme: Mathematical Methods in Natural Sciences).

From micro to macro model

We show in this section the ℓ P contraction estimate of periodic solutions of (1.3)-(1.4), namely we give the proof of Proposition 1.4.

Proof of Proposition 1.4. Let X = (x i ) i∈Z and Y = (y i ) i∈Z be a N -periodic (i.e.

We proceed as in Section 4, Step 1. First, without loss of generality, we may transform (1.1) into the following equation

with the function g(x, y) = k∈Z f (x, y + kN ) uniformly bounded in x. Thus, we calculate

x i (t)-y i (t) p-2 x i (t)-y i (t) g(x j (t)-x i (t), j-i)-g(y j (t)-y i (t), j-i)

Let us mention here that due to the fact that the function f = f (x, y), defined by (1.4a), is symmetric in y and antisymmetric in x, the function g = g(x, y) possesses such property as well. Moreover, as a result of the boundedness of the function g(•, y) (which comes form the Lipschitz condition of f (•, y)) and the fact that only finite sums are considered, we are allowed to use Fubini's theorem and change the order of summation. These facts justify the third equality.

Furthermore, we notice that for fixed y ∈ Z the function f (x, y) is nondecreasing in the variable x provided |x| 3 -2 √ 2|y|. Hence, by definition the function g(x, y) is also nondecreasing in the variable x under the same