Djamal Ait-Akli

EXISTENCE RESULT FOR A MODEL COUPLING A QUASI-LINEAR PARABOLIC EQUATION AND A LINEAR HYPERBOLIC SYSTEM

Keywords: 2010 Mathematics Subject Classification. 35A01, 35K20, 35L20, 35L53 Hyperbolic system, Stokes equation, Fluid-Structure Interaction, Inverse energy estimate, Fixed point theory. 1

We prove globally-in-time existence of solution for a problem coupling the linear Lamé system and the quasi-linear Stokes equation. A solution of this global coupled problem is viewed as the fixed point of some non-linear operator T . We construct, using a regularization procedure, a sequence (T ) of auxiliary approximating compact operators. Then we establish, using a combination of Banach and Schaeffer fixed point theorems, the existence of fixed points to every operator T . Finally we prove that these fixed points converge to the fixed point of T .

Introduction

We address the issue of existence of solution for the coupled system which reads:

                                 ∂ t v f -νdiva(t, ∇v f ) -ν∆v f + ∇π = F in Ω T f , div v f = 0 in Ω T f , v f = 0 on (0, T ) × (∂Ω f -Σ), v f (0, .) = v 0 f in Ω f , t 0 v f (r)dr = u s (t), (S(v f , π) + a(t, ∇v f )) • - → n = σ(u s ) • - → n on Σ T ,
∂ tt u s -div σ(u s ) = 0 in Ω T s , u s = 0 on (0, T ) × (∂Ω s -Σ), u s (0, .) = 0, ∂ t u s (0, .) = 0 in Ω s .

Let us describe the components and the notations in system (1.1)-(1.7). We let T > 0 to be a positive real number. We have denoted ω T := (0, T ) × ω. The function u s denotes the displacement of the solid structure, the function v f denotes the velocity of the fluid and π denotes its pressure. System (1.1)-(1.7) is formed out of:

• The parabolic Stokes equation with a quasi-linear diffusion term that describes the motion of a fluid inside a fluid domain Ω f ⊂ R 2 , cf. equations (1.1)-(1.3). • The coupling condition, cf. equation (1.4).

• The second order linear hyperbolic Lamé system that describes the deformation, due to interaction with the fluid, of a structure occupying a solid domain Ω s ⊂ R 2 , cf. equations (1.5)-(1.7). We assume that Ω := Ω s ∪ Ω f (t) ⊂ R 2 , for all t ∈ (0, T ), that is the global domain Ω doesn't vary in time during the interaction. Moreover, the boundaries ∂Ω f resp. ∂Ω s of the fluid resp. the structure domains are both assumed to meet the minimal C 2 -regularity.

The solid domain Ω s and the fluid domain Ω f (t) share a common part of their respective boundaries, this contact interface is denoted Σ i.e. ∂Ω s ∩ ∂Ω f (t) = Σ(t), the variation in time of Σ isn't taken into account, cf. the conclusion at the end of this paper, we assume furthermore that Σ is connected. The coupling condition, relating the fluid and solid problems, is prescribed on (0, T ) × Σ. It consists of imposing, on the contact interface Σ, the two equalities:

t 0 v f (r, x)dr = u s (t, x) (S(v f , π) + a(t, ∇v f )) • - → n = σ(u s ) • - → n , (t, x) ∈ Σ T , (1.8)
where -→ n denotes the exterior unit normal defined at each point of Σ. The term S(v f , π) denotes the Cauchy stress tensor:

S(v f , π) := -πI + 2νε(v f ), (1.9)
and the function a : [0, T ] × R 2 → R 2 such that a := (a 1 , a 2 ) satisfy (3.2) and other assumptions that will be precised in section 3. On the other hand, σ(u s ) • -→ n denotes the normal component of the stress tensor: σ(u) = 2µε(u) + λ Trε(u) Id, with ε(u) = 1 2 ∇u + ∇ t u .

(1. [START_REF] Lasiecka | Nonhomogeneous boundary value problem for second order hyperbolic operators[END_REF] Unlike what is usually done regarding the coupling condition, we imposed the equality of the displacements functions, u s and u f , on the contact interface instead of the equality of the velocities. The condition (1.8) is completed by an homogeneous Dirichlet data (1.2), (1.6) on the remaining part of the boundaries of the domain and by initial time conditions (1.3) and (1.7). The fluid is assumed to be divergence-free. We emphasize that, except for the fluid initial condition, (1.3), the restriction to homogeneous data for both Stokes and Lamé problems are adopted only for the sake of simplicity of presentation, one can refer to [START_REF] Raymond | A fluid-structure model coupling the Navier-Stokes equations and the Lamé system[END_REF] and the references therein for the case where non homogeneous data are considered but with restrictive assumption on the geometry of the domain, namely the flatness of the contact interface.

It is worth noting that this kind of coupled system of PDEs has proven efficient for modeling various Fluid-Structure interactions (FSI) phenomenon. The main result of the present work is stated in the following theorem:

Theorem 1.1. ∀T > 0, ∀F ∈ L 2 (0, T ; H -1 (Ω f )) and ∀v 0 f ∈ L 2 (Ω f ), the coupled problem (1.1)-(1.7
) admits at least one solution:

(v f , π, u s ) ∈L 2 0, T, H 1 (Ω f ) ∩ H 1 (0, T ; H -1 (Ω f ) × L 2 0, T ; L 2 (Ω f )/R × H 1 ((0, T ) × Ω s ) ∩ H 2 (0, T ; H -1 (Ω s )) .
(1.11)

The focus is on establishing existence of globally-in-time solution to the coupled problem (1.1)-(1.7) which is is analogous to problem [5, Problem 2.10, p.556] The interest in the regularity of the solutions is secondary in our current considerations.

The novelty in this work is about two main things. The first one consists in proving the globally-in-time existence of at least one solution to the coupled problem while dealing with the non-linearity in the Stokes equation. The second new thing consists in considering domains with arbitrary geometry, that is no particular restriction is assumed on the contact surface Σ, regarding namely its geometric flatness, this extends the result proved by the authors in [START_REF] Raymond | A fluid-structure model coupling the Navier-Stokes equations and the Lamé system[END_REF]. To achieve that, we view the solution of (1.1)-(1.7) as a fixed point of some non-linear operator T and we use a regularization method in order to apply fixed point theory.

Organization of the paper. In the second section we establish a well-posedness result for the Dirichlet problem associated to the Lamé operator along with an inverse estimate. In the third section we derive an energy estimate for the quasilinear Stokes system and we introduce the operator T : X → X whose fixed point is a solution of problem (1.1)-(1.7), the space X is given by (2.6). Next we apply the regularization method to construct a sequence of auxiliary compact operators (T ) such that T = lim →0 T , then we establish the boundedness and compactness of T using the preceding estimates. In the fourth section, we prove the existence of a fixed point u to T by combining Banach and Schaeffer fixed point theorems.

Finally we conclude by showing that the fixed points u converge to a fixed point u 0 of T .

2.

Inverse estimate for the Lamé system.

Throughout this section, we let Ω s ⊂ R 2 to be a bounded planar domain with boundary ∂Ω s ∈ C 2 . We consider the auxiliary Dirichlet problem (2.2) associated to the time dependent second order Lamé operator H given by:

Hu := ∂ tt u -divσ(u).
(2.1)

We prescribe a non-homogeneous Dirichlet condition on (0, T ) × Σ and a homogeneous Dirichlet condition on the remaining part of the boundary:

     ∂ tt u s -divσ(u s ) = 0 in (0, T ) × Ω s , u s = u d s on (0, T ) × Σ, u s = 0 on (0, T ) × (∂Ω s -Σ) , (2.2)
where T > 0. Moreover, initial-time conditions are prescribed:

u s (0) = 0, ∂ t u s (0) = 0.
(2.

3)

The Dirichlet data u d s in (2.2) is assumed to be compatible with (2.3). Let us define the space:

D s := {v ∈ C ∞ (Ω s ), suppv ∩ (∂Ω s -Σ) = ∅},
where Σ denotes the topological interior of Σ. We set:

U := D s H 1 (2.4)
to be the completion of D s with respect to the H 1 (Ω s )-norm. Consider the space:

{γ Σ (v) : v ∈ U} ≡ H 1 2 0 (Σ), (2.5)
where γ Σ (v) denotes the trace, on the boundary Σ, of the function v. The space

H 1 2
0 (Σ) is defined as the completion of C ∞ 0 (Σ) with respect to the H 1 2 (Σ)-norm. Denote X to be the space:

X := H 1 2 0, T ; L 2 (Σ) ∩ L 2 0, T ; H 1 2 0 (Σ) ⊂ H 1 2 ((0, T ) × Σ), (2.6)
and denote [X ] * its topological dual space. Let us define the operator T 1 by:

T 1 : X → L 2 0, T ; H -1 2 (Σ) u d s → g s = T 1 (u d s ).
(2.7)

where H - Proposition 2.1. The operator T 1 , given by (2.7), is well defined and bounded i.e. there exists a constant C s > 0 such that:

||σ(u s ) • - → n || L 2 0,T ; H -1 2 (Σ) ≤ C s ||u s || X (2.8)
for all u s ∈ X , where X is defined by (2.6).

Before passing on to the proof of Proposition 2.1, we give a remark stating a lifting property within the context of the Bochner space L 2 (0, T ; U) : Remark 2.1. Consider the map γ Σ defined by:

γ Σ : L 2 (0, T ; U) → L 2 0, T ; H 1 2 0 (Σ) v → γ Σ v,
this map associates to every v ∈ L 2 (0, T ; U) its trace on (0, T ) × Σ. We claim that γ Σ is onto. Indeed, let v ∈ L 2 0, T ; H 1 2 0 (Σ) . The function can be extended by zero, to the rest of the boundary, into a function in L 2 0, T ; H 1 2 (∂Ω s ) which we still denote by v. By using the lifting property, one can easily find a family of functions (0, T ) t → ṽ(t, .) ∈ H 1 (Ω s ) such that γ ∂Ωs ṽ(t, .) = v(t, .), ∀t ∈ (0, T ), and such that one also has: ||ṽ(t, .)|| H 1 (Ωs) ≤ ||v(t, .)||

H 1 2 (Ωs)
, ∀t ∈ (0, T ). The existence of a ṽ satisfying such estimate can be established, for instance, within the Banach space {w(t, .)

∈ H 1 (Ω s ) : Ωs w(t, x)∆ x φ(x)dx = 0, ∀φ ∈ C ∞ c (Ω s ), ∀t ∈ (0, T )}.
Thus the function ṽ satisfy:

||ṽ|| L 2 (0,T ;U ) ≤ ||v|| L 2 0,T ;H 1 2 0 (Σ)
, and moreover, by combining [3, Theorem 1, p.518] and [2, Theorem 2.1, p.731], we have ṽ(t, .) ∈ U, ∀t ∈ (0, T ); we thus conclude the surjectivity of γ Σ .

Let us state a lemma about a useful existence and regularity result:

Lemma 2.1. Consider the problem      ∂ tt φ -divσ(φ) = F in (0, T ) × Ω s , φ = 0 on (0, T ) × ∂Ω s , φ(0, .) = 0, ∂ t φ(0, .) = 0, (2.9)
we claim that for every F ∈ L 2 (0, T ; H -1 (Ω s )), problem (2.9) admits a unique solution φ ∈ L 2 (0, T ; H 1 (Ω s )) and moreover σ(φ)

• - → n ∈ L 2 (0, T ; H -1 2 (∂Ω s )).
Proof. One can, by density, find a sequence

(F n ) n of elements in L 2 (0, T ; L 2 (Ω s )) ⊂ L 1 (0, T ; L 2 (Ω s )) such that: ||F -F n || L 2 (0,T ;H -1 (Ωs)) → 0. (2.10)
Next one can use [10, Theorem 2.1, p.151] to show the existence of a unique solution φ n ∈ L 2 (0, T ; H 1 (Ω s )) =: Λ to problem (2.9) with F n as a right hand side instead of F . Starting from the weak formulation of problem (2.9), one writes:

< ∂ t φ n , ∂ t ψ > + < ε(φ n ), ε(ψ) > L 2 (0,T ;L 2 (Ωs)) =< F n , ψ > Λ * ,Λ (2.11)
for all ψ ∈ L 2 (0, T ; H 1 0 (Ω)). Choosing ψ = φ n and using (2.10) we infer easily that

||∂ t φ n || L 2 (0,T ;L 2 (Ωs)) + ||ε(φ n )|| L 2 (0,T ;L 2 (Ωs)) < C (2.12)
for some C > 0, and deduce the existence of a unique solution lim n φ n := φ ∈ Λ to problem (2.9).

Now we pass into the proof of Proposition 2.1:

Proof. Define the subspace

A :=      v ∈ H 1 ((0, T ) × Ω s ) ∩ H 2 (0, T ; H -1 (Ω s )) : T 0 Ωs vH(φ)dxdt = 0, ∀φ ∈ C ∞ c ((0, T ) × Ω s ), ||v(0)|| L 2 (Ωs) = ||∂ t v(0)|| H -1 (Ωs) = 0,      (2.13)
where H is defined in (2.1). Recall that:

H 1 (0, T ; L 2 (Ω s )) ∩ L 2 (0, T ; H 1 (Ω s )) = H 1 ((0, T ) × Ω s ).
One sees that A is a Banach space when endowed with the norm:

||v|| A := ||v|| H 1 ((0,T )×Ωs) + ||∂ tt v|| L 2 (0,T ;H -1 (Ωs)) ,
moreover A is reflexive. The idea of the proof consists at writing T 1 as a composition

T 1 = N • γ -1
0 of two linear operators and then establishing their boundedness. The rest of the proof is divided into two main steps:

Step 1: first inverse estimate. Consider the trace operator:

γ 0 : A → H 1 2 ((0, T ) × ∂Ω s ) u s → γ 0 u s ,
this operator associates to every function u s ∈ A its trace on (0, T )×∂Ω s , the space A is given by (2.13). The linear operator γ 0 is clearly one-to-one, we claim that it is also onto. Indeed, let u s ∈ H 1 2 ((0, T ) × ∂Ω s ), we are going to show the existence of U s ∈ A such that u s = γ 0 U s . To do this, it suffices to prove that the problem:

     ∂ tt U s -divσ(U s ) = 0 in (0, T ) × Ω s , U s = u s on (0, T ) × ∂Ω s , U s (0, .) = 0, ∂ t U s (0, .) = 0, (2.14) admits a solution U s ∈ A. Let u n s ∈ C ∞ ((0, T ) × ∂Ω s ) be such that: ||u n s -u s || H 1 2 ((0,T )×∂Ωs) → 0; (2.
< U n s , ∂ tt φ > -< U n s , divσ(φ) >=< U n s , σ(φ) • - → n >, (2.16) ∀φ ∈ L 2 (0, T ; H 1 (Ω s )) solution of problem (2.9). Since, by Lemma 2.1, σ(φ) • - → n ∈ L 2 (0, T ; H -1 2 (∂Ω s )
) then using the convergence (2.15) we have:

< U n s , H(φ) >=< U n s , σ(φ) • - → n >→< u s , σ(φ) • - → n >, (2.17)
for every φ ∈ L 2 (0, T ; H 1 (Ω s )) solution of problem (2.9) i.e.

< U n s , F > converges ∀F ∈ L 2 (0, T ; H -1 (Ω s )). (2.18)
Given that L 2 (0, T ; H -1 (Ω s )) is reflexive, then (2.18) implies:

∃C > 0, ||U n s || L 2 (0,T ;H 1 (Ωs)) ≤ C, ∀n, (2.19)
and thus (U n s ) n converges weakly to lim n→∞ U n s =: U s ∈ L 2 (0, T ; H 1 (Ω s )). Furthermore, we have: → 0, this show that γ 0 (lim U n s ) = γ 0 (U s ) = u s , which concludes the surjectivity of γ 0 . We thus infer that the operator γ 0 is an isomorphism. Since the domain and codomain of the operator γ 0 are Banach spaces and since γ 0 is bounded, then by applying the Banach isomorphism theorem, we deduce that the inverse operator γ -1 0 is bounded i.e. ∃C γ -1 0 > 0 such that:

< ∂ tt U n s , φ >=< U n s , ∂ tt φ >; (2.20) letting n → ∞, we obtain < ∂ tt U s , φ >=< U s , ∂ tt φ >, < ∂ t U s , ∂ t φ >=< U s , ∂ tt φ >, (2.21
||u s || A ≤ C γ -1 0 ||u s || H 1 2 ((0,T )×Σ) , (2.22) for all u s ∈ X ⊂ H 1 2 ((0, T ) × Σ).
Step 2: second inverse estimate. Consider the following operator:

N : A → L 2 (0, T ; H -1 2 (Σ)) u s → N (u s ) = g s ,
where A is defined by (2.13). The operator N associates to every displacement u s ∈ A the corresponding Neumann data, g s := σ(u s ) • -→ n , on the boundary (0, T ) × Σ.

We claim that N (A) ⊂ L 2 (0, T ; H -1 2 (Σ)). Indeed, using the density of smooth functions in the space X , || ||

H 1 2 ((0,T )×Σ)
, then with the aid of estimate (2.22) we can construct a sequence (u n s ) n of elements in C ∞ ((0, T ) × Ω) ∩ A such that:

||u n s -u s || A → 0 as n → ∞. (2.23) Let φ ∈ L 2 (0, T ; U)
, where U is given by (2.4). We integrate by part the first equation in (2.2) against the test function φ(t, .) ∈ U to obtain:

T 0 < ∂ tt u n s , φ > [1] * ,[1] dt + T 0 Ωs ε(u n s )ε(φ) dxdt = T 0 < g n s , φ > [-1 2 ],[ 1 2 ],Σ dt, (2.24)
for every φ ∈ L 2 (0, T ; U), where

g n s := σ(u n s ) • - → n ∈ C ∞ ((0, T ) × Σ),
| T 0 < g n s , φ > -1 2 , 1 2 ,Σ dt| n is a Cauchy sequence, (2.25) the completeness of R yields sup n | T 0 < g n s , φ > -1 2 , 1 2 ,Σ dt| < ∞ for every φ ∈ L 2 0, T ; H 1 2
0 (Σ) . Using the uniform boundedness principle we infer that

sup n ||g n s || L L 2 0,T ;H 1 2 0 (Σ) ,R < ∞, (2.26)
given the completeness and the separability of L 2 0, T ; H 1 2 0 (Σ) , we infer using the fundamental theorem of weak * convergence and estimate (2.26) that the sequence (g α(n) s

) n converges weakly * to some g s ∈ L 2 0, T ; H -1 2 (Σ) , for some subsequence (α(n)) n . Actually one can easily remark, using (2.25), that the whole sequence converges to g s . We infer that the operator N is well defined. The reader should notice that we have only proved: N (A) ⊂ L 2 0, T ; H -1 2 (Σ) . Moreover, the operator N is bounded. Indeed, knowing (2.5), we easily infer from the above arguments that the operator N sends every weakly convergent sequence in A into a weakly * convergent sequence in L 2 0, T ; H -1 2 (Σ) . But given the reflexivness of the space L 2 0, T ; H 1 2 0 (Σ) , the weak * convergence and the weak convergence agree. This shows that N is sequentially weakly continuous. Since N is linear, we deduce that it is bounded i.e. ∃C N > 0 such that:

||g s || L 2 0,T ;H -1 2 (Σ) ≤ C N ||u s || A , (2.27)
for every u s ∈ A.

Finally, by combining (2.22) and (2.27), we infer that ∃C s > 0 such that:

||σ(u s ) • - → n || L 2 0,T ;H -1 2 (Σ) ≤ C s ||u s || H 1 2 ((0,T )×∂Ωs) , (2.28)
for every u s ∈ H 1 2 ((0, T ) × Σ). Estimate (2.28) holds for every u s ∈ X ⊂ H 1 2 ((0, T ) × Σ), thus we conclude immediately estimate (2.8).

D. AIT-AKLI

Estimates for the quasi-linear Stokes problem

Problem setting for the fluid part. Assume Ω f ⊂ R 2 to be a sufficiently smooth domain, say with boundary ∂Ω f = Σ ∪ (∂Ω f -Σ) ∈ C 2 . We consider the nonstationary Stokes operator with a quasi-linear diffusion term appearing in the first equation of (3.1). This operator is endowed, cf. system (3.1), with mixed boundary conditions. We prescribe a non-homogeneous Neumann condition on the contact interface, (0, T ) × Σ, via the Cauchy stress tensor, and prescribe a homogeneous Dirichlet condition on the remaining part of the boundary. The fluid is assumed to be divergence-free. Let F ∈ L 2 (0, T ;

H -1 (Ω f )), g f ∈ L 2 (0, T ; H -1 2 (Σ)) and v 0 f ∈ L 2 (Ω f ).
The fluid part of the problem reads:

               ∂ t v f -νdiv a(t, ∇v f ) -ν∆v f + ∇π = F in (0, T ) × Ω f , divv f = 0 in (0, T ) × Ω f , (a(t, ∇v f ) + S(v f , π)) • - → n = g f on (0, T ) × Σ, v f = 0 on (0, T ) × (∂Ω f -Σ), v f (0, .) = v 0 f in Ω f , (3.1)
where v f is the unknown fluid velocity vector, π denotes the unknown pressure and S(v f , π) denotes the Cauchy stress tensor given by (1.9). For the viscosity, we assume for simplicity that ν = 1. Let the vector function a := (a 1 , a 2 ) be such that the functions

a j : [0, T ] × R 2 → R, with j = 1,
(a(t, ξ) -a(t, η))(ξ -η) ≥ c m |ξ -η| 2 2 , ∀ξ, η ∈ R 2 , ∀t ∈ [0, T ]. (3.2)
Consider the space:

D f := {v ∈ C ∞ (Ω f ), div v = 0, supp v ∩ (∂Ω f -Σ) = ∅}. (3.3)
Denote V to be the closure of D f with respect to the H 1 (Ω f )-norm i.e.

V := D f H 1 , (3.4)
the closed subspace V is endowed with the H 1 -norm and thus it is a Hilbert space.

Well-posedness and energy estimate.

Proposition 3.1. ∀F ∈ L 2 (0, T ; H -1 (Ω f )), ∀v 0 f ∈ L 2 (Ω s ) and ∀g f ∈ L 2 (0, T ; H -1 2 (Σ))
, there exists a unique solution v f to problem (3.1) such that:

v f ∈ L 2 (0, T ; V) ∩ H 1 (0, T ; H -1 (Ω f )); (3.5)
moreover, one has the following energy estimate:

||∂ t v f || L 2 (0,T ;H -1 (Ω f )) + ||v f || L 2 (0,T ;V) ≤ C f (|| (S(v f , π) + a(t, ∇v f )) • - → n || L 2 0,T ;H -1 2 (Σ) + ||F || L 2 (0,T ;H -1 (Ω f )) + ||v 0 f || L 2 (Ωs) ). (3.6)
The well-posedness result stated in Proposition 3.1 is rather classic. An equivalent weak formulation of (3.1) can be derived by integrating the first equation in (3.1), against φ ∈ L 2 (0, T ; V), to obtain, cf. [11, Problem 3.15, p.371]:

(∂ t v f , φ) + A(v f )(φ)-< (a(t, ∇v f ) + S(v f , π)) • - → n , φ > [ 1 2 ] * ,[ 1 2
],Σ =< F, φ >, (3.7) for a.e. time 0 ≤ t ≤ T , where

A(v f )(φ) := Ω f a(t, ∇v f )∇φdx + Ω f ε(v f )ε(φ)dx, (3.8)
where we used the fact div(∇v f + ∇ t v f ) = 0 which holds since divv f = 0, consequently one has divS(v f , π) = ∆v f -∇π in this case, cf. [6, Problem 1.1, p.237-240].

Given the assumption (3.2), the operator A is strongly monotone. Problem (3.7) rewrites:

Find v f ∈ L 2 (0, T ; V) ∩ H 1 0, T ; H -1 (Ω f ) s.t.: - T 0 (v f , ∂ t φ)dt + T 0 A(v f )(t)(φ)(t)dt = T 0 < (a(t, ∇v f ) + S(v f , π)) • - → n , φ(t) > -1 2 , 1 2 ,Σ dt + T 0 < F, φ > dt, (3.9)
∀φ ∈ L 2 (0, T ; V) with ∂ t φ ∈ L 2 0, T ; H -1 (Ω f ) and φ(T ) = 0. Problem (3.9) fits in the class of quasi-linear parabolic problems. One deals with such a problem using classical arguments, see for instance [7, Porposition 5.1, p.129] and [7, Example 6.A, p.139]. Applying these last results we infer that problem (3.9) admits a unique solution v f ∈ L 2 (0, T ; V) for every g f ∈ L 2 0, T ; H -1 2 (Σ) . Regarding the existence issue, also cf. [START_REF] Solonikov | Solvability of a problem on the evolution of a viscous incompressible fluid, bounded by a free surface, on a finite time interval[END_REF]Theorem 1.1,p.225]. Furthermore, one can derive the energy estimate (3.6) from problem (3.9) by choosing as test function φ = v f ∈ L 2 (0, T ; V) and using the assumption (3.2).

Remark 3.1. Let v f ∈ L 2 0, T ; H 1 2 (Σ) → continuous L 2 0, T ; L 2 (Σ) . It is a classi- cal fact that v f can be arbitrarily approximated by an element v n f ∈ C ∞ 0 0, T ; H 1 2 (Σ)
w.r.t. the norm of the space L 2 0, T ; H 1 2 (Σ) . On another hand, by applying the Poincaré inequality in the time variable with v n , we can easily show that:

t 0 ||u n f (r)|| 2 L 2 (Σ) dr ≤ C p t 0 ||v n f (r)|| 2 L 2 (Σ) dr + ||u n f (0)|| L 2 (Σ) . (3.10)
On the other hand:

T 0 |u n f (t)| 2 H 1 2 (Σ) dt := Σ Σ T 0 |u n f (t, x) -u n f (t, y)| 2 |x -y| 2 dtdxdy ≤ C p Σ Σ T 0 |v n f (t, x) -v n f (t, y)| 2 |x -y| 2 dtdxdy + T |u n f (0)| 2 H 1 2 (Σ)
.

(3.11)

Combining estimates (3.10), (3.11) and using u f (0, x) = 0 for x ∈ Σ, then letting n → ∞:

T 0 ||u f (r)|| 2 H 1 2 (Σ) dr ≤ C p T 0 ||v f (r)|| 2 H 1 2 (Σ)
dr.

(3.12)

The same conclusion holds in case of fractional Sobolev spaces H s (Σ) with s ∈ R * + .

We infer from Remark 3.1, that the fluid displacement u f satisfies:

u f ∈ L 2 0, T ; H 1 2 (Σ) ∩ H 1 0, T ; L 2 (Σ) ⊂ H 1 2 ((0, T ) × Σ). (3.13)
The idea. Let us explain the main idea of this section. Define T 2 to be the operator:

T 2 : L 2 0, T : H -1 2 (Σ) → X ⊂ H 1 2 ((0, T ) × Σ) g f → T 2 (g f ) = u f , (3.14)
this operator associates to every Neumann data, on the fluid part of the contact interface, the displacement u f corresponding to the velocity v f which is a solution of (3.1). It is easily seen, by combing (3.6), (3.12) and by applying (3.13), that:

||T 2 (g 1 f ) -T 2 (g 2 f )|| H 1 2 ((0,T )×Σ) = ||u 1 f -u 2 f || H 1 2 (0,T ;L 2 (Σ)) + ||u 1 f -u 2 f || L 2 (0,T ;H 1 2 (Σ)) ≤ C p ||v 1 f -v 2 f || L 2 (0,T ;H 1 (Ω f )) ≤ C f ||g 1 f -g 2 f || L 2 0,T ;H -1 2 (Σ) (3.15) for every g 1 f , g 2 f ∈ L 2 0, T ; H -1 2 (Σ)
, and thus the operator T 2 is continuous. Let T 1 and T 2 be defined respectively by (2.7) and (3.14). Define the operator:

T : X → X ⊂ H 1 2 ((0, T ) × Σ) u s → T (u s ) := T 2 • T 1 (u s ) = u f , (3.16)
We remark that the global solution of the coupled problem (1.1)-(1.7) is a fixed point of T , then to show existence of a solution to (1.1)-(1.7) it suffices to prove existence of a fixed point of the of the operator T . To be able to use fixed point theory we need some compactness. However, T sends solid displacements from : X ⊂ H 1 2 ((0, T ) × Σ) into no more spatial-regular fluid displacements, that is into: X ⊂ H 1 2 ((0, T ) × Σ). In order to recover some compactness we need to consider a sequence of auxiliary operators T 2 . To achieve that, we proceed into a regularization of the Stokes problem (3.9) i.e. to define a sequence of problems depending on a small real parameter, > 0, in such a way that the new operator

T 2 • T 1 sends H 1 2 ((0, T ) × Σ) into
a more regular space in the spatial variable, this will ensure the needed compactness. The (solution of the) original problem will be recovered by letting → 0.

Regularized problem. Consider the space:

W := D f H 2 (3.17)
to be the completion of D f , defined by (3.3), with respect to the Sobolev H 2 -norm. It is indeed a Hilbert spaces. We denote by (., .) H 2 its canonical inner product, and by (., .) H 2 sn the part of (., .) H 2 that involves only the second derivatives. We denote W * its dual. Given g f ∈ L 2 0, T ; H -1 2 (Σ) , consider the regularized problem:

Find v f ∈ L 2 (0, T ; W) such that: T 0 (∂ t v f , φ) L 2 dt + A (v f )(φ) = T 0 < F (t), φ(t) > -1,1,Ω f dt T 0 < g f (t), φ(t) > -1 2 , 1 2 ,Σ dt, (3.18)
for all φ ∈ L 2 (0, T ; W), where

A (v f )(φ) := T 0 A(v f )(φ)dt + T 0 Ω f (v f , φ) H 2 sn dxdt,
and where A is given by (3.8). It is easily seen that the quasi-linear elliptic operator A : W → W * satisfy the assumptions of [7, Proposition 5.1, p.129] and that of [START_REF] Showalter | Monotone Operators in Banach Space and Nonlinear Partial Differential Equations[END_REF]Theorem 5.1,p.128]. Then for every g f ∈ L 2 0, T ; H -1 2 (Σ) , problem (3.18) admits a unique solution:

v f ∈ L 2 (0, T ; W) (3.
19) for all > 0, where W is given by (3.17). An energy estimate can be derived by using the strong monotony of the operator A . We thus obtain the estimate:

||v 1, f -v 2, f || L 2 (0,T ;W) ≤ C f ||g 1 f -g 2 f || L 2 0,T ;H -1 2 (Σ) . (3.20)

Estimate (3.20) is obtained by choosing as test function

φ = v 1, f -v 2, f , in (3.18), corresponding to g 1 f , g 2 f .
Using the continuity of the trace operator

γ 0 : H 2 (Ω f ) → H 3 2 (Σ), we infer from (3.20): ||v 1, f -v 2, f || L 2 0,T ;H 3 2 (Σ) ≤ C f ||g 1 f -g 2 f || L 2 0,T ;H -1 2 (Σ) (3.21)
Remark 3.2. The same fact that was stated in Remark 3.1 holds with u f and v f in the space L 2 0, T ; H 3 2 (Σ) . Thus we obtain by combining (3.12) corresponding to the space H 3 2 (Σ) and estimate (3.21):

||u ,1 f -u ,2 f || L 2 (0,T ;H 1 (Σ)) + ||u ,1 f -u ,2 f || H 1 (0,T ;L 2 (Σ)) ≤ C f ||g 1 f -g 2 f || L 2 0,T ;H -1 2 (Σ) , (3.22)
for every

g 1 f , g 2 f ∈ L 2 0, T : H -1 2 (Σ) .
The auxiliary operator T . First define Z to be the space:

Z := H 1 0, T ; L 2 (Σ) ∩ L 2 0, T ; H 1 0 (Σ) ⊂ H 1 ((0, T ) × Σ), (3.23)
it is endowed with the standard H 1 ((0, T ) × Σ)-norm. Define the operator T 2 by:

T 2 : L 2 0, T ; H -1 2 (Σ) → Z ⊂ X g f → T 2 (g f ) = u f , (3.24)
i.e. it associates to every g f ∈ L 2 0, T ; H -1 2 (Σ) , the trace on (0, T )×Σ of the fluid displacement u f corresponding to the fluid velocity v f which is the solution of the regularized problem (3.18) with g f as a Neumann data. Estimate (3.22) translates the boundedness of T 2 . Let us introduce the auxiliary operator T := T 2 • T 1 , that is:

T : X → Z ⊂ X u s → u f = T 2 • T 1 (u s ), (3.25)
where X and Z are respectively given by (2.6) and (3.23), moreover the operators T 1 and T 2 are respectively defined by (2.7) and (3.24). The operator T associates to every solid displacement u s ∈ X , the fluid displacement u f ∈ X defined on the fluid part of the contact interface (0, T ) × Σ.

The auxiliary coupled problem is formed out of the Lamé solid problem (2.2) and the quasi-linear Stokes regularized problem (3.18). By combining the estimates (2.8) and (3.22) on one hand, and using the coupling condition (1.8) on the other hand, we infer the estimate:

∀ > 0, ||T (u 1 s ) -T (u 2 s )|| X ≤ C s C f ||u 1 s -u 2 s || X , (3.26)
for all u 1 s , u 2 s ∈ X i.e. the non-linear operator T is Lipschitz for every > 0. One should notice that the constant C s C f doesn't depend on .

4. Existence of solution for the main coupled problem 4.1. Existence result for the auxiliary coupled problem. Let T be as defined by (3.25) and X be as defined by (2.6). We propose to prove the following proposition: We are going now to prove Proposition 4.1:

Proof. We deal at first with a contraction mapping. Fix ρ ∈ R such that ρC s C f < 1. The coefficient ρ doesn't depend on . Consider the operator ρT defined by:

ρT : X → X ∩ Z ⊂ X u s → u f = ρT 2 • T 1 (u s ),
where Z is defined by (3.23). Applying estimate (3.26), we have:

||ρT (u 1 s ) -ρT (u 2 s )|| X ≤ ρC s C f ||u 1 s -u 2 s || X , (4.1)
for all u 1 s , u 2 s ∈ X . Let us note the following four facts: • Continuity of the operators T . Estimate (3.26) means that T is Lipschitz and thus continuous. Moreover, estimate (4.1) implies that ρT is a contraction mapping for every > 0. Consequently, since X is a Banach space, we deduce, by applying the Banach fixed point theorem, that ρT admits a unique [0, T ]-globally defined fixed point u ρ ∈ X .

• Boundedness condition. We deduce from the preceding point that, ∀ > 0, the set {u ∈ X , u = ρT u } reduces to the unique fixed point u ρ , and thus it is bounded. • Stability of the operators T . According to definition (3.25), we have T (X ) ⊂ X , thus T is stable for every > 0. • Compactness of the operator T . Let > 0. By combining estimates (2.8), (3.22) and the boundedness of ||T (0)|| Z , we remark that the operator T sends every bounded subset E ⊂ X into an

H 1 -bounded sub- set T (E) ⊂ H 1 ((0, T ) × Σ). Using the compact embedding H 1 ((0, T ) × Σ) → compact H 1 2 ((0, T ) × Σ), we deduce that T (E) is compact in X , thus T is compact ∀ > 0.
Thus we checked for T the sufficient conditions of Theorem 4.1; this proves Proposition 4.1.

4.2.

Existence of a solution to the coupled problem (1.1)-(1.7). We now establish the existence of a solution for the global coupled problem which, as pointed out above, amounts at establishing the existence of a fixed point of the operator T given by (3.16). According to Proposition 4.1, ∀ > 0, there exists u ∈ X such that T u = u , (4.2) where X is given by (2.6). Since (u ) >0 is an uncountable family of vectors belonging to the separable normed vector space (X , || || X ), then it must have a limit point with respect to the topology induced by the norm of X i.e. there exists u 0 ∈ X such that:

||u -u 0 || X → 0, (4.3) as → 0. To be able to pass to the limit → 0 in (4.2), we need Lemma 4.1: Lemma 4.1. Let T and T be respectively as defined by (3.25) and (3.16), then:

T u → strongly in X T u, ∀u ∈ X . (4.4)
Proof. Fix u ∈ X and pose g f = T 1 u, where T 1 is given by (2.7). Choose in (3.18) φ = u f where u f = T 2 (g f ) and where T 2 is given by (3.24). Using (3.22), we infer immediately that T u

weakly in X U f with U f ∈ X , let us show that U f = T u.
Denote v f to be the velocity v f := ∂ t u f . It is clear, by using (3.20) with g 2 f = 0 and g

1 f = g f , that ||v f || L 2 (0,T ;H 1 (Ω f )) is uniformly bounded, thus: v f weakly V f in L 2 (0, T ; H 1 (Ω f )). ( 4.5)
Moreover, by combining (3.20) and (4.5) we infer that both

T 0 |v f (t)| 2 H 2 sn (Ω f ) dt := T 0 (v f , v f ) H 2 sn dt and ||∂ t v f || L 2 (0,T ;H -1 (Ω f ))
are also uniformly bounded with respect to > 0, consequently:

∂ t v f weakly ∂ t V f in L 2 (0, T ; H -1 (Ω f )), ∂ 2 xixj v f weakly 0 in L 2 (0, T ; L 2 (Ω f )). (4.6)
Combining the weak problems (3.9), (3.18) and applying the last convergence in (4.6) on one hand, and by considering the auxiliary function u n corresponding to problem (3.9) with f n an dg n as right hand sides such that ||f n -f || → 0 and ||g n -g|| → 0, we deduce that:

||∇v f -∇v f || L 2 (0,T ;L 2 (Ω f )) → 0, (4.7)
it follows that V f = v f . Given that a is Lipschitz in the second variable, then (4.7) yields:

a(t, ∇v f ) → a(t, ∇V f ) converges strongly. ( 4.8)
Finally, letting → 0 in (3.18) by mean of combining (4.5), (4.6) and (4.8), one infers that V f is the fluid velocity field corresponding to the solid displacement u ∈ X . Consequently we have ∂ t u = V f and U f = T u, this yields convergence (4.4).

Proof of Theorem 1.1. Now we are ready to present a proof of the main result:

Proof. Let u 0 be such as defined by (4.3), we have:

||T u -T u 0 || X ≤ ||T u -T u 0 || + ||T u 0 -T u 0 ||. (4.9)
On one hand we have:

||T u -T u 0 || X ≤ C s C f ||u -u 0 || X ,
where we used (3.26). Applying (4.3), we infer:

||T u -T u 0 || X → 0 as → 0. that is T has a fixed point u 0 ∈ X . This completes the proof of Theorem 1.1.

Regarding the regularity claimed in Theorem 1.1, one can use estimate (2.22) to infer the regularity of the solid displacement u s ∈ X on Ω s . Furthermore, one combines the estimates (3.5) and (2.8) to infer the regularity of the fluid velocity v f on Ω f . The regularity of the pressure π can be inferred in the following fashion: one considers the regularity of the solution u 0 stated in (3.5), namely that ∂ t u 0 ∈ L 2 0, T ; H -1 (Ω f ) and then applies the energy estimate in [START_REF] Amrouche | Elliptic Differential Equations: Linear Theory[END_REF]Theorem 25,p.226] and thus infer that π ∈ L 2 0, T ; L 2 (Ω f )/R .

Conclusion

According to the procedure adopted by the authors in [START_REF] Raymond | A fluid-structure model coupling the Navier-Stokes equations and the Lamé system[END_REF], cf. the bottom of [5, Proof of Theorem 5.1, p.571], the globally-in-time existence of a solution can be inferred by using an iterative method which is based on the linearity of the problem they considered. We emphasize that this method is no longer applicable in the nonlinear context of the present work.

It should be noted that there is no memory effect in the (iterative) resolution of the presently addressed coupled problem in the sense that the result (in particular the displacement) obtained at a time t does not depend on the displacement at an earlier time. This is notably due to the fact that the deformation of the geometry (in particular that of the contact surface Σ), within time incrementing i.e. during the coupling, is not taken into account, although we assumed in the introduction that the domain Ω f (t) depends upon time.

That said, the main theorem established in this paper remains relevant. Indeed, it can be incorporated as an auxiliary result to demonstrate a more complex wellposedness result like for instance [START_REF] Raymond | A fluid-structure model coupling the Navier-Stokes equations and the Lamé system[END_REF]Theorem 2.1,p.555]. If one is willing to prove a result analogous to the later in the framework considered in this paper, then one has, in a first step, to generalize Theorem 1.1 to the case when the data in system (1.1)-(1.7) are non-homogeneous along with a non necessary free divergence condition. We believe that this step can be achieved with some slight modification of the procedure described in the present work, one obtains up to this step a result analogous to [START_REF] Raymond | A fluid-structure model coupling the Navier-Stokes equations and the Lamé system[END_REF]Theorem 5.1,p.570]. Then one should follow the same method as in the proof of [5, Theorem 2.1, p.555] which deals with the well-posedness of system [5, Problem 2.1, p.551-552] and which requires taking into account the deformation of the geometry during the interaction, especially that of the contact interface.

Proposition 4 . 1 .

 41 For every > 0, the operator T admits a fixed point u ∈ X .The idea for provingProposition 4.1 is to combine the Banach and the Schaefer fixed point theorems. Let us recall Schaefer's theorem, cf. [8, Theorem 4.3.2, p.29]:Theorem 4.1. (Schaefer). Let K be a Banach space. Let T : K → K be a continuous and compact mapping. Assume that the set {v ∈ K : v = ρT v for some ρ, 0 ≤ ρ ≤ 1} is bounded, then T admits a fixed point in K.

(4. 10 )

 10 On another hand, using (4.4), we obtain:||T u 0 -T 0 u 0 || → 0, → 0. (4.11) Combining (4.10) and (4.11) it yields ||T u -T u 0 || X ,

  the bracket < ., . > [β] * ,β,E denotes the duality pairing between [H β (E)] * and H β (E). The tensor ε is defined by (1.10). Using (2.23), the expression (2.24), the claim stated in Remark 2.1 and (2.5), we deduce that ∀φ ∈ L 2 0, T ; H

	1
	2 0 (Σ) :

L2CSP laboratory, Mouloud Mammeri University of Tizi-Ouzou, Tizi-Ouzou, 15000, Algeria. * Corresponding author: djamel.aitakli@ummto.dz


Antonia Ettorre

Franck Michel

Catherine Faron

Prediction of Students' performance in E-learning Environments based on Link Prediction in a Knowledge Graph

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

Prediction of Students' performance in E-learning Environments based on Link Prediction in a Knowledge Graph

Antonia Ettorre [0000-0003-4868-2584] , Franck Michel [0000-0001-9064-0463] , and Catherine Faron [0000-0001-5959-5561] Université Côte d'Azur, CNRS, Inria, I3S, Sophia Antipolis, France {aettorre,fmichel,faron}@i3s.unice.fr

Abstract. In recent years, the growing need for easily accessible highquality educational resources, supported by the advances in AI and Web technologies, has stimulated the development of increasingly intelligent learning environments. One of the main requirements of these smart tutoring systems is the capacity to trace the knowledge acquired by users over time, and assess their ability to face a specific Knowledge Component in the future with the final goal of presenting learners with the most suitable educational content. In this paper, we propose a model to predict students' performance based on the description of the whole learning ecosystem, in the form of a RDF Knowledge Graph. Subsequently, we reformulate the Knowledge Tracing task as a Link Prediction problem on such a Knowledge Graph and we predict students' outcome to questions by determining the most probable link between each answer and its correct or wrong realizations. Our first experiments on a real-world dataset show that the proposed approach yields promising results comparable with state-of-the-art models.

Introduction

The increasingly easy access to high-quality educational resources combined with the recent advances in AI and Web technologies have fostered the development of smart user-centered learning environments. The success of such systems is mainly due to their ability to assist users in their learning process, by offering real-time automated tutoring and personalized revision suggestions. One of the main requirements for these systems is the capacity to trace the knowledge acquired by users over time and assess their ability to face a specific knowledge concept in the future with the final goal of presenting learners with the pedagogical content that will most effectively improve their skills. The challenge of predicting students' outcomes when interacting with a given educational resource, also known as Knowledge Tracing (KT), has been largely investigated and several approaches have been proposed throughout the years, e.g. Bayesian Knowledge Tracing (BKT) [START_REF] Corbett | Knowledge tracing: Modeling the acquisition of procedural knowledge[END_REF], Additive Factors Model (AFM) [START_REF] Cen | Learning Factors Analysis -A General Method for Cognitive Model Evaluation and Improvement[END_REF] and Deep Knowledge Tracing (DKT) [START_REF] Piech | Deep Knowledge Tracing[END_REF]. Although very different, these approaches present a major commonality: their predictions rely on very limited and simply structured information, i.e. the student approaching the question, the question being answered, and the list of the skills (or knowledge components) involved in the question. Conversely, in real-world scenarios, students' performance can be influenced by several additional factors that are miss-represented or missing in the previously mentioned models, such as type of questions, number of possible answers, assignment or test length, hierarchical organization of knowledge components, etc.

In this paper, we present an approach to represent and exploit this heterogeneous information to provide reliable predictions for students' outcomes to questions. Firstly, we rely on the expressiveness offered by Semantic Web models to represent the whole learning ecosystem as an RDF Knowledge Graph (KG). Then, we reformulate the KT task as a Link Prediction (LP) problem on such a KG and we determine the most probable links between the answers whose outcomes need to be predicted and their correct or wrong results, and we convert these predictions into binary labels for answers' correctness. We think that reformulating the problem of predicting students' outcomes to questions in such a way allows us to take advantage of a much wider amount and variety of information about the learning environment while reusing widely-known and well-established Deep Learning methods for KGs, and avoiding the burden of features engineering. To empirically confirm the validity of the proposed approach, we apply it on a real-world dataset and compare its performance with state-of-the-art KT models.

Link Prediction for Students' Outcomes

The approach presented in this paper is mainly based on the hypothesis that it is possible to turn the KT task into a LP problem, after modeling the learning environment as a KG. In other words, instead of predicting the probability that a given student correctly answers a specific question, we evaluate the possibility that an implicit link (i.e. a triple) exists in the KG between the expected student's answer and its correct or incorrect result. Finally, an answer is labeled as 1 (correct) if the link towards the correct result has a higher score than the link towards the incorrect one, while it is labeled as 0 (incorrect) in the opposite case. For example, to predict the positive or negative outcome of the answer given by student A to question 1 (fig. 1), we compute the score of the two triples ⟨answer1, has result, correct⟩ and ⟨answer1, has result, incorrect⟩, and we predict that A's answer will be correct if the first triple has a higher score than the second one. To empirically validate our hypothesis, we designed and developed an end-to-end pipeline depicted in fig. 2, which takes as input the traces of the students' learning history, possibly enriched with contextual knowledge, and the list of the student-question interactions whose outcomes must be predicted. The framework implements four steps:

1. Graph Building: create a KG representing the learning ecosystem; 2. Graph Augmentation for Prediction: inject into the previously created KG new nodes representing the new answers we aim to predict;

Evaluation

Experimental Setup. To validate the proposed approach and to be able to compare it with state-of-the-art KT models, we decided to test our method on a widely-used benchmarking dataset: the ASSISTment 2009-2010 skill builder dataset [START_REF] Feng | Addressing the assessment challenge with an online system that tutors as it assesses[END_REF] that stores the learning logs of the users of the ASSISTments platform. For each attempt of a user to a problem, it contains information such as user and problem identifiers, skills required for the problem, problem type (main or scaffolding), answer type (open answer, multiple-choice, etc.), assignment and assistment in which the problem was faced, response time, etc. The first step to apply our approach is to model this information as a KG. For the modeling of the user-problem interactions, we linked each problem to two nodes, one for the positive and one for the negative result, and users' answers are connected to such nodes based on their result. The second step is the computation of the KGEs which has been carried out using TransE [START_REF] Bordes | Translating embeddings for modeling multi-relational data[END_REF] to obtain embeddings of dimension 100.

Results and Discussion. Table 1 shows that the newly proposed LP approach achieves an improvement of 2% in terms of bACC, when compared to DKT. We believe that the main reason for this improvement is the ability of our method to consider a greater variety of knowledge about the learning ecosystem. It is also interesting to point out that, for both models, there is a strong difference between negative and positive F1 scores, with Link Prediction achieving slightly more balanced results. This gap in the F1 values can be explained by the highly unbalanced distribution of the target values in the subject dataset, which contains about 70% of correct answers.

Conclusion

In this work, we reformulated the KT task into a KG LP problem able to take advantage of all the information commonly available in a smart learning system. Using a well-known benchmarking dataset that we transformed into a KG based on Semantic Web models, our empirical evaluation showed that LP performs slightly better in predicting students' outcomes to questions results, when compared to DKT. Although modest, this improvement suggests that rich context information usually ignored by traditional KT approaches can help achieve better prediction. In the future, we wish to further explore this lead, notably by measuring and enriching the information encoded in KGEs.

Fig. 1 :Fig. 2 : 3 .

 123 Fig. 1: Example of a KG representing the answer of a student to a question. The dashed lines represent the links we aim to predict.

Table 1 :

 1 Results of Link Prediction and DKT approach.

	KT model	F1 (0) F1 (1) bACC
	Link Prediction 0.544 0.734 0.657
	DKT	0.512 0.755 0.640