B J Jin

A Novotný May

Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids

Keywords: One velocity Baer-Nunziato type system, Weak solutions, Strong solutions, Relative energy, Weak-strong uniqueness

We investigate a version of one velocity Baer-Nunziato type system with dissipation describing the motion of a mixture of two compressible fluids. We define for this system weak solutions on one hand and dissipative weak solutions on the other hand, and recall the theorem about their existence on a large time interval. We investigate strong solutions and show their existence on a short time interval. Finally, we prove that any weak solution satisfies a relative energy inequality and prove for this system the weak-strong uniqueness principle. This is the main result of the paper.

Introduction

There is no general agreement about the modeling of the mixture of several compressible fluids, and from the general point of view, about the two phase flow modeling. One of the acceptable models is the so called two velocity Baer-Nunziato model. The equations of the Baer-Nunziato model with dissipation [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF], [18, Section 1] are as follows:

∂ t α ± + v I • ∇α ± = 0, (1.1)
∂ t (α ± ± ) + div(α ± ± u ± ) = 0, (1.2)

∂ t (α ± ± u ± ) + div(α ± ± u ± ⊗ u ± ) + ∇(α ± P ± ( ± )) -P I ∇(α ± ) (1.3) = α ± µ ± (∆u ± ) + α ± (µ ± + λ ± )∇divu ± 0 ≤ α ± ≤ 1, α + + α -= 1.
(1.4)

In the above (α ± , α ± ± ≥ 0, u ± ∈ R d ) -concentrations, densities, velocities of the ± species -are unknown functions of time t ∈ I = (0, T ), T > 0, and x ∈ Ω ⊂ R d , d = 2, 3, while µ ± > 0, λ ± + 2 d µ ± ≥ 0 are given constant shear and bulk viscosities of the ± species, P ± are two (different) given functions defined on [0, ∞) and P I , v I are conveniently chosen quantities -they represent the pressure and the velocity at the interface. In the multifluid modeling, there are many possibilities how the quantities v I , P I could be chosen, and there is no consensus about this choice. We refer the reader to [START_REF] Bresch | Multifluid models including compressible fluids[END_REF] for the overview of multifluid models from the mathematical point of view and to [START_REF] Drew | Theory of multicomponent fluids[END_REF], [START_REF] Ishii | Thermo-fluid dynamics of two-phase flow[END_REF] for the physical background of the multifluid modeling.

In [START_REF] Novotny | Weak solutions for a bi fluid model of a mixture of two compressible non interacting fluids[END_REF] the authors consider the Baer-Nunziato system with µ ± := µ, λ ± := λ, v I = u ± := u (1.5) αP ± (s) = P ± (f ± (α)s) for all α ∈ (0, 1), s ∈ [0, ∞),

where P ± are given functions defined on [0, ∞) and f ± are given functions defined on (0, 1). With this choice, the two velocity Baer-Nunziato system reduces to the following system (which we will call the one velocity Baer-Nunziato type system):

∂ t α + (u • ∇)α = 0,
(1.7) ∂ t + div( u) = 0, (1.8) ∂ t z + div(zu) = 0, (1.9) ∂ t (( + z)u) + div(( + z)u ⊗ u) + ∇p(f (α) , g(α)z) = divS(∇u), (1.10) in I × Ω, where Ω ⊂ R d , d = 2, 3. Here p : [0, ∞) 2 → [0, ∞) as well as f, g : [0, 1] → [0, ∞) are given functions, S(∇u) = µ(∇u + ∇ T u) + λdivuI is the stress tensor with I the identity tensor in R d . The constant viscosity coefficients satisfy the standard physical assumptions, µ > 0, λ + 2 d µ ≥ 0. The system is endowed with the initial conditions α| t=0 = α 0 , | t=0 = 0 , z| t=0 = z 0 , ( + z)u| t=0 = ( 0 + z 0 )u 0 , (1.11) and the no slip boundary conditions, u| ∂Ω = 0.

(1.12) Indeed, under the assumptions (1.5-1.6), if we set

α = α + , = α + + , z = α --,
in the two velocity Baer-Nunziato system (1.1-1.4), we obtain equations (1.7-1.10) with

p(R, Z) = P + (R) + P -(Z), f (s) = f + (s) s , g(s) = f -(1 -s) 1 -s .
In fact, equations (1.1), (1.2) ± written in the new variables yield (1.7-1.9) while equation (1.10) is obtained as the sum of the momentum equations for the species ±, (1.3) ± . We notice, that assumption (1.6) is certainly true in the classical situation of two isentropic gases when P ± (s) = a ± s γ ± , γ ± > 0; (1.13) indeed, in this case

p(R, Z) = a + R γ + + a -Z γ -, f (s) := s 1-γ + γ + , g(s) = (1 -s) 1-γ - γ -. (1.14)
This gives at least one example among others when the two velocity Baer-Nunziato system under hypotheses (1.5) reduces to the one velocity Baer-Nunziato type system (1.7-1.10).

It is not without interest that equations (1.7-1.10) correspond to the barotropic and viscous version of the five-equation model of two phase flows derived by Allaire, Clerc and Kokh in [START_REF] Allaire | Kokh A five-equation model for the numerical simulation of interfaces in two-phase flows CRAS[END_REF], [START_REF] Allaire | A five-equation model for the numerical simulation of interfaces between compressible fluids[END_REF] via different considerations.

The aim of the paper is to investigate the stability issues for a version of the one velocity Baer-Nunziato system (1.7-1.12). We prove that strong solutions (if they exist on the given time interval) are stable in the class of weak solutions, provided the constitutive law for the pressure gives rise to a strictly convex Helmohltz function. In particular, under such circumstances, the weak-strong uniqueness principle holds: any weak solution emanating from the sufficiently regular data coincides with the strong solution emanating from the same initial data as long as the latter exists.

The organization of the paper is inspired by [START_REF] Feireisl | Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system[END_REF], where the weak strong uniqueness is investigated in the "simple" mono-fluid case: In Section 2 we introduce the weak solutions, the relative energy functional and the dissipative solutions for the system (1.7-1.12). In Section 3 we announce the theorem about global existence of weak solutions, cf. Theorem 3.1, in the form proved in [START_REF] Novotny | Weak solutions for a bi fluid model of a mixture of two compressible non interacting fluids[END_REF]. In Section 4 we show the existence of strong solutions for the system (1.7-1.12) at least on a short time interval, see Theorem 4.1. Any weak solution with square integrable densities satisfies the relative energy inequality with arbitrary test functions and is therefore dissipative. This statement is announced in Theorem 5.2 and proved in Section 5. If we take in the relative energy inequality as a test function the strong solution of the same system, the inequality takes a particular form with the remainder which is quadratic in the difference of weak and strong solutions. This statement is formulated in Theorem 6.1 and proved in Section 6. The inequality derived in Theorem 6.1 is employed in Section 7 to show the weak strong uniqueness principle for the system (1.7-1.12), see Theorem 7.2.

We close the introduction by introducing the basic notation and the functional spaces used throughout this paper. In what follows, the scalar-valued functions will be printed with the usual font, the vector-valued functions will be printed in bold, and the tensor-valued functions with a special font, i.e. stands for the density, u for the velocity field and S for the stress tensor. We use standard notation for the Lebesgue and Sobolev spaces equipped by the standard norms • L p (Ω) and • W k,p (Ω) , respectively. We will sometimes distinguish the scalar-, the vector-and the tensor-valued functions in the notation, i.e. we use L p (Ω) for scalar quantities, L p (Ω; R 3 ) for vectors and L p (Ω; R 3×3 ) for tensors. The indication of the R or tensor character of the fields (here ; R 3 or ; R 3×3 ) may be omitted, when there is no danger of confusion. The Bochner spaces of integrable functions on I with values in a Banach space X will be denoted L p (I; X); likewise the spaces of continuous functions on I with values in X will be denoted C(I; X). The norms in the Bochner spaces will be denoted • L p (I;X) and • C(I;X) , respectively. In most cases, the Banach space X will be either the Lebesgue or the Sobolev space. Finally, we use vector spaces C weak (I; X) of functions defined on [0,T] belonging to the space L ∞ (I; X) and continuous in I with respect to the weak topology of X (meaning that f ∈ C weak (I;

X) if f ∈ L ∞ (I; X) and t → F(f (t)) belongs to C(I) for any F ∈ X * ).
The generic numbers in estimates will be denoted by c, c, c, C, C, C and their value may change even in the same formula or in the same line. They may depend on the parameters characterizing the problem; this dependence is always indicated in the text, where they appear.

Weak and dissipative solutions

We begin with the definition of weak solutions to system (1.7-1.12). Definition 2.1 Let O be an open subset of (0, ∞) 2 and let 0 ≤ α < α ≤ 1. Suppose that α 0 (x) ∈ [α, α] and (f (α 0 ) 0 (x), g(α 0 )z 0 (x)) ∈ O for a.a. x ∈ Ω. We say that the quartet (α, , z, u) is a bounded energy weak solution of problem (1.7-1.12) with densities ranging in O and concentration in [α, α] if:

1. It belongs to functional spaces:

α ∈ C(I; L 1 (Ω)), ( , z) ∈ C weak ([0, T ]; L γ (Ω)) ∩ C(I, L 1 (Ω)), γ > 1, u ∈ L 2 (0, T ; W 1,2 0 (Ω; R d )), (2.1) for all t ∈ I, α ≤ α(t, x) ≤ α, (f (α) (t, x), g(α)z(t, x)) ∈ O for a.a. x ∈ Ω.

The integral identity

Ω r(τ, •)ϕ(τ, •) dx - Ω r 0 (•)ϕ(0, •) dx = τ 0 Ω r∂ t ϕ + ru • ∇ϕ dxdt (2.2) holds for any τ ∈ [0, T ] and ϕ ∈ C 1 c ([0, T ] × Ω)
, where r stands for and z.

The integral identity

Ω α(τ, •)ϕ(τ, •) dx - Ω α 0 (•)ϕ(0, •) dx = τ 0 Ω α∂ t ϕ + αu • ∇ϕ + αϕdivu dxdt (2.3) holds for any τ ∈ [0, T ] and ϕ ∈ C 1 c ([0, T ] × Ω).

The function

( + z)u ∈ C weak ([0, T ], L 2γ γ+1 (Ω))
, and the integral identity

Ω ( + z)u(τ, •) • ϕ(τ, •) dx - Ω ( 0 + z 0 )u 0 (•)ϕ(0, •) dx (2.4) = τ 0 Ω ( + z)u • ∂ t ϕ + ( + z)u ⊗ u : ∇ϕ + p(f (α) , g(α)z)div x ϕ -S(∇u) : ∇ϕ dxdt holds for any τ ∈ [0, T ] and any ϕ ∈ C 1 c ([0, T ] × Ω; R d ).
6. There exists a function

H ∈ C(O) ∩ C 1 (O), a solution of first order partial differential equation R∂ R H(R, Z) + Z∂ Z H(R, Z) -H(R, Z) = p(R, Z) in O, (2.5)
such that the energy inequality

Ω 1 2 ( + z)|u| 2 + H(f (α) , g(α)z) (τ ) dx + τ 0 Ω S(∇u) : ∇u dxdt (2.6) ≤ Ω 1 2 ( 0 + z 0 )|u 0 | 2 + H(f (α 0 ) 0 , g(α 0 )z 0 ) dx
holds for a.a. τ ∈ I. Function H is called the Helmholtz function corresponding to the pressure p.

We associate to the Helmholtz function (2.5) the relative energy function

E α,β ( , z|r, z) = E(f (α) , g(α)z|f (β)r, g(β)z) (2.7)
where

E(R, Z|R, Z) = H(R, Z) -∂ R H(R, Z)(R -R) -∂ Z H(R, Z)(Z -Z) -H(R, Z).

Definition 2.2

We say that the quartet (α, , z, u) is a dissipative weak solution to the one velocity Baer-Nunziato type system (1.7 -1.12) with densities in O and concentration in [α, α] if :

1. It fulfills all statements in Items 1.-5. in the Definition 2.1;

2. It satisfies the relative energy inequality,

Ω 1 2 ( + z)|u -U| 2 + E α(τ ),β(τ ) ( , z|r, z) (τ, •) dx (2.8) + τ 0 Ω S(∇u -∇U) : (∇u -∇U) dxdt ≤ Ω 1 2 ( 0 + z 0 )|u 0 -U(0, •)| 2 + E α 0 ,β(0) ( 0 , z 0 |r(0, •), z(0, •)) dx + τ 0 R α,β ( , z, u|r, z, U)dt
for a.a. τ ∈ (0, T ) with any

(β, r, z, U) ∈ C 1 c ([0, T ] × Ω; R 6 ), α ≤ β ≤ α, (r(t, x), z(t, x)) ∈ O, U| ∂Ω = 0, (2.9)
where

R α,β ( , z, u|r, z, U) = Ω (U -u) • ( + z) ∂ t U + (u • ∇)U + S(∇U) : ∇(U -u) dx + Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z) divUdx + Ω f (β)r -f (α) ∂ t ∂ R H(f (β)r, g(β)z)dx + Ω g(β)z -g(α)z ∂ t ∂ Z H(f (β)r, g(β)z)dx + Ω f (β)rU -f (α) u • ∇∂ R H(f (β)r, g(β)z)dx + Ω g(β)zU -g(α)zu • ∇∂ Z H(f (β)r, g(β)z)dx.

Existence of weak solutions

Starting from now, we shall limit ourselves to the three dimensional flows (meaning that d = 3). The treatment of the situation with d = 2 is similar; we let its details to the interested reader. Let 0 < α < α < 1, 0 < a < a, and set

O = O a,a := {(R, Z) ∈ R 2 | R > 0, aR < Z < aR} (3.1)
The existence of bounded energy weak solutions to problem (1.7-1.12) with the densities in the range O a,a and the concentration in the range [α, α] has been proved in [START_REF] Novotny | Weak solutions for a bi fluid model of a mixture of two compressible non interacting fluids[END_REF]Theorem 1], in particular, under the following hypotheses:

1. Hypothesis on the domain

Ω ⊂ R 3 is a bounded domain in the regularity class C 2,ν , ν ∈ (0, 1). (3.2)
2. Hypotheses on initial data:

(f (α 0 ) 0 (x), g(α 0 )z 0 (x)) ∈ O a,a , α 0 (x) ∈ [α, α], (3.3)
3. Regularity and growth of the pressure function: Pressure function is such that

p ∈ C([0, ∞) 2 ) ∩ C 1 ((0, ∞) 2 ), (3.5)
and there is a number C ≥ 1 (dependent on α, α, a a) such that for all (R, Z) ∈ O a,a

C -1 (R γ + Z β -1) ≤ p(R, Z) ≤ C(R γ + Z β + 1), (3.6)
with γ ≥ 9 5 , β > 0. Moreover

|∂ Z p(R, Z)| ≤ C(R -Γ + R Γ-1 ) in O a,a (3.7)
with some 0 ≤ Γ < 1, and with some 0 < Γ < γ + γ BOG , where γ BOG = min{ γ 2 , 2 3 γ -1}. Finally, the functions → p( , Z), Z > 0 resp. Z → ∂ Z p( , Z), > 0 are Lipschitz on (Z/a, Z/a)∩ (r, ∞) resp. (a , a ) ∩ (r, ∞) for all r > 0 with the Lipschitz constants

L p ≤ C(r)(1 + Z A ) resp. L P ≤ C(r)(1 + A ) (3.8)
with some non negative number A. The number C(r) may diverge to +∞ as r → 0 + .

4.

Structure of the pressure: It is assumed that

p(R, Rs) = P(R, s) -R(R, s), (3.9)
where [0, P (R, Rs) ≤ cR B with some c > 0 and B > 0.

∞) R → P(R,
(3.11)

5. Regularity and monotonicity of functions f and g: Functions f, g ∈ C 1 ((0, 1)) and they are both strictly monotone and non vanishing on the interval (0, 1).

The theorem on existence of weak solutions to the one velocity Baer-Nunziato type system (1.7-1.12) reads (cf. [22, Theorem 1]): Theorem 3.1. Under assumptions enumerated in items 1.-5. above, problem (1.7-1.12) admits at least one bounded energy weak solution (α, , z, u) with the densities ranging in O a,a and the concentration in [α, α] in the sense of Definition 2.1, where O a,a is defined in (3.1), satisfying moreover: for all t ∈ I, (f (α) (t, x), g(α)z(t, x)) ∈ O for a.a. x ∈ Ω, for all t ∈ I, α ≤ α(t, x) ≤ α for a.a.

x ∈ Ω, α, , z ∈ C([0, T ]; L 1 (Ω)), , z ∈ L 2 (Q T ), z ∈ C weak ([0, T ); L β (Ω)) if β > γ
, and P ( , Z) ∈ L q (I × Ω) for some q > 1.

Remark 3.1

1. A convenient Helmholtz function H(R, Z) corresponding to p in the energy inequality (2.6) in Theorem 3.1 can be calculated from the explicit formula

H(R, Z) = R R 1 p(s, s Z R ) s 2 ds, if R = 0, H(0, 0) = 0.
(3.12)

We notice that condition (3.11) guarantees namely the continuity of H in O a,a (i.e. notably at (0, 0)) and assumption (3.6) guarantees that its growth is the same as that one of p:

C(R γ + Z β -1) ≤ H(R, Z) ≤ C(R γ + Z β + 1) in O a,a . (3.13)
2. Pressure function p introduced in (1.14) and corresponding functions f, g originated in P ± introduced in formula (1.13) represent one example (among others) which satisfies all assumptions on p, f, g requested by Theorem 3.1.

3. The domain Ω in Theorem 3.1 can be taken Lipschitz. See [START_REF] Feireisl | On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic fluid[END_REF] for the methods allowing this generalization.

4. Theorem 3.1 holds also with slip (Navier) boundary conditions

u • n| ∂Ω = 0, (S(∇u)n) × n| ∂Ω = 0
instead of (1.12), provided the definition of weak solutions is modified accordingly, cf. [START_REF] Novotny | Weak solutions for a bi fluid model of a mixture of two compressible non interacting fluids[END_REF]. It also holds, with appropriate modifications, provided Ω is a periodic cell.

5. Existence theorems for (even simple) multifluid models are in a short supply in the mathematical literature. Theorem 3.1 is one of such examples. We refer to [START_REF] Maltese | Existence of weak solutions for compressible Navier-Stokes equations with entropy transport[END_REF], [START_REF] Vasseur | Global weak solution to the viscous two-fluid model with finite energy[END_REF], [START_REF] Bresch | Finite-Energy Solutions for Compressible Two-Fluid Stokes System[END_REF], [START_REF] Novotny | Weak solutions for some compressible multicomponent fluid models[END_REF] for another relevant examples.

Local in time existence of a strong solution

The final goal of the paper is to compare the weak solutions of the system (1.7-1.12) on (an arbitrary large) time interval [0, T ) -whose existence, under certain assumptions, is guaranteed by Theorem 3.1 -with a strong solution of the same system on the same time interval, provided the latter exists. The existence of strong solutions for the system (1.7-1.12) on an arbitrary large time interval is, however, not known. The aim of this section is to show that the strong solutions exist at least on a short time interval. This result is the subject of the following theorem.

Theorem 4.1. Let Ω ∈ C 3 be a bounded domain, 0 < r < r < ∞, 0 < z < z < ∞, 0 < β < β < 1 be constants. Assume that p ∈ C 2 ((0, ∞) 2 ), f, g ∈ C 2 ((0, 1
)) are non vanishing on (0, 1).

Suppose that

u 0 ∈ W 1,2 0 (Ω), β 0 , r 0 , z 0 ∈ W 2,2 (Ω), β ≤ β 0 ≤ β, r ≤ f (β 0 )r 0 ≤ r, z ≤ g(β 0 )z 0 ≤ z, 1 r 0 + z 0 -∇p(f (β 0 )r 0 , g(β 0 )z 0 ) + µ∆u 0 + (µ + λ)∇divu 0 -(r 0 + z 0 )u 0 ∇u 0 ∈ W 1,2 0 (Ω).
1. Then there exists an interval I * = [0, T * ) and numbers r, r, z, z, 0 < r < r < r < r < ∞, 0 < z < z < z < z < ∞ such that the problem (1.7-1.12) admits in the class

(β, r, z) ∈ C(I * ; W 2,2 (Ω)), ∂ t (β, r, z) ∈ C(I * ; W 1,2 (Ω)), (4.1)
u ∈ L 2 (I * ; W 3,2 (Ω; R 3 )), ∂ t u ∈ L 2 (I * ; W 2,2 (Ω, R 3 )), ∂ 2 t u ∈ L 2 (I * ; L 2 (Ω, R 3 )), β ≤ β ≤ β, r ≤ f (β)r ≤ r, z ≤ g(β)z ≤ z, (4.2)
u(0) = u 0 , u| (0,T )×∂Ω = 0 a unique strong solution (β, r, z, u). 2. If moreover bf (β 0 )r 0 ≤ g(β 0 )z 0 ≤ bg(β 0 )r 0 , with some 0 < b < b < ∞, then bf (β)r ≤ g(β)z ≤ bg(β)r. (4.3)
In view of what is known from the "mono-fluid" case, cf. [START_REF] Valli | Zajaczkovski Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of solutions in the general case[END_REF], [START_REF] Cho | Unique solvability of the initial boundary value problems for compressible viscous fluids[END_REF], Theorem 4.1 is not surprising. However, to the best of our knowledge, there is no reference to its proof, the closest relevant reference beeing [START_REF] Yao | Zi Incompressible limit of of viscous liquid-gas two-face flow model[END_REF]. We will perform the proof for the sake of completeness. In this proof, we follow closely the line of the proofs of local existence of weak solutions in the "mono-fluid" case, see e.g. [START_REF] Valli | Zajaczkovski Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of solutions in the general case[END_REF].

Proof of Theorem 4.1

We shall proceed in several steps.

Step 1: Strategy of the proof Given an interval

I = (a, b), -∞ < a < b < ∞, we define Banach spaces V (a,b) := {(β, r, z) ∈ C([a, b]; W 2,2 (Ω)), ∂ t (β, r, z) ∈ C([a, b]; W 1,2 (Ω))} Y (a,b) := {u ∈ L 2 (a, b; W 3,2 (Ω; R 3 )), ∂ t u ∈ L 2 (a, b; W 2,2 (Ω, R 3 )), ∂ 2 t u ∈ L 2 (a, b; L 2 (Ω, R 3
))} (they are endowed with the natural norms denoted β, , z V (a,b) and u Y (a,b) , respectively) and their (closed, convex) subsets

V (a,b) := {(β, r, z) ∈ V (a,b) | β ≤ β ≤ β, r ≤ f (β)r ≤ r, z ≤ g(β)z ≤ z}, Y (a,b),B = {u ∈ Y (a,b) | u(0) = u 0 , u| (a,b)×∂Ω = 0, u Y (a,b) ≤ B}.
In the above, the number B > 0 (and T > 0) will be specified later.

1. Let B > 0 be fixed and suppose that ũ ∈ Y (0,T ),B , T > 0 is given. We shall first solve the following system of transport equations:

∂ t β + ũ • ∇ β = 0, (4.4) ∂ t ˜ + div(˜ ũ) = 0, (4.5) ∂ t z + div(z ũ) = 0 (4.6)
with the initial conditions

β| t=0 = β 0 , ˜ | t=0 = r 0 , z| t=0 = z 0 , (4.7)
for the unknown functions ( β, ˜ , z) ∈ V (0,T ) . We shall show that there is a unique solution ( β, ˜ , z) ∈ V (0,T ) of equations (4.4-4.6) emanating from the initial conditions (β 0 , r 0 , z 0 ).

2. We shall next find u ∈ Y (0,T ) a solution of the following parabolic system:

(˜ + z)∂ t u -µ∆u + (µ + λ)∇divu = F (4.8)
with the right hand side given by

F = -(˜ + z)ũ • ∇ũ -∇p(f (α)˜ , g(α)z), (4.9)
with the initial conditions u| t=0 = u 0 , (4.10)

and the boundary conditions u| ∂Ω = 0. Step 2: Existence, uniqueness and estimates for the problem (4.4-4.7)

The vector field ũ can be extended for t < 0 in such a way that the new vector field (denoted again ũ) belongs to Y (-T,T ),B . It has enough regularity, so that a solution of system (4.4-4.7) can be found by the method of characteristics. Indeed, it can be shown by using contraction mapping argument and then continuation principle (while using the rules of differential calculus of composed mappings in Sobolev spaces, cf. Brezis, Bourgignon [START_REF] Bourgignon | Remarks on the Euler equation[END_REF]Appendix]) that the integral equation for characteristics

X(t; x) = x + t 0 ũ(s; X(s; x))ds (4.12) admits a unique solution X ∈ C([-T, T ]; W 3,2 (Ω)), ∂ t X ∈ Y (-T,T ) provided ũ ∈ Y (-T,T ),B and there is C > 0 such that X C([0,T ];W 3,2 (Ω)) ≤ C(1 + √ T ũ Y (-T,T ) ) and ∂ t X Y (-T,T ) ≤ C ũ Y (-T,T ) .
Moreover, this solution is such that X(t, •) as well as

X -1 (t, •) = X(-t, •) are bijections on Ω. It is well known that r(t, x) = r 0 (X(-t; x))exp - t 0 divũ(s; X(s -t; x))ds ∈ V (-T,T )
solves the continuity equation (4.5) resp. (4.6) -if r 0 stands for r 0 resp. for z 0 -a.e. in (-T, T ) × Ω (i.e. in particular, in Q T ) with the initial data r 0 resp. z 0 . Likewise,

β(t, x) = β 0 (X(-t, x))
solves the transport equation (4.4) a.e. in (-T, T ) × Ω (i.e. in particular, in Q T ) with the initial datum β 0 . We can readily derive from the above formulas the following estimates:

∀(t, x) ∈ Q T , β ≤ β(t, x) ≤ β, (4.13)
∀(t, x) ∈ Q T , r 0 exp -K √ T B ≤ ˜ (t, x) ≤ r 0 exp K √ T B , (4.14)
z 0 exp -K √ T B ≤ z(t, x) ≤ z 0 exp K √ T B
where r 0 , z 0 are (strictly positive) infimums of r 0 and z 0 over Ω while r 0 and z 0 are corresponding supremums. Here and hereafter, the positive number K is a universal constant dependent solely on the Sobolev imbeddings (and is, in particular, independent of T , B and of the initial data). Moreover, multiplying equation (4.5) by ˜ , ∇(4.5) by ∇˜ , ∇ 2 (4.5) by ∇ 2 ˜ (and effectuating the same operations with (4.6) and z), adding everything and integrating over (0, τ ) × Ω, we obtain, after a long calculation and an application of the Gronwall lemma (cf. e.g. [START_REF] Valli | Zajaczkovski Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of solutions in the general case[END_REF]Lemma 2

.4]), r L ∞ (0,T ;W 2,2 (Ω)) ≤ K r 0 W 2,2 (Ω) exp K √ T B , (4.15)
∂ t r L ∞ (0,T ;W 1,2 (Ω)) ≤ KB r 0 W 2,2 (Ω) exp K √ T B ,
where r stands for ˜ and z, respectively, while r 0 stands for r 0 and z 0 , respectively. In the derivation of these estimates, we have again used the rules [START_REF] Bourgignon | Remarks on the Euler equation[END_REF]Appendix]. By the same token,

β L ∞ (0,T ;W 2,2 (Ω)) ≤ K β 0 W 2,2 (Ω) exp K √ T B , (4.16
)

∂ t β L ∞ (0,T ;W 1,2 (Ω)) ≤ KB β 0 W 2,2 (Ω) exp K √ T B .
It is rudimentary to show that ( β, ˜ , z) are unique solutions to (4.4-4.7) in class V (0,T ) .

Seeing the regularity of f , g, ˜ , z and the range of β we deduce from (4.4-4.7) that ( β, f ( β)˜ , g( β)z) ∈ V (0,T ) and f( β)˜ , g( β)z satisfy equations

∂ t (f (α)˜ ) + div(f (α)˜ ũ) = 0, (4.17) ∂ t (g(α)z) + div(g(α)z ũ) = 0 (4.18)
with the initial data f (α 0 )r 0 , g(α 0 )z 0 , respectively. Consequently, in particular, for all (t, x)

∈ Q T , r exp -K √ T B ≤ f ( β)˜ (t, x) ≤ r exp K √ T B , (4.19)
z exp -K √ T B ≤ g( β)z(t, x) ≤ z exp K √ T B .
We thus observe that there is

T 1 = T 1 (B) > 0, qe KB √ T 1 ≤ q (4.20)
-where q is, in order, r, z, 1 r , 1 z , when q is, in order, r, z, 1 r , 1 z -such that for all T ∈ (0, T 1 ), ( β, ˜ , z) satisfies (4.2) on Q T . In the same manner, we get that this triplet satisfies (4.3) on Q T . Under these circumstances, we certainly have at least

( β, ˜ , z) V (0,T ) ≤ CP (I 0 , B) exp P (I 0 , B) √ T , (4.21)
where

I 0 = β 0 , r 0 , z 0 W 2,2 (Ω) + u 0 W 3,2 (Ω) .
Here and in the sequel, P is a polynomial, C > 0, and C and the coefficients of P are non negative independent of T , B, I 0 (but they may depend, in particular, on µ, λ, Ω, β, β, r, r, z, z, r, r, z, z,

sup (r,z)∈[r,r]×[z,z] |∇ r,z p(r, z)|, sup β∈[β,β] (|f (β)| + |g (β)|).
They can be different in different formulas.

Step 3: Existence, uniqueness and estimates for the equations (4.8-4.10) Existence of a unique solution to this system (with coefficients dependent on ( β, ˜ , z) ∈ V (0,T ) ) in the regularity class Y (0,T ) follows from the maximal regularity theory for parabolic equations (cf. e.g. Denk, Hieber, Prüss [START_REF] Denk | Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data[END_REF]), provided

F ∈ W (0,T ) ∩ C([0, T ]; W 1,2 (Ω)), W (0,T ) := {F ∈ L 2 ([0, T ]; W 1,2 (Ω)), ∂ t F ∈ L 2 (Q T )}.
Indeed, one can first solve the problem

(˜ + z)∂ t U -µ∆U -(µ + λ)∇divU + ∂ t (˜ + z)U = ∂ t F U| (0,T )×∂Ω = 0, U(0) = F(0) + µ∆u 0 + (µ + λ)∇divu 0 (r 0 + z 0 ) .
which is (formally) the problem deduced from (

U L ∞ (0,T ;W 1,2 (Ω)) + U L 2 (0,T ;W 2,2 (Ω)) + ∂ t U L 2 (Q T ) ≤ C 1 + P (I 0 , B) √ T exp P (I 0 , B)T e P (I 0 ,B) √ T ∂ t F L 2 (Q T ) + Q(I 0 ) .
Here and in the sequel Q is a polynomial of one variable with non negative coefficients that are independent of T , B and I 0 (but may depend on the same variables as the coefficients of the polynomial P , cf. to which we may apply the standard elliptic estimates, cf. [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. Resuming all these calculations, we infer that

u Y (0,T ) ≤ C 1 + P (I 0 , B) √ T exp P (I 0 , B)T e P (I 0 ,B) √ T F W (0,T ) + Q(I 0 ) , (4.22)
where

F W (0,T ) = ∂ t F L 2 (Q T ) + ∇F L 2 (Q T ) .
Step Choose small T 1 = T 1 (B) > 0 and T 2 = T 2 (B) small enough so that exp P (I 0 , B)T 1 e P (I 0 ,B)

√ T 1 ≤ 2 and CP (I 0 , B) T 2 1 + Q(I 0 ) + P (I 0 , B) T 2 ≤ 1 4 B.
Once B, T 1 and T 2 are fixed in this way, we take

T ≤ min{T 1 , T 2 }. (4.24)
With this choice of B and T , Φ(Y (0,T ),B ) ⊂ Y (0,T ),B and ( β, r, z) ∈ V (0,T ) . (4.25)

2. Φ :

Y (0,T ),B :→ C(0, T ; L 2 (Ω)) is continuous on Y (0,T ),B .
Let ũ1 , ũ2 ∈ Y (0,T ),B for T and B satisfying the hypothesis (4.24). Let ( βi , ˜ i , zi ), i = 1, 2, be the solutions of the transport equations:

∂ t βi + (ũ i • ∇) βi = 0, (4.26) ∂ t ˜ i + div(˜ i ũi ) = 0, ( 4
, ˜ i , zi ) ∈ V (0,T ) .
Finally let u 1 , u 2 ∈ Y (0,T ),B be the solutions to

(˜ i + zi )∂ t u i -µ∆u i + (µ + λ)∇divu i = F i (4.30)
with the initial conditions

u i | t=0 = u 0 (4.31)
and the boundary conditions

u i | ∂Ω = 0, (4.32)
where

F i = -(˜ i + zi )ũ i • ∇ũ i -∇p(f ( βi )˜ i , g( βi )z i ).
Their existence has been established in Step 3.

We shall now write the equations for the differences:

∂ t ( β1 -β2 ) + (ũ 1 • ∇)( β1 -β2 ) = -(ũ 1 -ũ2 ) • ∇ β2 , (4.33) ∂ t (˜ 1 -˜ 2 ) + div((˜ 1 -˜ 2 )ũ 1 ) = -div(˜ 2 (ũ 1 -ũ2 )), (4.34) ∂ t (z 1 -z2 ) + div((z 1 -z2 )ũ 1 ) = -div(z 2 (ũ 1 -ũ2 )), (4.35) (˜ 1 + z1 )∂ t (u 1 -u 2 ) -(µ∆(u 1 -u 2 ) + (µ + λ)∇div(u 1 -u 2 )) (4.36) = -(˜ 1 + z1 -˜ 2 -z2 )∂ t u 2 + F 1 -F 2 with the initial conditions α1 -α2 | t=0 = 0, ˜ 1 -˜ 1 | t=0 = 0, z 1 -z 2 | t=0 = 0, u 1 -u 2 | t=0 = 0 (4.37)
and the boundary conditions

(u 1 -u 2 )| ∂Ω = 0. (4.38)
Testing (meaning multiplying and then integrating over (0, τ ) × Ω, τ ∈ (0, T )) the equation (4.33) by β1 -β2 , the equation (4.34) by ˜ 1 -˜ 2 , the equation (4.35) by z1 -z2 (in order to evaluate β1 -β2 resp. ˜ 1 -˜ 2 resp. z1 -z2 in terms of the difference ũ1 -ũ2 ), the equation (4.36) scalarly by u 1 -u 2 , using many times Hölder's inequality, Young's inequality and the Poincaré inequality, exploiting the bounds (4.22), (4.25) and the structure of F (to evaluate F 1 -F 2 in terms of the differences of β1 -β2 , ˜ 1 -˜ 2 , z1 -z2 , ũ1 -ũ2 ), we obtain, after a straightforward but lengthy calculation

(r + z) (u 1 -u 2 )(τ ) 2 L 2 (Ω) + µ τ 0 ∇u 1 -u 2 2 L 2 (Ω) dt ≤ D τ 0 ∇u 1 -∇u 2 2 L 2 (Ω) dt,
where the number D may depend on the parameters of the problem listed in (4.21) (and also on T , B, I 0 , which are now, however, fixed according to (4.23-4.24)) but it is independent of βi , ˜ i , zi , ũi . The latter formula implies the continuity of Φ on Y (0,T ),B in the topology of C([0, T ]; L 2 (Ω)).

Conclusion

By virtue of the Lions-Aubin lemma, the set Y (0,T ),B is a compact subset of C([0, T ]; L 2 (Ω)). Since it is clearly also convex, we conclude that Φ admits a fixed point u ∈ Y (0,T ),B . According to Step 2., the solution (β, r, z) of system (4.26-4.28) with ũ = u belongs to V (0,T ) . We have thus constructed a (unique) strong solution (β, r, z, u) to problem (1.7-1.12) belonging to V (0,T ) ×Y (0,T ),B . Theorem 4.1 is proved.

Relative energy inequality(with the general test functions)

The goal of this section is to show that any bounded energy weak solution to the Bauer-Nunziato type system (1.7-1.12) is dissipative. We shall start with an auxiliary lemma dealing with the continuity and transport equations, cf.

(Ω)) ∩ L ∞ (I; L γ (Ω)) ∩ L 2 (Q T ), γ > 1, s ∈ C(I; L 1 (Ω)), for all t ∈ I, 0 ≤ s ≤ s(t, x) ≤ s < ∞ for a.a. x ∈ Ω, u ∈ L 2 (I; W 1,2 0 (Ω, R 3
)). Suppose that couple (r, u) satisfies continuity equation

Ω (rϕ)(τ, •) dx - Ω (rϕ)(0, •) dx = τ 0 Ω r∂ t ϕ + ru • ∇ϕ dx dt (5.1)
for all τ ∈ I and ϕ ∈ C 1 c (I × Ω) and couple (s, u) satisfies transport equation

Ω (sϕ)(τ, •) dx - Ω (sϕ)(0, •) dx = τ 0 Ω s∂ t ϕ + su • ∇ϕ -sdivuϕ dx dt (5.2)
for all τ ∈ I and ϕ ∈ C 1 c (I × Ω). Then for any b ∈ C 1 ([s, s]), rb(s) ∈ C(I; L 1 (Ω)) and couple (rb(s), u) satisfies continuity equation Let (α, , z, u) with ( , z) ∈ L 2 (Q T ) be a bounded energy weak solution to the problem (1.7-1.12) with the densities ranging in O and the concentration in [α, α] according to the Definition 2.1, where

Ω (rb(s)ϕ)(τ, •) dx - Ω (rb(s)ϕ)(0, •) dx = τ 0 Ω rb(s)∂ t ϕ + rb(s)u • ∇ϕ dx dt (5.
p ∈ C(O), H ∈ C(O) ∩ C 2 (O), f, g ∈ C 1 (0, 1) ∩ C 1 [α, α].
Then it is a dissipative solution of the same system. In particular, it satisfies the relative energy inequality (2.8) with any set of test functions (β, r, z, U) in the class (2.9).

Remark 5.1

As in the "mono-fluid" case, one can show by a density argument that the test functions (β, r, z, U) can be taken in a slightly broader class than (2.9), namely

(β, r, z, U) ∈ C([0, T ] × Ω), ∂ t β ∈ L 2 (Q T ), ∇β ∈ C(Q T ), ∂ t (r, z, U), ∇(r, z, U) ∈ L 2 (0, T ; C(Ω)), (5.4) α ≤ β ≤ α, (r(t, x), z(t, x)) ∈ O, U| ∂Ω = 0.
The rest of this section is devoted to the proof of Theorem 5.2

Proof of Theorem 5.2 Throughout this proof (β, r, z, U) is any quadruple in the regularity class (2.9). If we take in (2.2) r= and in (2.2) r=z , the function ϕ = |U| 2 2 as the test function, we obtain the identity

Ω ( + z) |U| 2 2 dx τ 0 = τ 0 Ω ( + z)U • ∂ t U + (u • ∇)U dxdt.
(5.5)

The equation (2.4) with the test function ϕ = U, reads

- Ω ( + z)u • Udx τ 0 = - τ 0 Ω ( + z)u • ∂ t U + (u • ∇)U +p(f (α) , g(α)z)divU -S(∇u) : ∇U dxdt. (5.6)
In view of Lemma 5.1, the equation (2.2) r= with the test function

ϕ = ∂ R H(f (β)r, g(β)z) yields, - Ω f (α)∂ R H(f (β)r, g(β)z)dx τ 0 = - τ 0 Ω f (α) ∂ t + (u • ∇) ∂ R H(f (β)r, g(β)z)dxdt, (5.7)
while, by the same token, the equation (2.2) r=z with the test function ϕ = ∂ Z H(f (β)r, g(β)z) gives, in particular,

- Ω zg(α)∂ Z H(f (β)r, g(β)z)dx τ 0 = - τ 0 Ω g(α)z ∂ t + (u • ∇) ∂ Z H(f (β)r, g(β)z)dxdt.
(5.8)

We deduce from the equation (2.5) written in the form

f (β)r∂ R H(f (β)r, g(β)z) + g(β)z∂ Z H(f (β)r, g(β)z) -H(f (β)r, g(β)z) = p(f (β)r, g(β)z), the integral identity Ω f (β)r∂ R H(f (β)r, g(β)z) + g(β)z∂ Z H(f (β)r, g(β)z) -H(f (β)r, g(β)z) dx τ 0 (5.9) = τ 0 Ω ∂ t p(f (β)r, g(β)z)dxdt.
Summing up the energy inequality (2.6) and the identities (5.5-5.9), we arrive at the inequality

Ω ( + z) 2 |u -U| 2 + H(f (α) , g(α)z) -H(f (β)r, g(β)z) +(f (β)r -f (α) )∂ R H(f (β)r, g(β)z) + (g(β)z -g(α)z)∂ Z H(f (β)r, g(β)z) dx τ 0 + τ 0 Ω S(∇u) : ∇(u -U)dxdt ≤ τ 0 Ω ( + z) ∂ t U + u • ∇U • (U -u)dxdt - τ 0 Ω p(f (α) , g(α)z)divUdxdt - τ 0 Ω f (α) ∂ t + u • ∇ ∂ R H(f (β)r, g(β)z)dxdt - τ 0 Ω g(α)z ∂ t + u • ∇ ∂ Z H(f (β)r, g(β)z)dxdt + τ 0 Ω ∂ t p(f (β)r, g(β)z)dxdt.
(5.10)

Adding to the both sides of (5.10) the term -

τ 0 Ω S(∇U) : ∇(u -U)dxdt one gets Ω ( + z) 2 |u -U| 2 + H(f (α) , g(α)z) -H(f (β)r, g(β)z) +(f (β)r -f (α) )∂ R H(f (β)r, g(β)z) + (g(β)z -g(α)z)∂ Z H(f (β)r, g(β)z) dx τ 0 + τ 0 Ω S(∇u -∇U) : ∇(u -U)dxdt ≤ τ 0 Ω ( + z)(U -u) • ∂ t U + u • ∇U + S(∇U) : ∇(U -u) dxdt - τ 0 Ω p(f (α) , g(α)z)divUdxdt - τ 0 Ω f (α) ∂ t + u • ∇ ∂ R H(f (β)r, g(β)z)dxdt - τ 0 Ω g(α)z ∂ t + u • ∇ ∂ Z H(f (β)r, g(β)z)dxdt + τ 0 Ω ∂ t p(f (β)r, g(β)z)dxdt. (5.11)
Observing that

∂ t H(f (β)r, g(β)z) = ∂ R H(f (β)r, g(β)z)∂ t (f (β)r) + ∂ Z H(f (β)r, g(β)z)∂ t (g(β)z)
we verify that

∂ t p(f (β)r, g(β)z) = ∂ t -H((f (β)r, g(β)z) + f (β)r∂ R H(f (β)r, g(β)z) + g(β)z∂ Z H(f (β)r, g(β)z) = f (β)r∂ t ∂ R H(f (β)r, g(β)z) + g(β)z∂ t ∂ Z H(f (β)r, g(β)z).
(5.12)

Likewise, observing that

∇H(f (β)r, g(β)z) = ∂ R H(f (β)r, g(β)z)∇(f (β)r) + ∂ Z H(f (β)r, g(β)z)∇(g(β)z),
we verify the identity

- Ω p(f (β)r, g(β)z)divUdx = Ω H(f (β)r, g(β)z) -f (β)r∂ R H(f (β)r, g(β)z)-g(β)z∂ Z H(f (β)r, g(β)z) divUdx = - Ω U • ∇ H(f (β)r, g(β)z) -f (β)r∂ R H(f (β)r, g(β)z)-g(β)z∂ Z H(f (β)r, g(β)z) dx = Ω f (β)rU • ∇∂ R H(f (β)r, g(β)z) + g(β)zU • ∇∂ Z H(f (β)r, g(β)z) dx.
(5.13) Employing (5.12) and (5.13) we finally arrive at

- Ω p(f (α) , g(α)z)divUdx - Ω f (α) ∂ t + u • ∇ ∂ R H(f (β)r, g(β)z)dx - Ω g(α)z ∂ t + u • ∇ ∂ Z H(f (β)r, g(β)z)dx + Ω ∂ t p(f (β)r, g(β)z)dx = Ω f (β)r -f (α) ∂ t ∂ R H(f (β)r, g(β)z) + g(β)z -g(α)z ∂ t ∂ Z H(f (β)r, g(β)z)dx + Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z) divUdx + Ω f (β)rU -f (α) u • ∇∂ R H(f (β)r, g(β)z) + g(β)zU -g(α)zu • ∇∂ Z H(f (β)r, g(β)z)dx.
(5.14)

With this identity at hand, the right hand side of (5.11) can be rewritten as follows

τ 0 Ω ( + z)(U -u) • ∂ t U + u • ∇U + S(∇U) : ∇(U -u) dxdt + τ 0 Ω f (β)r -f (α) ∂ t ∂ R H(f (β)r, g(β)z)dxdt + τ 0 Ω g(β)z -g(α)z ∂ t ∂ Z H(f (β)r, g(β)z)dxdt + τ 0 Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z) divUdxdt + τ 0 Ω f (β)rU -f (α) u • ∇∂ R H(f (β)r, g(β)z)dxdt + τ 0 Ω g(β)zU -g(α)zu • ∇∂ Z H(f (β)r, g(β)z)dxdt. (5.15)
This completes the proof of Theorem 5.2.

Relative energy inequality with the strong solution

We employ in the relative energy inequality (2.8) the test functions (β, r, z, U), where (β, r, z, U) is a strong solution of system (1.7-1.12) in the class (2.9) with the initial data (β 0 , r 0 , z 0 , U 0 ). In this case, the remainder in the relative energy inequality becomes quadratic in the difference of the weak and the strong solution. This property is formulated rigorously in the following theorem: Theorem 6.1. Let all assumptions of Theorem 5.2 be satisfied and suppose moreover that (β, r, z, U) is a strong solution to the one velocity Baer-Nunziato type system (1.7-1.12) in the class (2.9) emanating from the initial data (β 0 , r 0 , z 0 , U 0 ). Then the remainder in the relative energy inequality (2.8) takes the form:

R α,β ( , z, u|r, z, β, U) = Ω (U -u) • ( + z -r -z)∂ t U + ( + z)u -(r + z)U • ∇U dx + Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z)-∂ R p(f (β)r, g(β)z)(f (β)r -f (α) ) -∂ Z p(f (β)r, g(βz)(g(β)z -g(α)z) divUdx + Ω f (α) -f (β)r (U -u) • ∇∂ R H(f (β)r, g(β)z)dx + Ω g(α)z -g(β)z (U -u) • ∇∂ Z H(f (β)r, g(β)z)dx. (6.1)
Remark 6.1

The strong solution in Theorem 6.1 can be taken in the class (5.4) which is slightly broader than (2.9), cf. Remark 5.1.

The rest of this section will be devoted to the proof of Theorem 6.1. Proof of Theorem 6.1

Recall that R α,β ( , z, u|r, z, U) = Ω (U -u) • ( + z) ∂ t U + (u • ∇)U -divS(∇U) dx + Ω f (β)r -f (α) ∂ t ∂ R H(f (β)r, g(β)z)dx + Ω g(β)z -g(α)z ∂ t ∂ Z H(f (β)r, g(β)z)dx + Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z) divUdx 20 + Ω f (β)rU -f (α) u • ∇∂ R H(f (β)r, g(β)z)dx + Ω g(β)zU -g(α)zu • ∇∂ Z H(f (β)r, g(β)z)dx = I + II + III + IV + V + V I, (6.2)
where we have used (2.8) and the integration by parts which gives the identity

Ω (U -u) • divS(∇U)dx = -Ω S(∇U) : ∇(U -u)dx. Indeed, in view of (2.9), divS(∇U) ∈ C([0, T ] × Ω) by virtue of (1.10). Since (r + z)(∂ t U + U • ∇U) -divS(∇U) + ∇p(f (β)r, g (β)
z) = 0, I can be rewritten as

I = Ω (U -u) • ( + z) ∂ t U + u • ∇U -(r + z)(∂ t U + U • ∇U) -∇p(f (β)r, g(β)z) dx = Ω (U -u) • ( + z -r -z)∂ t U + ( + z)u -(r + z)U • ∇U dx - Ω (U -u) • ∇p(f (β)r, g(β)z)dx = I 1 + I 2 . (6.3)
Since ∂ t (f (β)r) + div(f (β)rU) = 0 and ∂ t (g(β)z) + div(g(β)zU) = 0, we have

∂ t ∂ R H(f (β)r, g(β)z) = -U • ∇∂ R H(f (β)r, g(β)z) -∂ R p(f (β)r, g(β)z)divU and
∂ t ∂ Z H(f (β)r, g(β)z) = -U • ∇∂ Z H(f (β)r, g(β)z) -∂ Z p(f (β)r, g(β)z)divU.
Here we used the fact that

R∂ 2 R H(R, Z) + Z∂ Z ∂ R H(R, Z) = ∂ R p(R, Z) and R∂ R ∂ Z H(R, Z) + Z∂ 2 Z H(R, Z) = ∂ Z p(R, Z).
Hence II and III can be rewritten as follows

II = - Ω f (β)r -f (α) U • ∇∂ R H(f (β)r, g(β)z)dx - Ω f (β)r -f (α) ∂ R p(f (β)r, g(β)z)divUdx = II 1 + II 2 (6.4)
and

III = - Ω g(β)z -g(α)z U • ∇∂ Z H(f (β)r, g(β)z)dx - Ω g(β)z -g(α)z ∂ Z p(f (β)r, g(β)z)divUdx = III 1 + III 2 . (6.5)
Combining IV , II 2 and III 2 , we have

IV + II 2 + III 2 = Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z) -∂ R p(f (β)r, g(β)z)(f (β)r -f (α) )-∂ Z p(f (β)r, g(β)z)(g(β)z -g(α)z) divUdx. (6.6)
Observe that

V + II 1 = Ω f (α) (U -u) • ∇∂ R H(f (β)r, g(β)z)dx (6.7)
and

V I + III 1 = Ω g(α)z(U -u) • ∇∂ Z H(f (β)r, g(β)z)dx. (6.8) Since ∇H(f (β)r, g(β)z) = ∂ R H(f (β)r, g(β)z)∇(f (β)r) + ∂ Z H(f (β)r, g(β)z)∇(g(β)z), we get ∇p(f (β)r, g(β)z) = ∇ -H(f (β)r, g(β)z) + f (β)r∂ R H(f (β)r, g(β)z) + g(β)z∂ Z H(f (β)r, g(β)z) = f (β)r∇∂ R H(f (β)r, g(β)z) + g(β)z∇∂ Z H(f (β)r, g(β)z).
Hence, we have

V + II 1 + V I + III 1 + I 2 = Ω f (α) -f (β)r (U -u) • ∇∂ R H(f (β)r, g(β)z)dx + Ω g(α)z -g(β)z (U -u) • ∇∂ Z H(f (β)r, g(β)z)dx. (6.9)
From I 1 in (6.3), (6.6) and (6.9) we conclude that

R α,β ( , z, u|r, z, U) = Ω (U -u) • ( + z -r -z)∂ t U + ( + z)u -(r + z)U • ∇U dx + Ω p(f (β)r, g(β)z) -p(f (α) , g(α)z)-∂ R p(f (β)r, g(β)z)(f (β)r -f (α) ) -∂ Z p(f (β)r, g(β)z)(g(β)z -g(α)z) divUdx + Ω f (α) -f (β)r (U -u) • ∇∂ R H(f (β)r, g(β)z)dx + Ω g(α)z -g(β)z (U -u) • ∇∂ Z H(f (β)r, g(β)z)dx. (6.10)
Theorem 6.1 is proved.

The weak-strong uniqueness

We start with the following simple algebraic lemma dealing with functions of two variables which will be systematically used in the weak-strong uniqueness proof.

Lemma 7.1. Let L, K ⊂ O be two compact sets such that K ⊂ intL, where O is a convex subset of (0, ∞) 2 . Let H ∈ C(O)∩C 2 (O) have a (strictly) positive Hessian matrix. Suppose that there are numbers C > 0 and R > 1 such that

∀(R, Z) ∈ O, |R| + |Z| ≥ R, R ξ + Z ξ + p(R, Z) ≤ CH(R, Z) with some ξ > 1. (7.1)
Then there exists a positive number c = c(K, L, δ, C, R) such that for all (R, Z) ∈ K and all (R, Z) ∈ O,

(R -R) 2 + (Z -Z) 2 1 L (R, Z) + 1 + R ξ + Z ξ + p(R, Z) 1 O\L (R, Z) ≤ cE(R, Z | R, Z).
Recall that E is defined in (2.7).

Proof of Lemma 7.1

We have for the Hessian of H,

∀h ∈ R 2 , inf (R,Z)∈L h T D 2 H(R, Z)h ≥ c|h| 2 with some c > 0.
Consequently, the inequality

(R -R) 2 + (Z -Z) 2 1 L (R, Z) ≤ cE(R, Z | R, Z)
is a consequence of (2.7) and the second order Taylor formula. Due to the strict convexity of H the map (R, Z) → E(R, Z|R, Z) has in O a unique global minimum in the point (R, Z) ∈ K, which is equal to 0. Consequently, there is c > 0 such that inf (R,Z)∈O\L,(R,Z)∈K E(R, Z|R, Z) ≥ c (7.2) so that we have for all (R, Z)

∈ K max (R,Z)∈O∩[0,R] 2 p(R, Z) := p ≤ p c E(R, Z|R, Z).
This yields the inequality

1 O\L p(R, Z) ≤ cE(R, Z | R, Z). (7.3)
Finally, by virtue of the assumption (7.1), we have for all R + Z ≥ R,

p(R, Z) + R ξ + Z ξ ≤ CE(R, Z|R, Z) + C ∂ R H(R, Z)(R -R) +∂ Z H(R, Z)(Z -Z) + H(R, Z) ≤ CE(R, Z|R, Z) + A(R + Z) + B.
where we have denoted

A := C max (R,Z)∈K ∂ R H(R, Z) + ∂ R H(R, Z) B := C max (R,Z)∈K ∂ R H(R, Z)R + ∂ Z H(R, Z)Z + H(R, Z) .
Now we may employ at the right hand side of the last inequality the Young inequality in order to "absorb" the term A(R + Z) at the left hand side, and then use estimate (7.2). This yields, in particular,

p(R, Z) + R ξ + Z ξ 1 O\L ≤ CE(R, Z|R, Z)
and finishes the proof of Lemma 7.1.

Next theorem compares a weak solution on interval (0, T ) emanating from the initial data (α 0 , 0 , z 0 , u 0 ) with a strong solution on the same interval emanating from the initial data (β 0 , r 0 , z 0 , U 0 ) (provided it exists on interval [0, T )). It yields, in particular, the weak-strong uniqueness principle for the one velocity Baer-Nunziato type system (1.7-1.12). Theorem 7.2. Let Ω be a bounded Lipschitz domain. Let O be an open convex subset of (0, ∞)

2 , 0 ≤ α < α ≤ 1. Assume that f, g ∈ C 1 (0, 1) ∩ C 1 ([α, α]), p ∈ C(O) ∩ C 2 (O), H ∈ C(O) ∩ C 2 (O)
where the Hesian matrix of H is (strictly) positive on O and satisfies relation (7.1), and f, g are nonvanishing on [α, α].

Let (α, , z, u) be a weak solution with densities ranging in O and concentration in [α, α] according to Definition 2.1 emanating from initial data

α ≤ α 0 ≤ α, (f (α 0 ) 0 , g(α 0 )z 0 ) ∈ O, 0 ∈ L γ (Ω), γ > 1, u 0 ∈ L 1 (Ω), ( 0 + z 0 )u 2 0 ∈ L 1 (Ω).
Let (β, r, z, U) be a strong solution of the same system in the class (5.4) emanating from initial data (β 0 , r 0 , z 0 , U 0 ) = (β(0), r(0), z(0), U(0)). Then there exists a number C > 0 independent of the weak solution and its initial data (but dependent possibly on the strong solution through the norms, parameters and sets indicated in (5.4)), such that for a.a. τ ∈ (0, T ),

E α, , z, u β, r, z, U (τ ) + 1 2 τ 0 Ω S(∇(u -U)) : ∇(u -U) dxdt ≤ CE α 0 , 0 , z 0 , u 0 β 0 , r 0 , z 0 , U 0 , where E α, , z, u β, r, z, U := Ω 1 2 |α -β| 2 + 1 2 ( + z)|u -U| 2 + E(f (α) , g(α)z | f (β)r, g(β)z) (τ, x) dx
In particular, if (β 0 , r 0 , z 0 , U 0 ) = (α 0 , 0 , z 0 , u 0 ) then (α, , z, u) = (β, r, z, U).

L := {(R, Z) | Rb/2 ≤ Z ≤ 2bR, r/2 ≤ R≤2r}.
Clearly, this set is compact in O. The couple (f (β)r, g(β)z) created from the strong solution (β, r, z, U) is ranging, according to Theorem 4.1, in the compact set

K = {(Z, R)|bR ≤ Z ≤ bR, r ≤ R≤r}
which is included in the interior of the set L.

2. Theorem 7.2 is valid also with the slip (Navier) boundary conditions for velocity, or if Ω is a periodic cell (with periodic boundary conditions), cf. Remark 3.1.

3. Function p with γ ± > 1 introduced in formula (1.14) provides an example of a pressure stemming from partial pressure constitutive laws P ± , cf. (1.13), which satisfies together with its Helmoholtz function H given in (3.12) all assumptions of Theorem 7.2.

Before starting the proof of Theorem 7.2, we shall need an auxiliary lemma dealing with the estimates of differences of concentrations.

Lemma 7.3. Let (α, u), (β, U), α, β ∈ L ∞ (Q T ) ∩ C(I; L 1 (Ω)), u, U ∈ L 2 (I; W 1,2 0 (Ω))
be solutions of the pure transport equation in D (Q T ), cf. (1.7). Suppose moreover that

∇β ∈ L ∞ (Q T ), divU ∈ L 1 (I; L ∞ (Ω)).
Then for any δ > 0 and for all τ ∈ I,

Ω (α -β) 2 (τ, •) dx - Ω (α -β) 2 (0, •) dx ≤ δ τ 0 u -U 2 W 1,2 (Ω) dt + c δ τ 0 a(t) Ω (α -β) 2 dxdt, (7.4) where a = α 2 L ∞ (Ω) + β 2 L ∞ (Ω) + ∇β 2 L ∞ (Ω) + divU L ∞ (Ω) ∈ L 1 (0, T ) and c > 0 is a universal constant.
Proof of Lemma 7.3. To prove the lemma we shall use the nowadays classical DiPerna-Lions regularizing technique, cf. [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]. According to the assumptions, the couples (α, u) and (β, U) satisfy

∂ t α + u • ∇α = 0 in D (Q T ), ∂ t β + u • ∇β = (u -U) • ∇β a.e. in Q T .
Since u admits a zero trace on ∂Ω, the first equation holds in D (I × R 3 ) provided we extend u and α by 0 outside Ω. We may therefore regularize the first equation by using the standard mollifiers over the space variable in order to get, in particular,

∂ t [α] ε + u • ∇[α] ε = r ε ( , u) a.e. in Q T ,
where [α] ε denotes the mollified α and r ε ( ,

u) = u • ∇[α] ε -[u • ∇α] ε .
Combining the third and the second equation, we deduce

∂ t ([α] ε -β) + U • ∇([α] ε -β) = r ε + f ε a.e.in Q T , where f ε = (U -u) • ∇[α] ε . Multiplying the latter identity by [α] ε -β and integrating over (0, τ ) × Ω, we get 1 2 Ω ([α] ε -β) 2 (τ ) dx - 1 2 Ω ([α] ε -β) 2 (0) dx (7.5) - 1 2 τ 0 Ω divU([α] ε -β) 2 dxdt = τ 0 Ω (r ε + f ε )([α] ε -β) dxdt
for all τ ∈ I.

Due to the chain of identities,

Ω f ε ([α] ε -β) dx = 1 2 Ω (U -u) • ∇([α ε ] -β) 2 dx + Ω (U -u) • ∇β([α ε ] -β) dx = - 1 2 Ω div(U -u)([α ε ] -β) 2 dx + Ω (U -u) • ∇β([α ε ] -β) dx, we have Ω f ε ([α ε ] -β) dx ≤ δ 2 div(U -u) 2 L 2 (Ω) + δ 2 U -u 2 L 2 (Ω) + 1 δ ( [α] ε -β 2 L ∞ (Ω) + ∇β 2 L 2 (Ω) ) Ω ([α ε ] -β) 2 dx,
with any δ > 0, where we have used the Hölder and Young inequalities, and where

[α] ε L ∞ (Ω) ≤ α L ∞ (Ω)
. Moreover, by virtue of the Friedrichs lemma about commutators, cf. [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF],

r ε → 0 in L 2 (Q T ) as ε → 0+; whence τ 0 Ω r ε ([α ε ] -β) dx → 0 as ε → 0+.
Consequently, the identity (7.5) yields the inequality

1 2 Ω ([α] ε -β) 2 (τ ) dx - 1 2 Ω ([α] ε -β) 2 (0) dx ≤ δ 2 τ 0 div(U -u) 2 L 2 (Ω) dt + δ 2 τ 0 U -u 2 L 2 (Ω) dt + τ 0 1 δ ( [α] ε -β 2 L ∞ (Ω) + ∇β 2 L 2 (Ω) ) + divU L ∞ (Ω) Ω ([α] ε -β) 2 dx dt.
We get the inequality (7.4) after the limit passage ε → 0 in the latter relation. The proof of Lemma 7.3 is complete.

Proof of Theorem 7.2.

We define the essential and residual sets corresponding to (α, , z) with respect to compact set L as follows Ω ess (t We are now in position to estimate the right hand side of the relative energy inequality (6.1). We shall do it in four steps.

) := {x ∈ Ω | (f (α) (t), g(α)z(t)) ∈ L}, Ω res (t) := Ω \ Ω ess (t). ( 7
Step 1: The material derivative term (the first term). First, we split the first term in the remainder (6.1) as follows: In order to handle the first term on the right hand side of (7.8), we write . We shall treat separately the "essential part" and the "residual part". We estimate the essential part as follows where a is given in (7.12). Finally, where δ > 0 is an arbitrary number, c = c(f, g, α, α, r, z, r, z) > 0, and

τ 0 Ω (U -u) • ( + z -r -z)∂ t U + ( + z)u -(r + z)U • ∇U dxdt (7.8)
-r = 1 f (α) f (α) -f (β)r + 1 f (α) - 1 f (β) r
τ 0 Ω 1 { >D} ( )[1] res ( -r)(∂ t U + U • ∇U) • (U -u) dxdt ≤ 2 τ 0 Ω 1 { >D} ( )[1] res √ ∂ t U + U • ∇U √ U -u dxdt ≤ τ 0 ∂ t U + U • ∇U L ∞ (Ω;R 3 ) [ ] res 1/2 L 1 (Ω) u -U 2
a = ∇U L ∞ (Ω;R 9 ) + ∂ t U + U • ∇U 2 L ∞ (Ω;R 3 ) ∈ L 1 (0, T ).
Step 2:The pressure/divergence U term (the second term) Similarly as in the first step, we split the second term in the remainder (6.1) to the esssential and residual parts. The essential part is bounded by with any δ > 0, where a(t) = 1

(4. 11 ) 3 .

 113 Next, we shall consider the map Φ : Y (0,T ),B → C([0, T ]; L 2 (Ω)), u = Φ(ũ). (a) We shall show that this map maps the set Y (0,T ),B into itself for any fixed B > 0 and some T = T (B) > 0 sufficiently small. (b) We shall show that this map is continuous. Since the set Y (0,T ),B is a closed convex subset of the space C([0, T ]; L 2 (Ω)), Φ admits a fixed point u ∈ Y (0,T ),B by the Schauder fixed point theorem. 4. Finally, define the map A : Y (0,T ),B → V (0,T ) by A(ũ) = ( β, ˜ , z) and set A(u) = (β, r, z), where u ∈ Y (0,T ),B is a fixed point of Φ. Then (β, r, z, u) is a solution of the nonlinear problem (1.7-1.12).

0 U

 0 (4.21)). It may be different in different formulas. Then u(t, •) = t (s, •)ds satisfies equation (4.8), i.e.µ∆u + (µ + λ)u = U -F, u| ∂Ω = 0,

4 :

 4 Fixed point of the map Φ 1. Φ maps Y (0,T ),B into itself In view of the form of F , we find easily (but laboriously) that for all ũ ∈ Y (0,T ),B F W (0,T ) ≤ P (I 0 , B) √ T . Revisiting (4.22) with this information, we get u Y (0,T ) ≤ C 1 + P (I 0 , B) √ T exp P (I 0 , B)T e P (I 0 ,B) √ T Q(I 0 ) + P (I 0 , B) √ T .For given initial data (characterized by I 0 ), we can thus choose B in such a way that B/4 ≥ CQ(I 0 ).(4.23)

. 27 )

 27 ∂ t zi + div(z i ũi ) = 0 (4.28) with the initial conditions βi | t=0 = β 0 , ˜ i | t=0 = r 0 , zi | t=0 = z 0 . (4.29) Their existence and uniqueness has been established in Step 2. According to (4.25), ( βi

[ 24 ,Lemma 5 . 1 .

 2451 Proposition 6],[START_REF] Novotny | Weak solutions for a bi fluid model of a mixture of two compressible non interacting fluids[END_REF] Proposition 4]. Let Ω be a bounded Lipschitz domain. Let r ∈ C(I; L 1

3 )

 3 for all τ ∈ I and ϕ ∈ C 1 c (I × Ω) With Lemma 5.1 at hand, we can prove the main theorem of this section. Theorem 5.2. Let Ω be a bounded Lipschitz domain. Let O ⊂ (0, ∞) 2 be an open set and 0 ≤ α < α ≤ 1.

Remark 7. 1 1.

 1 The assumptions of Theorem 7.2 are not void. Indeed, there is at least one setting (characterized by O) in which the weak solutions exist on an arbitrary large interval I and the strong solutions exist at least on a short time interval I * . Indeed: The weak solutions satisfying the assumptions of Theorem 7.2 on an arbirary large interval I = (0, T ) with the convex set O = O a,a , 0 < a < a defined in (3.1) have been constructed[22, Theorem 1] provided 0 < α < α < 1 and f, g are strictly monotone non vanishing on (0, 1). Their existence is recalled in Theorem 3.1 (and Remark 3.1). The strong solutions satisfying the assumptions of Theorem 7.2 have been constructed on a short time interval I * = [0, T * ) in Theorem 4.1, provided the domain Ω is of class C 3 and the initial data are sufficiently regular and verify the compatibility conditions at the boundary. Indeed, if we take in Theorem 4.1 β = α, β = α and 0 < b < b < a/2 we may define the set L ⊂ O in Theorem 7.2 as

  u) • ( + z -rz)∂ t U + ( + z) -(r + z) U • ∇U dxdt + τ 0 Ω ( + z)(u -U) • ∇U • (U -u)dxdt, )(u -U) • ∇U • (U -u)dxdt (7.9) ≤ τ 0 a(t)E α, , z, u β, r, z, U dt, where a = 2 ∇U L ∞ (Ω;R 9 ) ∈ L 2 (0, T ).

  meaning that | -r| ≤ c f (α) -f (β)r + |α -β| ,(7.10)where c > 0 depends on α, α, r,max ζ∈[α,α] f (ζ) f 2 (ζ). Similarly,|z -z| ≤ c g(α)z -g(β)z + |α -β|(7.11)where c > 0 depends on α, α, z, max ζ∈[α,α]g (ζ) g 2 (ζ)

[ 1 ]≤ c τ 0 ∂ 0 Ω[ 1 ] 0 Ω[ 1 ][ 1 ][ 1 ]2D τ 0 ∂

 100101110 ess ( -r) + (z -z) ∂ t U + U • ∇U • (U -u) dxdt (7.12) t U + U • ∇U L ∞ (Ω;R 3 ) u -U L 2 (Ω;R 3 ) × f (α) -f (β)r L 2 (Ωess(t)) + g(α)z -g(β)z L 2 (Ωess(t)) + α -β L 2 (Ω) dt )E α, , z, u β, r, z, U dt, where a = ∂ t U + U • ∇U 2 L ∞ (Ω;R 3 ) ∈ L 1 (0, T ).Concerning the residual part, we may writeτ res ( -r) + (z -z) ∂ t U + U • ∇U • (U -u) dxdt τ res ( -r)1 { ≤D} ( ) + ( -r)1 { >D} ( ) + (z -z)1 { ≤D} (z) + (z -z)1 {r>D} (z) × ∂ t U + U • ∇U • (U -u)dxdtwith any D > 0 which we take sufficiently large (larger than r+z). The typical terms of this development will be estimated as follows:res 1 { ≤D} ( )( -r)(∂ t U + U • ∇U) • (U -u) dxdt res ∂ t U + U • ∇U Uu dxdt ≤ t U + U • ∇U L ∞ (Ω;R 3 ) [1] res L 2 (Ω) )E α, , z, u β, r, z, U dt,

1 / 2 L 1 (Ω) dt ≤ c τ 0 a 7

 12107 (t)E α, , z, u β, r, z, U dt with the same a as before. In all above three formulas, we have employed Lemma 7.1 in the passage to their last lines.Resuming the first step, the material derivative term obeys the boundτ 0 Ω (U -u) • ( + z -rz)∂ t U + ( + z)u -(r + z)U • ∇U dxdt ()E α, , z, u β, r, z, U dt,

[ 1 ]c τ 0 Ω[ 1 ] 2 + 2 dxdt ≤ c τ 0 a[ 1 ][ 1 ]Step 3 : 0 u -U 2 L 2 (

 101220113022 ess p(f (β)r, g(β)z) -p(f (α) , g(α)z)-∂ R p(f (β)r, g(β)z)(f (β)r -f (α) )-∂ Z p(f (β)r, g(β)z)(g(β)z -g(α)z) dxdt ≤ ess f (α) -f (β)r) g(α)z -f (β)z) (t)E(α, , z, u|β, r, z, U)dtby virtue of the second order Taylor formula, where we have used Lemma 7.1 to get the last inequality.Employing Lemma 7.1 (and the Young inequality), we deduce the pointwise bound, res p(f(β)r, g(β)z) -p(f (α) , g(α)z)-∂ R p(f (β)r, g(β)z)(f (β)r -f (α) ) -∂ Z p(f (β)r, g(β)z)(g(β)z -g(α)z) ≤ c[1] res E(α, , z, u|β, r, z, U)in order to estimate the residual part, res p(f(β)r, g(β)z) -p(f (α) , g(α)z)-∂ R p(f (β)r, g(β)z)(f (β)r -f (α) ) -∂ Z p(f (β)r, g(β)z)(g(β)z -g(α)z) dxdt ≤ c τ 0 a(t)E(α, , z, u|β, r, z, U)dt.In the above two formulas, c is a positive number dependent of R, r, r, z, z and|∂ R p| C 1 (L) + |∂ Z p| C 1 (L) (β)r, g(β)z) -p(f (α) , g(α)z)-∂ R p(f (β)r, g(β)z)(f (β)r -f (α) ) -∂ Z p(f (β)r, g(β)z)(g(β)z -g(α)z) dxdt ≤ c τ 0 a(t)E(α,, z, u|β, r, z, U)dt (7.14) The terms containing the Helmholtz function (the third and fourth terms) By the same token as in Steps 1 and 2,τ0 Ω f (α) -f (β)r (U -u) • ∇∂ R H(f (β)r, g(β)z)dxdt + Ω g(α)z -g(β)z (U -u) • ∇∂ Z H(f (β)r, g(β)z)dx ≤δ τ )E(α, , z, u|β, r, z, U)dt, (7.15)

  4.8-4.11) for U = ∂ t u by using [8, Theorem 2.1]. One obtains for this problem estimates by testing the above evolution equation first by U, then by -µ∆U -(µ + λ)∇divU and finally by ∂ t U and employing estimate (4.21). They are resumed in the following formula (see Valli, Zajaczkowski [27, Theorem 2.4] for details),

∈ L γ (Ω), z 0 ∈ L β (Ω) if β > γ, ( 0 + z 0 )|u 0 | 2 ∈ L

(Ω).(3.4)

* Bum Ja Jin was supported by the grant NRF-2016R1D1A1B03934133 1

and c > 0 depends on α, α, r, r, z, z and |∂ R H| C 1 (L) + |∂ Z H| C 1 (L) .

Step 4: Conclusion

Summarizing Steps 1-3, we get the following bound for the remainder (6.1),

Coming back with this estimate to relative energy inequality (2.8) with the remainder given by (6.1) and adding to it the inequality (7.4) we end up with the relative energy inequality presented in Theorem 7.2. This completes the proof.


Paul Monsarrat
email: paul.monsarrat@univ-tlse3.fr

Gabriel Fernandez De Grado

Arnaud Constantin

Claire Willmann

Cathy Nabet

Michel Sixou

Alain Cantagrel

Thomas Barnetche

Nadia Mehsen

Thierry Schaeverbeke

Elise Arrivé

Jean-Noel Vergnes

Easpera Group

Sandrine Borg- Santos

Marie-Hélène Cerato

Michel Laroche

Delphine Nigon

The effect of periodontal treatment on patients with rheumatoid arthritis

Keywords: Periodontal Diseases, Periodontal Medicine, Periodontitis, Randomised Controlled Trial, Rheumatoid Arthritis, Quality of Life

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

Introduction

Periodontitis is the most prevalent chronic immune/infectious disease and the global prevalence of severe periodontitis is estimated at 10% [START_REF] Papapanou | Periodontitis epidemiology: is periodontitis under-recognized, over-diagnosed, or both?[END_REF]. This disease is characterised by the loss of tooth-supporting tissues [START_REF] Papapanou | Periodontitis epidemiology: is periodontitis under-recognized, over-diagnosed, or both?[END_REF]. If left untreated, periodontitis can result in tooth loss with a significant negative impact on oral health and overall quality of life [START_REF] Kassebaum | Global, Regional, and National Prevalence, Incidence, and Disability-Adjusted Life Years for Oral Conditions for 195 Countries, 1990-2015: A Systematic Analysis for the Global Burden of Diseases, Injuries, and Risk Factors[END_REF]. Dozens of systemic diseases have been believed to be associated with periodontitis [START_REF] Monsarrat | Clinical research activity in periodontal medicine: a systematic mapping of trial registers[END_REF] and the potential link between rheumatoid arthritis (RA) and periodontitis has been brought to light over the past decades [START_REF] Monsarrat | Clinical research activity in periodontal medicine: a systematic mapping of trial registers[END_REF]. RA is a chronic autoimmune disease with a prevalence of 0.5% to 1% in Europe and North America (overall prevalence of 0.31% in France), characterised by progressive articular destruction and associated with several comorbidities and a subsequent decline in quality of life [START_REF] Smolen | Rheumatoid arthritis[END_REF]. Although the underlying mechanisms are not fully understood, the characteristics and pathogenic processes of RA mirror those of periodontitis, and evidence supports the notion of a bidirectional relationship. RA and periodontitis have the same genetic susceptibility and contributing environmental factors, including some HLA genotypes, smoking, socioeconomic status, lifestyle and stress [START_REF] Potempa | The case for periodontitis in the pathogenesis of rheumatoid arthritis[END_REF][START_REF] Marotte | Tooth-brushing: An impact on rheumatoid arthritis[END_REF]. It has also been suggested that the oral microbiome (periopathogens Porphyromonas gingivalis and Aggregatibacter actinomycetemcomitans) promotes and/or worsens RA by affecting tissue protein citrullination. Periodontal therapy may help to reduce local inflammation and bacterial load, which in turn reduces the putative systemic entry of proinflammatory molecules (e.g. TNFα, IL-1, IL-17) and bacterial by-products or enzymes (e.g. lipopolysaccharides from gram negative bacterial walls, peptidylarginine deiminases [START_REF] Potempa | The case for periodontitis in the pathogenesis of rheumatoid arthritis[END_REF][START_REF] Marotte | Tooth-brushing: An impact on rheumatoid arthritis[END_REF]).

Clinical trials on this subject suggest a positive effect of periodontal treatment on biological and clinical measures for RA [START_REF] Pinho M De | Relationship between periodontitis and rheumatoid arthritis and the effect of non-surgical periodontal treatment[END_REF][START_REF] Ortiz | Periodontal therapy reduces the severity of active rheumatoid arthritis in patients treated with or without tumor necrosis factor inhibitors[END_REF][START_REF] Bissada | Control of periodontal infection reduces the severity of active rheumatoid arthritis[END_REF][START_REF] Okada | Periodontal treatment decreases levels of antibodies to Porphyromonas gingivalis and citrulline in patients with rheumatoid arthritis and periodontitis[END_REF]. However, a 2014 systematic review and meta-analysis drew attention to the fact that some methodological aspects should be improved in future trials. In particular, there is a need for a better-controlled intervention and more rigorous clinical and biochemical measurements [START_REF] Kaur | Does periodontal treatment influence clinical and biochemical measures for rheumatoid arthritis? A systematic review and meta-analysis[END_REF]. In addition, more emphasis should be given to patient-centred outcomes such as quality of life parameters. The oral health-related quality of life (OHRQoL) has be shown to be highly affected in RA patients [START_REF] Blaizot | Oral health-related quality of life among outpatients with rheumatoid arthritis[END_REF][START_REF] Mühlberg | Oral health-related quality of life depending on oral health in patients with rheumatoid arthritis[END_REF].

The main objective of this randomised trial was to assess the effect of periodontal treatment on clinical and biological parameters of rheumatoid arthritis in patients also diagnosed with periodontitis. Secondary objectives were to determine the effect of periodontal treatment on periodontal health and quality of life and whether it is safe in this context.

Methods

The full study protocol was published [START_REF] Monsarrat | Effect of periodontal treatment on the clinical parameters of patients with rheumatoid arthritis: study protocol of the randomized, controlled ESPERA trial[END_REF] and registered before the first participants were enrolled (ISRCTN79186420). This study followed the CONSORT guidelines [Appendix A, Table S1; See the supplementary material associated with this article online].

Trial design, setting and ethics

The ESPERA (Experimental Study of Periodontitis and Rheumatoid Arthritis) trial is a prospective, open-label, randomised controlled clinical study. This is a two-centre trial with two parallel groups ("immediate" periodontal treatment for the treatment group versus "delayed" periodontal treatment for the control group, with a balanced 1:1 treatment allocation) and follow-up of three months.

The trial was conducted in accordance with the provisions of the Declaration of Helsinki and French Good Clinical Practice guidelines. All participants gave their written informed consent. The trial was approved by an independent ethics committee (ID-RCB-2010-A00533-36, CPP Sud-Ouest Outremer 1 on 11-JUL-2011) and regulatory agencies (AFSSAPS No.2010-A01193-36 on 04-NOV-2010).

Participants

The patients were recruited between 06 March 2012 and 15 June 2015 in the Rheumatology Departments of two teaching hospitals in southwestern France (Toulouse-Purpan and Bordeaux-Pellegrin). Subjects were included if: they had been diagnosed with RA at least one year before the date of inclusion (according to the 1987 revised American Rheumatism Association criteria for the classification of RA); their RA was moderately active on the date of inclusion (Disease Activity Score 28 using the Erythrocyte Sedimentation Rate, i.e., DAS28-ESR score between 3.2 and 5.1); and their RA treatment was unchanged over the previous three months (no change in dosage, formulation or method of administration). To be included, patients needed to have at least six permanent natural teeth and a diagnosis of periodontitis attested by the presence of at least four teeth with at least one probed site with a clinical attachment loss (CAL) ≥ 3 mm and periodontal pocket depth (PPD) ≥ 4 mm [START_REF] Armitage | Periodontal diagnoses and classification of periodontal diseases[END_REF].

Study intervention

Periodontal treatment was administered over the ten days following the inclusion visit for the immediate treatment group, and at the end of the 3-month follow-up period for the delayed control group (Appendix A, Fig. S1), for ethical reasons. Periodontal treatment included full-mouth disinfection with non-surgical scaling and root planing (SRP), systemic antibiotics (amoxicillin 1.5g/day or, in case of allergy to this molecule, 1200 mg clindamycin, for seven days after periodontal treatment), and oral hygiene instructions. Briefly, SRP was performed during a single, 2-hour, full-mouth ultrasonic and hand instrument debridement session by dental surgeons who had received special training from an experienced periodontist prior to the launch of the study. SRP was combined with subgingival irrigation with an antiseptic mouth rinse (chlorhexidine 0.12%). After SRP, subjects received personalised oral hygiene instructions, antibiotics and a free pack containing all the dental care products needed for three months of periodontal maintenance.

Primary outcome assessment

The primary outcome of the ESPERA trial was the change in RA activity, measured by the DAS28-ESR between the inclusion visit (V1) and the 3-month follow-up visit (V4 endpoint). The DAS28-ESR was computed from the number of tender and swollen joints (out of the 28 evaluated joints), the ESR in one hour, and the patient's overall assessment of their health (0 to 100).

Secondary outcomes

RA activity improvement was also evaluated using the American College of Rheumatology 20%, 50% and 70% improvement criteria (ACR 20, 50 and 70) [START_REF] Felson | American College of Rheumatology. Preliminary definition of improvement in rheumatoid arthritis[END_REF]. For example, ACR 20 corresponds to a 20% improvement in the number of tender or swollen joints as well as a 20% improvement in at least three of the five other criteria: patient's overall health assessment, physician's assessment, pain scale, disability/functional questionnaire and acute phase reactant (ESR or CRP).

The change in RA disability (Health Related QoL -HRQoL) was measured by the Health Assessment Questionnaire (HAQ) between V1 and V4. Each of the eight subdomains (dressing, rising, eating, walking, hygiene, reach, grip and activity) were also considered. The scores for each domain were grouped and calculated with a mean global score between 0, indicating full ability to perform the activities, and 3, indicating total incapacity [START_REF] Guillemin | Measurement of the functional capacity in rheumatoid polyarthritis: a French adaptation of the Health Assessment Questionnaire (HAQ)][END_REF]. Another patient-centred outcome was based on the General Oral Health Assessment Index (GOHAI), a self-administered questionnaire validated in French [START_REF] Tubert-Jeannin | Validation of an oral health quality of life index (GOHAI) in France[END_REF][START_REF] Perneger | Validation of a French-language version of the MOS 36-Item Short Form Health Survey (SF-36) in young healthy adults[END_REF] to study the OHRQoL. The GOHAI contains 12 items that assess physical functioning, pain and discomfort and psychosocial impacts. The scores for the answers were re-coded when necessary so that responses that indicate good conditions and no problems had the highest scores. GOHAI was analysed as an additive score (Add-GOHAI) and included the three dimensions [START_REF] Blaizot | Oral health-related quality of life among outpatients with rheumatoid arthritis[END_REF]. For a better evaluation, these scores were expressed as percentages. The safety and efficacy of periodontal treatment on the periodontal health of RA patients were also considered. All adverse health events during the 3-month follow-up period were recorded for both groups after each participant was interviewed. Special emphasis was placed on reporting any potential adverse oral health events.

Data collection

Clinical investigators collected sociodemographic data, medical history and treatment, RA outcomes, QoL outcomes and oral hygiene habits. RA outcomes were measured by rheumatologists who were blinded to the subjects' group assignment. Full-mouth periodontal charting included CAL, PPD and bleeding on probing (BoP). Initial periodontal charting was performed before randomisation, using a constant pressure probe (Florida Probe®; Gainesville, FL, USA) with six measurements per tooth [START_REF] Gupta | Comparative evaluation of accuracy of periodontal probing depth and attachment levels using a Florida probe versus traditional probes[END_REF] at inclusion and after months. Moreover, investigators were trained before the launch of the study (overall inter-examiner kappa was as high as 0.9 for CAL, PPD and BoP). Periodontal investigators could not be blinded to the subjects' group assignment during follow-up visits.

Using the individual periodontal measurements, the total periodontal inflammatory burden was also estimated at the participant level, taking into account periodontitis severity, extent and activity. The total gingival epithelium surface (Periodontal Epithelial Surface Area -PESA) and the total surface area of inflamed gingiva (Periodontal Inflammation Surface Area -PISA) were also calculated [START_REF] Nesse | Dose-response relationship between periodontal inflamed surface area and HbA1c in type 2 diabetics[END_REF] and periodontitis was classified at baseline according to the extent and severity [START_REF] Armitage | Development of a classification system for periodontal diseases and conditions[END_REF]. Severity was based on the CAL: mild (mean of 1-2 mm periodontal attachment loss), moderate (mean of 3-4 mm periodontal attachment loss) or severe (mean ≥ 5 mm periodontal attachment loss). Extent was characterised as localised (≤ 30% of sites involved) or generalised (> 30%).

Statistical analyses

Sample size calculation was based on detecting a 0.6 point change in DAS28-ESR from baseline in the two groups; 0.6 was chosen because it is the minimum therapeutic response according to EULAR [START_REF] Van Gestel | Validation of rheumatoid arthritis improvement criteria that include simplified joint counts[END_REF]. Assuming a standard deviation of 0.6 [START_REF] Bissada | Control of periodontal infection reduces the severity of active rheumatoid arthritis[END_REF], a two-sided test at a 5% significance level with 16 participants per group would yield an 80% power. In anticipation of a 25% dropout rate, the initial target sample size was 40 participants (20 per centre, with independent randomisation between the two centres). The random allocation sequence was generated using a computer number generator to select random permuted blocks (block lengths of 2, 4, 6 and 8), stratified by clinical centre. Randomisation cards were printed and sealed in sequentially numbered, opaque envelopes by an independent research assistant prior to the beginning of the trial [START_REF] Monsarrat | Effect of periodontal treatment on the clinical parameters of patients with rheumatoid arthritis: study protocol of the randomized, controlled ESPERA trial[END_REF].

The analysis was performed as an intention to treat; the changes in DAS28-ESR during the follow-up period were compared between the two arms. The mean differences were estimated with their 95% confidence intervals and were analysed using an analysis of covariance (ANCOVA) model, adjusted for the baseline DAS28-ESR level and the clinical centre (aMD). Residual values were tested for an approximated normal distribution. To assess the efficacy of periodontal treatment in periodontal health and other secondary outcomes, variables were also analysed using an ANCOVA model adjusted for the baseline level of the variable and the clinical centre. The proportion of subjects who had one or more adverse event in the two groups was compared using Fisher's Exact test. The level of statistical significance was set at 5% (p<0.05).

Results

Participants

A total of 139 individuals were screened, particularly because these subjects presented adequate RA inclusion criteria during their previous rheumatology appointment. Nevertheless, 117 were not included: 31 subjects declined to participate, and 86 subjects did not meet inclusion criteria for oral or rheumatological reasons (Fig. 1). A total of 22 participants were randomised (Fig. 1, Table 1). The management of RA according to tight control, treat-to-target strategies may explain the difficulty of recruitment with frequent changes in therapeutics if the patient remained in an active disease state. After very active inclusion between 2012 and 2014, only three patients were included between 2015 and 2016. The main reason was the change in recommendations for the treatment of active RA, with a more frequent and earlier use of biologics among patients and dosing modifications in current treatment.

Additionally, an interim analysis demonstrated a low conditional power (threshold below 35%) and an effect size very close to a null effect (mean difference close to zero, Table 2). The trial's scientific committee therefore decided to end recruitment due to futility reasons [START_REF] Jitlal | Stopping clinical trials early for futility: retrospective analysis of several randomised clinical studies[END_REF]. The control and treatment groups consisted of 11 subjects each.

The effect of periodontal treatment on clinical and biological parameters of RA

No statistically significant differences in DAS28-ESR were found between the groups (Table 2, Fig. 2a). The ANCOVA, adjusted on the baseline value and clinical centre, also showed no difference between the two groups (aMD -0.03 [-0.98; 0.92], p = 0.95). No difference was detected for the components of the DAS28-ESR, C-Reactive Protein level or the binary ACR 20, 50 and 70 outcomes.

Evidence of improved periodontal health after periodontal treatment

One patient in the control group and two patients in the treatment group refused to undergo the final periodontal examination. Patients were anxious about the potential triggering of an RA crisis. Mean PPD, mean CAL and % BoP sites improved three months after SRP (adjusted mean difference with a 95% confidence interval (aMD) of -0.17 mm [-0.37; 0.03], -0.17 mm [-0.53; 0.18] and -11.8% [-24.7; 0.01], respectively, although this improvement was not statistically significant. The mean PESA and PISA significantly decreased after periodontal treatment (aMD of -127.7 mm² [-246.5; -8.9] and -91.8 mm² [-172.2; -11.3], respectively) (Table 3, Figure 2b).

Periodontal treatment does not improve oral health

No significant effect was found for the OHRQoL, despite a positive trend in the "psychological impacts" domain (aMD 0.13 [-0.07; 0.33], p = 0.20; Table 4). No effect was found for the HAQ or any of the subdomains (Table 4).

Safety of periodontal treatment

During the 3-month follow-up visit, six subjects in the control group and eight subjects in the treatment group reported having been hospitalised or having had a health problem that might affect the course of the clinical trial (p = 0.22). Six control subjects and three treatment group subjects experienced oral disorders (p = 0.25).

Discussion

In this 3-month randomised controlled clinical trial, periodontal treatment had no effect on disease activity in patients with active rheumatoid arthritis. Although other clinical trials on this subject have been published in recent years [START_REF] Kaur | Does periodontal treatment influence clinical and biochemical measures for rheumatoid arthritis? A systematic review and meta-analysis[END_REF], our study provides new and important information.

First, although the final sample size is fairly small and therefore underpowered, this study gathered a wide range of clinical and biological data. These results will fuel future meta-analyses.

Second, this trial is consistent with current evidence that the effect of periodontal treatment on systemic diseases is, at best, clinically modest, as it was recently demonstrated for diabetes in a large-scale clinical trial and confirmed by the decrease in effect sizes in a recent Cochrane systematic review [START_REF] Engebretson | The effect of nonsurgical periodontal therapy on hemoglobin A1c levels in persons with type 2 diabetes and chronic periodontitis: a randomized clinical trial[END_REF][START_REF] Simpson | Treatment of periodontal disease for glycaemic control in people with diabetes mellitus[END_REF]. In this study, the care given to the randomisation process and the subsequent formation of the control group with methodological rigor may also explain the "negative results" we obtained [START_REF] Kaur | Does periodontal treatment influence clinical and biochemical measures for rheumatoid arthritis? A systematic review and meta-analysis[END_REF][START_REF] Monsarrat | Effect of periodontal treatment on the clinical parameters of patients with rheumatoid arthritis: study protocol of the randomized, controlled ESPERA trial[END_REF]. Despite the fact that our control group was a true "no periodontal treatment" control arm, there was no loss of opportunity for the participants. In fact, the 3-month delay between the diagnosis of periodontitis and the treatment in our trial corresponded with the average waiting time for the treatment of chronic periodontitis in our routine hospital practice. Futility cannot be explained by ineffective periodontal treatment since periodontal health significantly improved after periodontal treatment (periodontal inflammation significantly diminished in the intervention group). Furthermore, the addition of antibiotics to the periodontal treatment argues in favour of optimal non-surgical periodontal treatment. The statistically insignificant improvement in PPD and CAL can be blamed on the lack of final power in the study. Nevertheless, the levels of periodontal improvements (PPD improvements of 0.3-0.4mm) are consistent with other clinical studies [START_REF] Pinho M De | Relationship between periodontitis and rheumatoid arthritis and the effect of non-surgical periodontal treatment[END_REF][START_REF] Ortiz | Periodontal therapy reduces the severity of active rheumatoid arthritis in patients treated with or without tumor necrosis factor inhibitors[END_REF][START_REF] Bissada | Control of periodontal infection reduces the severity of active rheumatoid arthritis[END_REF][START_REF] Okada | Periodontal treatment decreases levels of antibodies to Porphyromonas gingivalis and citrulline in patients with rheumatoid arthritis and periodontitis[END_REF].

The ESPERA participants had an initial mean PPD of approximately 2.3 mm and 13% BoP. Periodontal conditions are different from the 3 mm PPD and 20-50% BoP in other clinical trials [START_REF] Pinho M De | Relationship between periodontitis and rheumatoid arthritis and the effect of non-surgical periodontal treatment[END_REF][START_REF] Ortiz | Periodontal therapy reduces the severity of active rheumatoid arthritis in patients treated with or without tumor necrosis factor inhibitors[END_REF][START_REF] Bissada | Control of periodontal infection reduces the severity of active rheumatoid arthritis[END_REF][START_REF] Okada | Periodontal treatment decreases levels of antibodies to Porphyromonas gingivalis and citrulline in patients with rheumatoid arthritis and periodontitis[END_REF]. This particularly low level of BoP might be explained by the fact that 15 of the 17 patients were being treated with Abatacept. Abatacept is a selective T-cell co-stimulation inhibitor [START_REF] Durez | Switching of biologics in RA patients who do not respond to the first biologic[END_REF]. No study has been published on the consequences of this specific biologic agent on observed periodontal clinical conditions. We could make an analogy with lower baseline BoP values observed with an anti-IL6 monoclonal antibody compared to tumour necrosis factor inhibitors [START_REF] Kobayashi | Interleukin-6 receptor inhibitor tocilizumab ameliorates periodontal inflammation in patients with rheumatoid arthritis and periodontitis as well as tumor necrosis factor inhibitors[END_REF]. Better periodontal health might also be linked to the encouragement to practice oral hygiene before biotherapies are introduced. If gingival inflammatory status is in fact lower than in other studies, the margin for periodontal inflammation improvement may have been insufficient to obtain a clinical improvement in RA [START_REF] Neto | An Evaluation of Non-Surgical Periodontal Therapy in Patients with Rheumatoid Arthritis[END_REF].

Unlike other studies with moderately active RA [START_REF] Ortiz | Periodontal therapy reduces the severity of active rheumatoid arthritis in patients treated with or without tumor necrosis factor inhibitors[END_REF][START_REF] Bissada | Control of periodontal infection reduces the severity of active rheumatoid arthritis[END_REF][START_REF] Erciyas | Effects of periodontal therapy on disease activity and systemic inflammation in rheumatoid arthritis patients[END_REF][START_REF] Bıyıkoğlu | Periodontal therapy in chronic periodontitis lowers gingival crevicular fluid interleukin-1beta and DAS28 in rheumatoid arthritis patients[END_REF], we did not demonstrate a decrease in DAS28, swollen joints, CRP or ESR despite a clinical reduction in periodontal inflammation. Another reason could be the type of RA patients included, since approximately 90% of the participants had long-term and severe RA, with HAQ ≥ 0.5 (Table 1). In other studies, RA duration was usually shorter [START_REF] Erciyas | Effects of periodontal therapy on disease activity and systemic inflammation in rheumatoid arthritis patients[END_REF][START_REF] Bıyıkoğlu | Periodontal therapy in chronic periodontitis lowers gingival crevicular fluid interleukin-1beta and DAS28 in rheumatoid arthritis patients[END_REF][START_REF] Ribeiro | Periodontal infection as a possible severity factor for rheumatoid arthritis[END_REF] and baseline HAQ values were hardly ever reported [START_REF] Ribeiro | Periodontal infection as a possible severity factor for rheumatoid arthritis[END_REF]. The complexity of the intrinsic pathophysiological mechanisms of RA [START_REF] Potempa | The case for periodontitis in the pathogenesis of rheumatoid arthritis[END_REF], the chronic ailments with significant sequelae, and the many confounding factors between periodontitis and polyarthritis, may explain the lack of effect of periodontal treatment in such a study; the focal infection theory as a causal relationship still being in debate [START_REF] Da Silva | Arthritis and Periodontitis: An Association Debated for Over Two Centuries[END_REF].

Third, to our knowledge, this trial is the first to investigate the effect of periodontal treatment on quality of life in RA patients. The GOHAI was used to evaluate the OHRQoL [START_REF] Tubert-Jeannin | Validation of an oral health quality of life index (GOHAI) in France[END_REF]; it highlights the subject's daily experience, and can determine functional problems, discomfort and pain [START_REF] Blaizot | Oral health-related quality of life among outpatients with rheumatoid arthritis[END_REF]. In some previous studies the oral quality of life in RA patients was worse [START_REF] Mühlberg | Oral health-related quality of life depending on oral health in patients with rheumatoid arthritis[END_REF][START_REF] Monsarrat | Effect of periodontal treatment on the clinical parameters of patients with rheumatoid arthritis: study protocol of the randomized, controlled ESPERA trial[END_REF]. It is known that periodontitis affects OHRQoL [START_REF] Buset | Are periodontal diseases really silent? A systematic review of their effect on quality of life[END_REF] and that non-surgical periodontal treatment can improve OHRQoL [START_REF] Goel | A Comparison of Impact of Chronic Periodontal Diseases and Nonsurgical Periodontal Therapy on Oral Health-Related Quality of Life[END_REF], particularly in terms of function, psychology and pain [START_REF] Shanbhag | The impact of periodontal therapy on oral health-related quality of life in adults: a systematic review[END_REF]. In this study, there was a tendency towards psychological improvement. In terms of the patients' overall quality of life, the HAQ disability index was used. We found no significant changes in the HAQ index, as previously suggested [START_REF] Ribeiro | Periodontal infection as a possible severity factor for rheumatoid arthritis[END_REF], or in any of the eight HAQ subdomains.

Several limitations should be considered. First, it is likely that ESPERA participants are not representative of RA patients, even within France. This study mainly enrolled patients admitted to the outpatient Rheumatology units of the Regional University Hospitals of Toulouse and Bordeaux. These are referral units for RA diagnosis and management in the southwest of France. These patients have long-term, severe, ACPA positive, even refractory RA that is essentially treated with biological DMARDs (Table 1). It might be interesting to repeat a similar intervention on other categories of RA patients, such as early RA patients [START_REF] Wolff | Oral status in patients with early rheumatoid arthritis: a prospective, case-control study[END_REF]. Second, the main limitation of this work is the small sample size available for analysis, which did not reach the levels calculated a priori. We had to stop inclusions because no effect trend was found in an interim analysis, in a context where the inclusion of participants was difficult. According to the treat-to-target approach, tight control principles of RA management, frequent adjustments of therapeutics (in dosage or type of molecule) were therefore necessary if there was no improvement in three months at most after the start of treatment, or if the target had not been reached by six months (remission or low disease activity) [START_REF] Daien | Update of French Society for Rheumatology Recommendations for Managing Rheumatoid Arthritis[END_REF]. Consequently, it was very difficult to enrol patients with a DAS28 score between 3.2 and 5.1 and who had also had no change in medication, dose, or RA treatment formulation in the three months preceding inclusion [START_REF] Monsarrat | Effect of periodontal treatment on the clinical parameters of patients with rheumatoid arthritis: study protocol of the randomized, controlled ESPERA trial[END_REF]. Under these circumstances, it would not have been ethical to pursue inclusions, and amending the protocol to adapt to these new practices would have profoundly disrupted the analyses. The absence of significant improvement on the DAS28 after periodontal treatment could therefore be interpreted as insufficient power to demonstrate an effect. Nevertheless, even if we initially planned to enrol 16 participants per group to be able to detect a difference of 0.6 point in DAS28-ESR between the two groups, it would be unlikely that achieving this objective would have revealed a statistically significant difference according to the absence of any quantitative difference in the 3-month change of DAS28-ESR between the two groups of 11 participants (-0.03 [-0.98;0.92], p=0.95). Lastly, although the plaque control record is important to assess participant compliance with oral hygiene instructions, we did not consider this data to be essential because the focus was on the ITT analysis.

Although this trial found no clinical effect of periodontal treatment on RA, it provides important data to support periodontal care among RA patients. Periodontal treatment is safe and reduces oral inflammation with a possible effect on oral health QoL. Since both periodontitis and RA are complex and multifactorial chronic diseases, it is likely that person-centred approaches that involve both oral health professionals and rheumatologists will contribute to optimal patient management [START_REF] Monsarrat | Oral health status in outpatients with rheumatoid arthritis: the OSARA study[END_REF][START_REF] Mourgues | Economic analysis of a nurse-led programme for comorbidities management of rheumatoid arthritis patients[END_REF]. Results are presented as mean ± standard deviation for the first visit (V1) and the visit in three months (V4). a p-value of the non-parametric Mann-Whitney test for a difference between control and treatment groups.

b adjusted mean difference of the V4 outcome by ANCOVA analysis, adjusted on the baseline value (V1) and the clinical centre (Bordeaux or Toulouse). A negative value indicates that the value for the treatment group is lower than the value for the control group. c p-value of the adjusted mean difference.

FiguresFig. 1 :

 1 Figures

Fig. 2 :

 2 Fig. 2: (a) Scatter plot of the relationship between DAS28-ESR at V1 and V4 for the control (clear circles) and treatment groups (black circles). The linear regression line is shown in blue (solid for the treatment group and dashed for the control group) along with 95% confidence intervals. (b) Scatter plot of the relationship between PISA at V1 and V4 for the control (clear circles) and treatment groups (black circles). The linear regression line is shown in blue (solid for the treatment group and dashed for the control group) along with 95% confidence intervals.

Table 2 : Effect of the intervention on clinical and biological markers related to rheumatoid arthritis

 2 p-value of the non-parametric Mann-Whitney test for a difference between control and treatment groups. b adjusted mean difference of the V4 outcome by ANCOVA analysis, adjusted on the baseline value (V1) and the clinical centre (Bordeaux or Toulouse). A negative value indicates that the value for the treatment group is lower than the value for the control group.

	Medical outcomes	N	Control group Mean ± SD	N	Treatment group Mean ± SD	p a	adjusted Mean Difference b aMD [CI95%]	p c
	Tender joints (0-28)
	V1 V4	11 11	5.09 ± 4.35 3.55 ± 3.30	11 11	6.27 ± 3.95 6.27 ± 6.59	0.43 0.49	1.94 [-2.34; 6.23]	0.35
	Swollen joints (0-28)
	V1 V4	11 11	1.55 ± 2.07 0.82 ± 0.98	11 11	2.82 ± 2.27 2.36 ± 2.98	0.07 0.33	0.85 [-0.96; 2.65]	0.34
	Erythrocyte Sedimentation
	Rate (1-hour ESR, mm.h -1 )
	V1 V4	11 11	20.55 ± 14.87 21.45 ±14.29	11 11	21.73 ± 14.89 24.00 ± 18.25	0.77 0.90	2.30 [-12.77; 17.38]	0.75
	C-Reactive Protein (CRP,
	mg. L -1 )
	V1 V4	11 11	6.45 ± 3.34 7.42 ± 3.61	10 11	10.61 ± 7.03 12.23 ± 10.37	0.13 0.51	1.68 [-4.39; 7.75]	0.57
	Patient's overall
	assessment (0-100)
	V1 V4	11 11	40.91 ± 23.43 30.27 ± 21.51	11 11	32.91 ± 16.62 30.45 ± 22.19	0.43 0.95	6.25 [-7.20; 19.70]	0.34
	Physician's overall
	assessment (0-100)
	V1 V4	11 11	28.18 ± 18.88 30.00 ± 20.98	11 11	35.36 ± 15.76 30.45 ± 22.19	0.23 0.95	-4.34 [-22.23; 13.56]	0.62
	Pain scale (0-100)
	V1 V4	11 11	40.91 ± 23.43 31.82 ± 20.89	11 11	33.82 ± 17.30 31.36 ± 21.92	0.49 1.00	4.00 [-10.48; 18.47]	0.57
	Disease Activity Score 28
	(DAS28-ESR)
	V1 V4	11 11	3.82 ± 0.54 3.47 ± 0.93	11 11	4.24 ± 0.61 3.96 ± 1.40	0.12 0.22	-0.03 [-0.98; 0.92]	0.95
	American College of Rheumatology -ACR20	11	7 (64%)	11	4 (36%)	0.20	-	-
	ACR50	11	3 (27%)	11	3 (27%)	0.68	-	-
	ACR70	11	2 (18%)	11	2 (18%)	0.71	-	-
	Results are presented as mean ± standard deviation (SD) for the first visit (V1) and the visit in three months (V4).

a c p-value of the adjusted mean difference.

Table 3 : Effect of the intervention on periodontal outcomes

 3

	Periodontal outcomes	N	Control group Mean ± SD	N	Treatment group Mean ± SD	p a	adjusted Mean Difference b aMD [CI95%]	p c
	Periodontal Pocket Depth -PPD
	(mm)
	V1 V4	11 10	2.27 ± 0.53 2.22 ± 0.43	11 9	2.36 ± 0.59 2.01 ± 0.56	0.70 0.22	-0.17 [-0.37; 0.03]	0.09
	Clinical Attachment Loss -CAL
	(mm)
	V1 V4	11 10	2.71 ± 1.37 2.78 ± 1.29	11 9	3.03 ± 1.12 2.66 ± 1.20	0.27 0.81	-0.17 [-0.53; 0.18]	0.31
	Recession -REC (mm)
	V1 V4	11 10	0.52 ± 1.09 0.51 ± 0.85	11 9	0.66 ± 0.70 0.64 ± 0.89	0.32 0.18	0.12 [-0.23; 0.47]	0.49
	Bleeding on Probing -BoP (%)
	V1 V4	11 10	15.5 ± 12.1 16.5 ± 19.0	11 9	12.0 ± 13.3 4.0 ± 4.5	0.36 0.05	-11.8 [-24.7; 0.01]	0.07
	Periodontal Epithelial Surface
	Area -PESA (mm²)
	V1 V4	11 10	844.0 ± 188.2 828.5 ± 221.1	11 9	837.2 ± 244.8 677.4 ± 278.0	0.94 0.25	-127.7 [-246.5; -8.9]	0.04
	Periodontal Inflamed Surface
	Area -PISA (mm²)
	V1 V4	11 10	138.9 ± 130.3 122.3 ± 107.9	11 9	122.53 ± 160.0 30.6 ± 33.4	0.55 0.05	-91.8 [-172.2; -11.3]	0.03
	Results are presented as mean ± standard deviation for the first visit (V1) and the visit in three months (V4).

a p-value of the non-parametric Mann-Whitney test for a difference between control and treatment groups. b adjusted mean difference of the V4 outcome by ANCOVA analysis, adjusted on the baseline value (V1) and the clinical centre (Bordeaux or Toulouse). A negative value indicates that the value for the treatment group is lower than the value for the control group. c p-value of the adjusted mean difference.

Table 4 : Effect of the intervention on quality of life outcomes

 4

	Quality of life outcomes	N	Control group Mean ± SD	N	Treatment group Mean ± SD	p a adjusted Mean Difference b aMD [CI95%]	p c
	General Oral Health
	Assessment Index -GOHAI
	Add-GOHAI (%)
	V1 V4	11 11	0.74 ± 0.14 0.69 ± 0.24	11 11	0.63 ± 0.24 0.71 ± 0.27	0.37 0.74	0.10 [-0.12; 0.31]	0.37
	Physical functioning (%)
	V1 V4	11 11	0.77 ± 0.24 0.76 ± 0.26	11 11	0.70 ± 0.26 0.71 ± 0.28	0.46 0.67	-0.01 [-0.22; 0.21]	0.96
	Pain and discomfort (%)
	V1 V4	11 11	0.72 ± 0.25 0.67 ± 0.23	11 11	0.60 ± 0.19 0.71 ± 0.28	0.25 0.51	0.13 [-0.15; 0.40]	0.34
	Psychosocial impacts (%)
	V1 V4	11 11	0.74 ± 0.19 0.65 ± 0.28	11 11	0.60 ± 0.30 0.70 ± 0.31	0.20 0.64	0.13 [-0.07; 0.33]	0.20
	Health Assessment
	Questionnaire -HAQ
	Disability Index
	V1	11	0.91 ± 0.59	10	1.31 ± 0.71	0.17	0.06 [-0.54; 0.65]	0.85
	V4	10	0.75 ± 0.50	10	1.05 ± 0.70	0.32
	Dressing
	V1 V4	11 10	0.91 ± 0.94 0.70 ± 0.95	10 10	1.10 ± 0.74 0.60 ± 0.70	0.60 0.97	-0.23 [-1.03; 0.56]	0.54
	Rising
	V1 V4	11 10	0.64 ± 0.67 0.20 ± 0.42	9 10	1.00 ± 0.87 0.70 ± 0.82	0.33 0.13	0.41 [-0.16; 0.99]	0.15
	Eating
	V1 V4	11 10	0.73 ± 0.79 0.80 ± 0.63	10 10	1.30 ± 1.06 1.10 ± 0.99	0.21 0.54	0.21 [-0.68; 1.10]	0.62
	Walking
	V1 V4	11 9	0.91 ± 0.83 0.56 ± 0.73	10 10	1.00 ± 0.94 0.70 ± 0.82	0.82 0.72	0.00 [-0.67; 0.67]	1.00
	Hygiene
	V1 V4	11 10	0.91 ± 1.14 0.70 ± 1.06	10 10	1.80 ± 1.23 1.50 ± 1.18	0.09 0.09	0.20 [-0.78; 1.18]	0.67
	Reach
	V1 V4	11 10	1.55 ± 0.93 1.70 ± 0.95	10 10	1.50 ± 0.85 1.60 ± 0.84	0.79 0.81	0.01 [-0.63; 0.65]	0.97
	Grip
	V1 V4	11 10	0.73 ± 0.90 0.60 ± 0.97	10 10	1.20 ± 1.03 0.80 ± 0.79	0.28 0.45	-0.08 [-0.91; 0.75]	0.84
	Activity
	V1 V4	11 10	0.91 ± 0.70 0.70 ± 0.82	10 10	1.50 ± 0.53 1.40 ± 0.97	0.05 0.10	0.20 [-0.79; 1.20]	0.67

Acknowledgments

We are very grateful to A. Blaizot and O. Sechoy-Balussou for their help during the design of the trial; the entire medical, clinical and administrative staff of the respective Rheumatology Departments for their warm welcome and assistance in recruiting patients; and of course, all the trial participants.

Disclosure of interest

This study was supported by a grant from the French Ministry of Health Clinical Research Hospital Programme "2010" (Programme Hospitalier de Recherche Clinique inter-régional 1004608 [PHRCI]). The necessary equipment for scaling and root planing (supersonic motor, periodontal inserts and hand pieces) was kindly provided free of charge by the ACTEON® Group (Merignac, France).

Appendix A. Supplementary data

Supplementary data (Table S1, Fig. S1) associated with this article can be found in the online version at … Tables Table 1: