Laurent Chupin
email: laurent.chupin@math.univ-bpclermont.fr

Jordane Math
email: jordane.mathe@math.univ-bpclermont.fr

Existence theorem for homogeneous incompressible Navier-Stokes equation with variable rheology

Keywords: weak solution, existence result, variable rheology, Bingham fluids, Galerkin method, monotone graph

We look at a homogeneous incompressible fluid with a time and space variable rheology of Bingham type, which is governed by a coupling equation. Such a system is a simplified model for a gas-particle mixture that flows under the influence of gravity. The main application of this kind of model is pyroclastic flows in the context of volcanology. In order to prove long time existence of weak solutions, classical Galerkin approximation method coupled with a priori estimates allows us to partially solve the problem. A difficulty remains with the stress tensor, which must satisfy an implicit constitutive relation. Some numerical simulations of a flow of this type are given in the last section. These numerical experiments highlight the influence of the fluidization phenomenon in the flow.

Introduction

Dense pyroclastic flows are fast-moving density currents emitted by volcanoes that are particularly destructive. They are composed of high temperature crushed rocks ranging in size from micrometers to meters. They are granular flows but they show a very high fluidity: flows thinner than one meter can reach run-out distances longer than 10 kilometers and they are able to flow on horizontal surfaces. One objective of modern volcanology is to better understand the behavior of this type of flows for a better assessment of volcanic hazards. A possible explanation of the high mobility is the interaction between the fine particles of the flow and volcanic or atmospheric gases. Some physical experiments have been made on the collapse of a fluidized dense granular column in a U-shaped channel, see for instance [START_REF] Iverson | The physics of debris flows[END_REF][START_REF] Philippe | Start and stop of an avalanche in a granular medium subjected to an inner water flow[END_REF] or [START_REF] Roche | Pore fluid pressure and internal kinematics of gravitational laboratory air-particles flows: Insights into the emplacement dynamics of pyroclastic flows[END_REF]. In these experiments, the granular bed is initially fluidized by injection of air through the column, which generates high interstitial pore fluid pressure so that the released granular material flows like an inertial single-phase fluid. In this context, a lot of mathematical models have been proposed to take into account the change of rheology all along the flow: [START_REF] Aranson | Continuum theory of partially fluidized granular flows[END_REF][START_REF] Aranson | Nonlocal rheological properties of granular flows near a jamming limit[END_REF][START_REF] Drucker | Soil mechanics and plasticity analysis or limit design[END_REF][START_REF] Forterre | Flows of dense granular media[END_REF][START_REF] Iverson | Flow of variably fluidized granular masses across three-dimensional terrain[END_REF][START_REF] Mangeney | Avalanche mobility induced by the presence of an erodible bed and associated entrainment[END_REF][START_REF] Meruane | On the role of the ambient fluid on gravitational granular flows dynamics[END_REF][START_REF] Volfson | Partially fluidized shear granular flows: Continuum theory and molecular dynamics simulations[END_REF]. A first category concerns bi-fluids systems, in which each phase is described by a Navier-Stokes type equation added to a coupling term (see [START_REF] Meruane | On the role of the ambient fluid on gravitational granular flows dynamics[END_REF]). An important problem with this kind of model is their heaviness in terms of numerical computations. This is why plenty of models are of Shallow-Water type, provided the length of the flow is much bigger than the thickness, see [START_REF] Iverson | Flow of variably fluidized granular masses across three-dimensional terrain[END_REF]. This method allows to reduce a d-dimensional problem to a (d-1)-dimensional one, and is then much more efficient for computational experiments. But the cost is the loss of information in the direction which seems to be crucial to understand the behavior of fluidized granular flows. In order to overcome the problems above it is possible to consider a homogeneous incompressible fluid with a rheology depending on a parameter (called order parameter in [START_REF] Aranson | Nonlocal rheological properties of granular flows near a jamming limit[END_REF][START_REF] Mangeney | Avalanche mobility induced by the presence of an erodible bed and associated entrainment[END_REF][START_REF] Volfson | Partially fluidized shear granular flows: Continuum theory and molecular dynamics simulations[END_REF]), which is supposed to satisfy an equation describing its evolution. For such a model, the equations governing the hydrodynamics are the Navier-Stokes equations including an extra-stress tensor. They read ρ ∂ t v + v • ∇v + ∇p -η ∆v = div σ + ρ g,

div v = 0,
where v is the velocity, p the pressure and σ is the extra-stress tensor. The vector g characterizes the gravity forces, the coefficient η > 0 refers to the Newtonian viscosity and ρ > 0 is the constant density of the mixture. Note that the constant density of the mixture can be expressed as the linear combination of the constant density of each component of the mixture:

ρ = πρ s + (1 -π)ρ f ,
where ρ f is the constant density of the gas, ρ s the constant density of the solid particles and π is the mass fraction of gas in the mixture. In order to take into account the behaviour of dense granular flows, the extra-stress σ depends not only on the symmetric part of the velocity gradient Dv but also on an order-parameter q evolving with the flow (cf [14]):

σ = σ(q, Dv) with Dv = ∇v + (∇v) T 2 .
More precisely, the function σ is defined using a Bingham type relation:

   σ(q, Dv) = q Dv |Dv| if Dv = 0, |σ(q, Dv)| ≤ q if Dv = 0, (1)
where for any tensor A, |A| = √ A : A = i,j A 2 i,j . In other words, we want to impose a solid behaviour when the stress is smaller than a plasticity yield, named q. In the model presented in this paper, contrary to the ones in [START_REF] Aranson | Nonlocal rheological properties of granular flows near a jamming limit[END_REF][START_REF] Mangeney | Avalanche mobility induced by the presence of an erodible bed and associated entrainment[END_REF][START_REF] Volfson | Partially fluidized shear granular flows: Continuum theory and molecular dynamics simulations[END_REF], the order-parameter depends on the interstitial pore fluid pressure, denoted by p f . It means that in this model the gas present between solid particles only acts on the global dynamic via its local pressure. More precisely, our order parameter is in fact the plasticity yield defined as a function of the difference between p f and a solid pressure p s as follows:

q = q 0 (p s -p f ) + , q 0 > 0. (2)
This choice means that when the fluid pressure compensates the solid pressure then the mixture is Newtonian (q = 0), whereas when fluid pressure is lower then plasticity effects happen (q > 0). The model must be completed by the relation giving such pressures p f and p s .

• Let's begin with the pore fluid pressure p f . As in [START_REF] Roche | Depositional processes and gas pore pressure in pyroclastic flows: an experimental perspective[END_REF], some experiments show that, in the static case, the evolution of this pressure is given by a diffusion law. To understand what happens in a flowing case, one can follow Iverson and Denlinger's inference, which gives an advection-diffusion law for the behaviour of the basal pore pressure in a shallow-water type model, see [START_REF] Iverson | Flow of variably fluidized granular masses across three-dimensional terrain[END_REF]. Same phenomenological observations allow us to say that the fluid pressure p f verifies, in our model, an advection-diffusion equation like:

∂ t p f + v • ∇p f -K ∆p f = 0,
• We assume that the solid pressure p s only depends on the gravity. More precisely, it corresponds to the pressure that imposes a column of grain particles anywhere in the bed. Then it can be explicitly defined as the lithostatic solid pressure, i.e.

p s = p s0 + ρ |g| (y 0 -y), (3)
where p s0 ≥ 0 is a reference pressure given at the altitude y = y 0 (y begin the vertical coordinate).

Remark 1.1. Note that there are three pressures in the model: p s is the lithostatic (or hydrostatic) pressure, p f is the pressure of the gas which fills the gap between solid particles, and p is the hydrodynamic pressure of the mixture. As a result, in the particular case of a static state of the solid matter, p equals p s . In [START_REF] Roche | Depositional processes and gas pore pressure in pyroclastic flows: an experimental perspective[END_REF], this solid pressure represents the minimum fluidization pressure measured on the base of a static grain column during its fluidization. More precisely, this is the pressure of the gas once the velocity of injection is sufficient to impose an upward drag force which counterbalances the weight of particles. But there is no a priori relation between p s , p f and p in the general case.

Finally, the dynamic of such a fluid is governed by a system of three equations with three unknowns: v, p and p f , which is

           ρ ∂ t v + v • ∇v + ∇p -η ∆v = q 0 div (p s -p f ) + Dv |Dv| + f, div v = 0, ∂ t p f + v • ∇p f -K ∆p f = 0, (4)
where p s is the function given by ( 3) and f is a source term taking into account all exterior forces, for instance the gravity forces when f = ρg. In the case where the solid pressure and fluid pressure are equal (p s = p f ) then we obtain a pure viscous flow (modeled by the incompressible Navier-Stokes equations). If the difference between these two pressures is assumed to be constant then we obtain the classical Bingham model. These two regimes are already well known, see for example [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF]. In order to solve the existence problem for such complex fluids, a general theory has been developed and studied in [START_REF] Ček | On unsteady flows of implicitly constituted incompressible fluids[END_REF], in which the authors prove the existence of a weak solution for a general class of rheology including fluids of a Bingham and Herschel-Bulkley type and depending on both time and space. But this framework does not include our case based on coupling equations.

In this article, we generalize these results to our physical model by treating the coupling relation through the definition of the order-parameter. Even if the result given in [START_REF] Ček | On unsteady flows of implicitly constituted incompressible fluids[END_REF] is true in the threedimensional case, according to Remark 3.2, we will restrict the study to the two-dimensional case in this article.

This paper is organized as follows. In Section 2, we introduce the mathematical formulation of the problem and give the main result: the existence of a weak solution. The Section 3 is devoted to the proof of this result: a construction of approximated solutions is given in the Subsection 3.1. We next prove uniform estimates in the Subsection 3.2, and we finally deduce the existence result using a limit process in the Subsection 3.4. In the last section, we give some numerical results on the classical Poiseuille-type flow in a 2D-channel, that show the impact of the fluidization on the behaviour of a Bingham fluid.

Mathematical framework and main result

In order to describe the notion of solution to the system (4), we need to introduce suitable functional spaces.

Functional spaces

In the sequel, the fluid domain Ω ⊂ R 2 is a bounded connected open set with a smooth boundary denoted by ∂Ω. Its outward unit normal vector is denoted by n.

According to classical notations, we will denote by L m (Ω), m ≥ 1, Lebesgue spaces on Ω and H s (Ω), or more generally W s,q (Ω) (s > 0 and q ≥ 1), Sobolev spaces on Ω.

To deal with boundary conditions and incompressibility constraint, we consider the following notation

H 1 0 (Ω) = {g ∈ H 1 (Ω) 2 ; g| ∂Ω = 0}, H = {g ∈ L 2 (Ω) 2 ; div g = 0 and g • n = 0 on ∂Ω}, V = {g ∈ H 1 0 (Ω) 2 ; div g = 0}.
As usual, the dual space of H 1 0 (Ω) and V will be denoted by H -1 and V respectively.

D(Ω) = C ∞ c
(Ω) will be the notation for the set of infinitely differentiable functions with compact support on Ω.

For time-dependent functions, we denote by L m X the set also denoted by L m (0, T ; X), 0 < T < +∞ and m ≥ 1, defined by

L m X = {g(t, •) ∈ X for a. a. t ∈ (0, T ) ; t → g(t, •) X ∈ L m (0, T )},
with • X being a norm on X, where X is any separable functional space as above. In the same way, we denote by C 0 ([0, T ], X) the set of time-continuous functions on [0, T ] to the space X.

For a given basis (x i ) i of any functional space X from above, we will denote by X k the finite dimensional subspace of X of the form

X k = span(x 1 , . . . , x k ).
Finally, because lots of tensors used in this paper are symmetric, we will denote this with a subscript L m sym = {A ∈ L m (Ω) 2×2 ; A is symmetric}, and the set of 2 × 2 real symmetric matrices will be denoted by S 2 (R).

Rigorous formulation of the System (4)

In System (4), one element is, at this stage, not completely well defined. Indeed, the fraction Dv |Dv| makes no sense on the (rigid) set {Dv = 0}. Therefore, instead of using definition (1) for σ(q, Dv), it is preferred to use formulation below:

σ : Dv = q |Dv| , |σ| ≤ q. ( 5
)
Lemma 2.1. Both systems (1) and ( 5) are equivalent, more precisely for all (σ, Dv) ∈ R 2×2 2 and for all q ≥ 0, we have

   σ = q Dv |Dv| if Dv = 0 |σ| ≤ q if Dv = 0 ⇐⇒ σ : Dv = q |Dv| |σ| ≤ q.
Proof. Let's consider first the case with q = 0. In this case, both systems are reduced to σ = 0 so that they are equivalent. Let's suppose now that q > 0. It is obvious that the system on the left (1) implies the one on the right [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF]. To prove the converse, we assume that ( 5) is true and we split the proof into two cases:

• Case 1: Dv = 0.

If Dv = 0, then the second relation of the system (1) is obviously true.

• Case 2: Dv = 0. Else, taking the absolute value in the first relation of ( 5), we compute the scalar equality

|σ : Dv| = q |Dv| .
Whereas applying the Cauchy-Schwarz inequality and using the second relation of ( 5), we get

|σ : Dv| ≤ |σ| |Dv| ≤ q |Dv| .
This inequality, together with the previous equality, gives q |Dv| ≤ |σ| |Dv| ≤ q |Dv|. Then remember that Dv = 0, so that we have |σ| = q and thus |σ : Dv| = |σ| |Dv| .

In this case, using a corollary of Cauchy-Schwarz inequality, we also know that the tensors σ and Dv are linearly dependent. That means there is a scalar a such that σ = aDv.

Using this expression together with the first line of the system (5), we conclude that this scalar exactly corresponds to a = q |Dv| , which concludes the proof.

Thereby, the problem is formulated as follows. Being given initial data velocity v 0 , initial fluid pressure p f 0 , a source term f and a time T > 0, we look for a velocity vector field v, two scalar pressures p and p f , and a symmetric stress tensor σ such that

                         ρ ∂ t v + v • ∇v + ∇p -η ∆v = div σ + f in Ω × (0, T ), div v = 0 in Ω × (0, T ), ∂ t p f + v • ∇p f -K ∆p f = 0 in Ω × (0, T ), q = q 0 (p s -p f ) + where p s is defined by (3) in Ω × (0, T ), (σ, Dv, q) satisfies (5) in Ω × (0, T ), (v, p f )| t=0 = (v 0 , p f 0 ) in Ω, (v, p f ) = (0, 0) in ∂Ω × (0, T ). (6)
We would like to point out that, as an assumption, we restrict the boundary conditions for the interstitial pressure p f and the velocity v to homogeneous Dirichlet conditions. Note that this restriction is essentially due to mathematical convenience when working with the variational formulation and that adaptation to a more general case will be not treated in this paper.

Definition of a weak solution and existence theorem

In order to work with non-regular solutions, and then to show global existence in time result, we use a weak formulation of the system [START_REF] Brezis | Analyse fonctionnelle : Th éorie et applications[END_REF]. The weak interpretation of the Navier-Stokes equation being standard, we only detail the weak formulation of the condition (5). For any bounded function q, we introduce the following graph:

G[q] = (σ, Dv) ∈ L 2 L 2 sym × (L ∞ H -1 ∩ L 2 L 2 ) ; (σ, Dv, q) satisfies (5) a.e. , (7)
and we say that (σ, Dv, q) satisfies ( 5) in a weak sense if (σ, Dv) ∈ G[q]. One of the most important properties of the graph G[q] is its monotony. More exactly, we have the following result which has been first introduced in this context in [START_REF] Ček | On unsteady flows of implicitly constituted incompressible fluids[END_REF] and is fundamental at the end of the proof (see Section 3.4.2-Step 2 (b).3) to reach some limits in the graph G[q]. Lemma 2.2. Let q be a given bounded function.

If (σ 1 , D 1 ) and (σ 2 , D 2 ) are in G[q], then (σ 1 -σ 2 ) : (D 1 -D 2 ) ≥ 0 a.e.
Proof. Suppose that (σ 1 , D 1 , q) and (σ 2 , D 2 , q) satisfy ( 5). We have

(σ 1 -σ 2 ) : (D 1 -D 2 ) = σ 1 : D 1 + σ 2 : D 2 -σ 1 : D 2 -σ 2 : D 1 = q |D 1 | + q |D 2 | -σ 1 : D 2 -σ 2 : D 1 .
Using the Cauchy-Schwarz inequality we deduce

(σ 1 -σ 2 ) : (D 1 -D 2 ) ≥ q |D 1 | + q |D 2 | -|σ 1 | |D 2 | -|σ 2 | |D 1 | ≥ 0,
which concludes the proof.

Definition 2.1. Let f ∈ L 2 (R + ; V ) and (v 0 , p f 0 ) ∈ H × L ∞ be the initial conditions. For T > 0, we say that (v, p f , σ) is a weak solution of (6) on Ω × (0, T ) if v ∈ L ∞ H ∩ L 2 V, ∂ t v ∈ L 2 V , p f ∈ L ∞ L ∞ ∩ L 2 H 1 0 , ∂ t p f ∈ L 2 H -1 , σ ∈ L 2 L 2 sym ,
and satisfy, for any (ψ, r)

∈ V × H 1 0 (Ω):                    ρ ∂ t v, ψ V ,V + Ω (v • ∇v) • ψ + η Ω ∇v : ∇ψ + Ω σ : ∇ψ = f, ψ V ,V , ∂ t p f , r H -1 ,H 1 0 + Ω (v • ∇p f ) r + K Ω ∇p f • ∇r = 0, (σ, Dv) ∈ G[q] with q = q 0 (p s -p f ) + , (v, p f )| t=0 = (v 0 , p f 0 ) a.e. in Ω, (8)
where p s = p s0 + ρ |g| (y 0 -y) is a given function of the density and the vertical coordinate y.

The main result of the paper is the following theorem.

Theorem 2.3. If f ∈ L 2 (R + ; V ), v 0 ∈ H and p f 0 ∈ L ∞ (Ω) is non-negative,
then for all T > 0, there exists a weak solution (v, p f , σ) of (6) on Ω × (0, T ) in the sense of the definition 2.1, with p f ≥ 0 a.e. in (0, T ) × Ω.

Proof

The proof of the existence Theorem 2.3 consists of two main steps:

1. We first introduce an approximate solution using a sequence of well-posed problems. The first difficulty is common: we look for a solution (σ, p f , v) with values in infinite dimensional spaces (of kind L 2 ) and we use a Galerkin scheme to approach it by functions with values in finite dimensional spaces (for instance L 2 k ), and then use the classical Cauchy theorem for ODE. The second difficulty comes from the Bingham condition (σ, Dv) ∈ G[q]. Even in the finite dimensional case, this kind of condition is not usual and we introduce a regularization: we approach the graph G[q] by the graph of a regular function Dv -→ σ;

2. Next, we take the limit in the approximate solution to find a solution to the initial problem.

We then use the energy estimates of the model to deduce bounds. The main difficulty is to obtain the limit in the non-linear condition (σ, Dv) ∈ G[q]. This point is treated in the subsection 3.4.2.

Approximate model

Let k > 0 be an integer.

Galerkin approximation

In a classical way, we reduce the problem to finite dimensional spaces following Galerkin approximations. Let (w i ) i≥1 be the basis of V made of eigenfunctions of Stokes' operator. Let (r i ) i≥1 be the basis of H 1 0 (Ω) made of eigenfunctions of the operator -∆. Let P k be the orthogonal projection from V to V k , finite dimensional space spanned by the k first eigenfunctions of Stokes' operator {w 1 , . . . , w k }. In the same way, Q k is the orthogonal projection from H 1 0 to H 1 0,k := span(r 1 , . . . , r k ). We define the projected velocity and the projected fluid pressure by

v k = P k (v) and p f,k = Q k (p f ).
Graph regularization For any given q k , we define a regularized graph G k by

G k = q k Dv |Dv| + 1/k , Dv , v ∈ L 2 V k . ( 9
)
Regularization of the initial conditions The initial velocity v 0 is approached using the projector above:

v k 0 = P k (v 0
). Since the initial fluid pressure p f 0 is non-negative (and since the fluid pressure satisfies the maximum principle), in order to conserve this property at each step of the approximation, we define

p f k 0 = T Q k (p f 0 )
, where T is the truncation between 0 and ess sup Ω p f 0 .

Approximated problem Now, the approximated problem is the following:

Find (v k , p f,k , σ k ) ∈ (L ∞ H∩L 2 V k )×(L ∞ L ∞ ∩L 2 H 1 0,k )×L 2 L 2 sym such that, for all (ψ, r) ∈ V k ×H 1 0,k we have                    ρ ∂ t v k , ψ V ,V + Ω (v k • ∇v k ) • ψ + η Ω ∇v k : ∇ψ + Ω σ k : ∇ψ = f, ψ V ,V , ∂ t p f,k , r H -1 ,H 1 0 + Ω (v k • ∇p f,k ) r + K Ω ∇p f,k • ∇r = 0, (σ k , Dv k ) ∈ G k with q k = q 0 (p s -p f,k ) + , (v k , p f,k )| t=0 = (v k 0 , p f k 0 ). (10)
Thanks to the definition of the graph G k , see the equation ( 9), the stress tensor σ k is explicitly given as a function of q k and v k , so we have decoupled the problem. We can then solve problem (10) using the Schauder theorem and classical estimates (see [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF]).

A priori estimates

The aim of this subsection is to obtain suitable estimates on the solution (v k , p f,k , σ k ) to the approximated problem [START_REF] Forterre | Flows of dense granular media[END_REF], which do not depend on the parameter k.

Proposition 3.1. The solution (v k , p f,k , σ k ) to the problem (10) satisfies the following estimates (where c is a constant which does not depend on k):

1. Uniform bounds on the fluid pressure, plasticity yield and stress tensor

0 ≤ p f,k ≤ ess sup p f 0 , q k L ∞ L ∞ ≤ c, σ k L ∞ L ∞ ≤ c. (11)
2. Energy estimate on the velocity field

v k 2 L ∞ L 2 + ∇v k 2 L 2 L 2 + σ k : Dv k L 1 L 1 ≤ c. (12)
3. Estimate on the momentum

∂ t v k L 2 V ≤ c. ( 13
)
4. Estimate on the fluid pressure and its time derivative

p f,k L 2 H 1 ≤ c and ∂ t p f,k L 2 H -1 ≤ c. ( 14
)
Remark 3.1. In the sequel, we will want to use approximated solutions v k and p f k as testfunctions in [START_REF] Forterre | Flows of dense granular media[END_REF]. It is theoretically not possible because of the time dependency of these functions. In fact, it is possible to avoid the problem, remarking that these functions are linear combinations of space-dependent eigenfunctions, with time-dependent coefficients. For example, function v k reads as

v k (x, t) = k i=1 c i (t)w i (x).
Therefore, it suffices to choose the eigenfunctions as test-functions, multiply each equation by the appropriate coefficient and sum up the results to get the same equation we would have if we formally used v k as a test-function, see [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF], p.349.

Proof. Consider a solution (v k , p f,k , σ k ) to the approximated problem [START_REF] Forterre | Flows of dense granular media[END_REF].

1. The first estimate of (11) comes from to the maximum principle. We also use the definition of the truncation T and the bounds on the initial condition p f 0 in L ∞ . The second estimate of ( 11) is a consequence of the first estimate since we recall that q k = q 0 (p s -p f,k ) + . The last estimate of (11) comes from the definition of σ k .

2. In order to obtain a so-called energy estimate for the velocity we use the usual method: we chose ψ = 2v k as test-function in the weak formulation [START_REF] Forterre | Flows of dense granular media[END_REF]. After classical calculation we obtain

ρ Ω v 2 k (t) + η t 0 Ω |∇v k | 2 + 2 t 0 Ω σ k : Dv k ≤ ρ Ω v 2 k (0) + 2c f 2 L 2 V . (15)
Using the explicit expression of the stress tensor σ k given by the graph definition (9) and the fact that q k is non-negative, we have

σ k : Dv k = q k |Dv k | 2 |Dv k | + 1/k ≥ 0. (16)
We obtain the estimate [START_REF] Iverson | The physics of debris flows[END_REF], noting that the right-hand side member of ( 15) is bounded as follows

ρ Ω v 2 k (0) + c f 2 L 2 V ≤ ρ P k (v 0 ) 2 L 2 + c f 2 L 2 V ≤ ρ v 0 2 L 2 + c f 2 L 2 V ≤ c.
3. In order to evaluate the term ∂ t v k in V , we begin with any ψ ∈ V k and rewrite (10) 1 as

ρ ∂ t v k , ψ V ,V = -ρ Ω (v k • ∇v k ) • ψ -η Ω ∇v k : ∇ψ - Ω σ k : ∇ψ + f, ψ V ,V .
Then, using the divergence-free constraint in the first term of the right hand side, the following inequality holds

ρ ∂ t v k , ψ V ,V ≤ ρ Ω |v k | |∇ψ| |v k | + η Ω |∇v k | |∇ψ| + Ω |σ k | |∇ψ| + f, ψ V ,V .
Now applying H ölder and Poincar é inequalities, for all ψ in V k , ψ = 0, we get

ρ ∂ t v k , ψ V ,V ψ V ≤ ρ v k L 4 v k L 4 + η v k H 1 + σ k L 2 + f V .
Since we work in Ω ⊂ R 2 , the velocity v k belongs to ∈ H 1 . As a consequence of Poincar é inequality (see for example [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF])

we have v k 2 L 4 ≤ c v k L 2 v k H 1 . By integrating on (0, T ), we get ρ ∂ t v k L 2 V ≤ c T 0 v k L 2 v k H 1 + v k H 1 + σ k L 2 + f V 2 dt 1 2 ≤ c v k L ∞ L 2 v k L 2 H 1 + v k L 2 H 1 + σ k L 2 L 2 + f L 2 V .
We conclude thanks to ( 11) and ( 12).

4. The first estimate in ( 14) is obtained using r = 2p f,k as test-function in the weak formulation [START_REF] Forterre | Flows of dense granular media[END_REF]:

Ω p 2 f,k (t) + 2k t 0 Ω |∇p f,k | 2 = Ω p 2 f,k (0) ≤ c.
In order to obtain the second one, we proceed similarly to obtain estimates on the velocity time derivative: we begin with any r ∈ H 1 0,k , r = 0 in (10) 2 and we conclude thanks to the previous results [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF], ( 12) and (14) 1 .

Existence of limits

Limit for the velocity From ( 12) and ( 13), and using Aubin-Lions-Simon lemma (cf [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF]), we deduce that a velocity v exists such that, up to a sub-sequence, we have the following weak, weak and strong convergences:

(C1) v k v in L 2 V, (C1') v k v in L ∞ H, (C2) v k → v in L 2 H.
Limit for the fluid pressure Thanks to estimate (14) of Proposition 3.1 and Aubin-Lions-Simon lemma again, we can also extract a sub-sequence from (p f,k ) k≥1 , which converges to a certain function p f as follows

(C3) p f,k p f in L 2 H 1 , (C3') p f,k p f in L ∞ L ∞ , (C4) p f,k → p f in L 2 L 2 .
Limit related to the Bingham tensor As the domain Ω is bounded, using estimates [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF], there exist q and σ such that, up to a sub-sequence again:

(C5) q k → q in L 2 L 2 , (C6) σ k σ in L 2 L 2 sym .
Note here that convergences (C1') and (C3') prove that the limits v and p f respectively belongs to L ∞ H and L ∞ L ∞ . We do not use it again in the sequel.

Passage to the limit when k tends to infinity

We now prove that the limit (v, p f , σ) is a solution of the problem (8).

Limit in equations and initial conditions

Limit in equations In order to deal with non-linear terms at the limit when k tends to infinity, it suffices to obtain the compactness on approximated solution by establishing an estimate on the time derivative of the velocity. As for homogeneous Navier-Stokes equations, see for instance [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF] for details, we can estimate τ h v k -v k , where τ is a shift operator in time in a suitable space. Let j > 0 be an integer and ψ ∈ V j be a test-function. Then for all k ≥ j,

ρ ∂ t v k , ψ j V ,V + Ω (v k • ∇v k ) • ψ j + η Ω ∇v k : ∇ψ j + Ω σ k : ∇ψ j = f, ψ j V ,V .
The integer j being given, the aim is to let k tend to infinity. From (C1) and (C6), we already know that

• ∇v k ∇v ∈ L 2 L 2 , • σ k σ ∈ L 2 L 2 sym .
In order to treat the terms containing ∂ t v k and v k • ∇v k ,we first estimate (∂ t v k ) k≥1 . This quantity is bounded in L 2 V , then it has a weak limit, which must be ∂ t v by continuity of the time derivative. Consequently, we have

• ∂ t v k ∂ t v ∈ L 2 V .
Furthermore, we can deduce (see [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF], p.357-358 for details) from convergences on v k that

• v k • ∇v k v • ∇v ∈ L 4 3 L 6 5
.

We now take the limit when k tends to infinity and we get the following equality on σ and v :

ρ ∂ t v, ψ j V ,V + Ω (v • ∇v) • ψ j + η Ω ∇v : ∇ψ j + Ω σ : ∇ψ j = f, ψ j V ,V .
As (w i ) i≥1 is total, if ψ ∈ V, then with ψ j = P j (ψ), (ψ j ) j converges to ψ in V. So we can take the limit in each term when j tends to infinity, and prove that (v, σ) satisfies the equation (8) 1 for all ψ ∈ V. In a more classical way, we can prove that the fluid pressure p f satisfies (8) 2 for all r ∈ H 1 0 (Ω).

Initial conditions

We apply twice the theorem II.5.13 p.101 in [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF] to prove that the limits v and p f are continuous on [0, T ]. More precisely: for the velocity, thanks to ( 12) and ( 13) we deduce that v ∈ C 0 ([0, T ], H), and

v| t=0 = v 0 a.e.
For the fluid pressure, from ( 14) we deduce that p f ∈ C 0 ([0, T ], L 2 (Ω)), and

p f | t=0 = p f 0 a.e.

Passage to the limit in the graph G k

The last step of the proof is to show that (σ, Dv) ∈ G[q]. More precisely, we already know that Dv belongs to L ∞ H -1 ∩ L 2 L 2 and that σ belongs to L 2 L 2 sym . We will prove that (see the definition [START_REF] Ček | On unsteady flows of implicitly constituted incompressible fluids[END_REF] of the graph G[q]):

|σ| ≤ q and σ :

Dv = q |Dv| . ( 17
)
Step 1 -To prove the first relation of ( 17) we note that by definition of G k , see [START_REF] Drucker | Soil mechanics and plasticity analysis or limit design[END_REF], for all k ≥ 0, we have

σ k = q k Dv k |Dv k |+1/k , so that |σ k | ≤ q k in (0, T ) × Ω.
We easily deduce that for all (t, t + h) ⊂ (0, T ) and for all subset ω ⊂ Ω we have

t+h t ω |σ k | ≤ t+h t ω q k .
But we know (see (C5) and (C6)) that

q k → q ∈ L 2 L 2 and σ k σ ∈ L 2 L 2 sym .
As the domain Ω is bounded, we can take the limit when k tends to infinity:

∀(t, t + h) ⊂ (0, T ) ∀ω ⊂ Ω σ L 1 ((t,t+h)×ω) ≤ q L 1 ((t,t+h)×ω) .
So we finally conclude that |σ| ≤ q a.e. in (0, T ) × Ω.

Step 2 -Next steps consist of checking the second relation of [START_REF] Muravleva | Uzawa method on semi-staggered grids for unsteady Bingham media flows[END_REF]. Note first that the difference between q k |Dv k | and σ k : Dv k converges to 0 when k tends to infinity. Indeed, let's compute the difference:

q k |Dv k | -σ k : Dv k = q k 1 k |Dv k | |Dv k | + 1/k ,
and thanks to (11) the right-hand side is essentially bounded by 1 k c, and thus converges to 0 in L ∞ . Therefore, it suffices to prove that

(a) q k |Dv k | q |Dv| in L 1 L 1 and (b) σ k : Dv k σ : Dv in L 1 L 1 .
Step 2 (a) -Due to strong-weak convergence results, as q k → q ∈ L 2 L 2 (see (C5)) and Dv k Dv ∈ L 2 L 2 , see (C1), we have

q k |Dv k | q |Dv| ∈ L 1 L 1 .

Now we must prove (b).

Step 2 (b).1 -Using ψ = v k as a test-function in the first equation of [START_REF] Forterre | Flows of dense granular media[END_REF] and taking into account divergence-free constraint, we obtain

Ω σ k : Dv k = f, v k V ,V - 1 2 ρ d t Ω |v k | 2 -η Ω |∇v k | 2 . (18)
Integrating [START_REF] Philippe | Start and stop of an avalanche in a granular medium subjected to an inner water flow[END_REF] with respect to the time between 0 and t ∈ (0, T ), we have

t 0 Ω σ k : Dv k = t 0 f, v k V ,V + 1 2 ρ v k (0) 2 L 2 - 1 2 ρ v k (t) 2 L 2 -η t 0 ∇v k 2 L 2 .
Taking the limit k → +∞ (and using the fact that lim inf k→∞ g k X ≥ g X as soon as g k g ∈ X,

see [START_REF] Brezis | Analyse fonctionnelle : Th éorie et applications[END_REF]), we deduce

lim sup k→+∞ t 0 Ω σ k : Dv k ≤ t 0 f, v V ,V + 1 2 ρ v 0 2 L 2 - 1 2 ρ v(t) 2 L 2 -η t 0 ∇v 2 L 2 . (19)
If we now choose ψ = v as a test-function in ( 8) 1 (see Remark 3.2 for more precision) and use the divergence-free constraint, we obtain

Ω σ : Dv = f, v V ,V - 1 2 ρ d t Ω |v| 2 -η Ω |∇v| 2 . ( 20
)
Then we integrate in time the equality (20) from 0 to t:

t 0 Ω σ : Dv = t 0 f, v V ,V + 1 2 ρ v 0 2 L 2 - 1 2 ρ v(t) 2 L 2 -η t 0 ∇v 2 L 2 . (21)
Therefore, combining this equation ( 21) with the inequality (19) it follows

lim sup k→+∞ t 0 Ω σ k : Dv k ≤ t 0 Ω σ : Dv. (22)
Step 2 (b).2 -Now we use the fact that we have an idea of what σ must be. Indeed, thanks to (5), when Dv does not vanish, σ is equal to q Dv |Dv| , but when Dv vanishes we don't know the value of σ. The idea is that we can choose zero as a default value. In this way, let's define two explicit functions σ * = σ * (q, A) and

σ * k = σ * k (q, A) on R × S d (R) by σ * (q, A) = q A |A| if A = 0 0 either,
and for any k ∈ N * ,

σ * k (q, A) = q A |A|+1/k if A = 0 0 either.
By the density of D((0,

T ) × Ω) in L 2 L 2 , we consider a sequence (D i ) i≥1 ⊂ D((0, T ) × Ω), which satisfies (C7) D i i - → Dv ∈ L 2 L 2 .
Noting that (σ * (q, D i )) i≥1 is bounded in L 2 L 2 , we can extract a sub-sequence (again indexed by i ≥ 1), which converges in L 2 L 2 . Let's denote the limit by χ, so that we have

σ * (q, D i ) χ ∈ L 2 L 2 . ( 23
)
Furthermore, for any integer i, the sequence (σ * k (q, D i )) k≥1 converges and the limit when k tends to infinity is σ * (q, D i ).

In order to prove a convergence in L 1 L 1 , let's fix the index i and estimate the following quantity as k tends to infinity:

t 0 Ω (σ k -σ * k (q k , D i )) : (Dv k -D i ) = t 0 Ω σ k : Dv k - t 0 Ω σ k : D i + t 0 Ω σ * k (q k , D i ) : D i - t 0 Ω σ * k (q k , D i ) : Dv k .
We now look at the convergence for each term in the right-hand side except for the first one:

As σ k σ ∈ L 2 L 2 (see (C6)), we have lim sup k→∞ - t 0 Ω σ k : D i = - t 0 Ω σ : D i .
Thanks to the strong convergence of (q k ) k≥1 in L 2 L 2 , we can say that σ * k (q k , D i ) -→ k σ * (q, D i ) ∈ L 2 L 2 and we deduce that

lim sup k→∞ t 0 Ω σ * k (q k , D i ) : D i = t 0 Ω σ * (q, D i ) : D i .
From convergence (C1) and considering again that

σ * k (q k , D i ) -→ k σ * (q, D i ) ∈ L 2 L 2 , we
apply the weak-strong convergence theorem to deduce that lim sup

k→∞ t 0 Ω σ * k (q k , D i ) : Dv k = t 0 Ω σ * (q, D i ) : Dv.
Therefore, for any positive integer i we have

lim sup k→∞ t 0 Ω (σ k -σ * k (q k , D i )) : (Dv k -D i ) = lim sup k→∞ t 0 Ω σ k : Dv k - t 0 Ω σ : D i + t 0 Ω σ * (q, D i ) : (D i -Dv).
Now let i tend to infinity. Recalling that (D i ) i≥1 converges strongly to Dv in L 2 L 2 and using (23), the last term in the right-hand side tends to zero. Finally, thanks to (22), we have

lim i→∞ lim sup k→∞ t 0 Ω (σ k -σ * k (q k , D i )) : (Dv k -D i ) = lim sup k→∞ t 0 Ω σ k : Dv k - t 0 Ω σ : Dv ≤ 0. ( 24
)
Step 2 (b).3 -As (σ * k (q k , D i ), D i ) and (σ k , Dv k ) are in the same graph G k , thanks to Lemma 2.2 we have (σ k -σ * k (q k , D i )) : (Dv k -D i ) ≥ 0. Therefore, "lim sup k " which appears above reads "lim k " and inequalities in (24) are equalities, i.e.

lim i,k→∞ t 0 Ω |(σ k -σ * k (q k , D i )) : (Dv k -D i )| = 0, i.e. (σ k -σ * k (q k , D i )) : (Dv k -D i ) converges to 0 in L 1 L 1
when k and i tend to infinity, so that for any ϕ ∈ L ∞ , we have

lim i,k→∞ t 0 Ω ((σ k -σ * k (q k , D i )) : (Dv k -D i )) ϕ = 0. (25)
We now look for the limit of σ * k (q k , D i ). Note that for all i ≥ 1 and k ≥ 1, σ * k (q k , D i ) L 2 L 2 ≤ q L 2 L 2 , then we can chose any index between i and k first to compute the limit. For more convenience, we begin with i. Therefore, we set any k ≥ 1 and compute the difference

σ * k (q k , D i ) -σ * k (q k , Dv) = q k D i |D i | + 1/k -q k Dv |D i | + 1/k + q k Dv |D i | + 1/k -q k Dv |Dv| + 1/k = q k D i -Dv |D i | + 1/k + Dv |Dv| -|D i | (|D i | + 1/k)(|Dv| + 1/k) .
Thereby, thanks to (C7) we have

σ * k (q k , D i ) i - → σ * k (q k , Dv) ∈ L 2 L 2 . Then, we can observe that Dv |Dv| + 1/k converges almost surely in (0, T ) × Ω to Dv |Dv| if Dv = 0 0 if Dv = 0.
As the sequence Dv |Dv|+1/k k≥1 is uniformly bounded by 1, now let k tend to infinity and apply Lebesgue's dominated convergence theorem, so that

Dv |Dv| + 1/k k -→ Dv |Dv| if Dv = 0 0 if Dv = 0 ∈ L 2 L 2 .
Then, thanks to (C5) the product

q k Dv |Dv|+1/k = σ * k (q k , Dv) converges strongly in L 1 L 1 : σ * k (q k , Dv) k -→ σ * (q, Dv) ∈ L 1 L 1 .
But we know that σ * k (q k , Dv) -σ * (q, Dv) is uniformly essentially bounded in (0, T ) × Ω (see [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF] 2 ). Hence the difference converging to 0 also in L 2 L 2 , and we have

σ * k (q k , D i ) i,k -→ σ * (q, Dv) ∈ L 2 L 2 . (26)
Combining (C1) with (26), we automatically get

σ * k (q k , D i ) : (Dv k -D i ) i,k 0 ∈ L 1 L 1 .
Finally from (25) and for any ϕ ∈ L ∞ , the following convergence remains:

lim i,k→∞ t 0 Ω (σ k : (Dv k -D i )) ϕ = 0, i.e. lim k→∞ t 0 Ω (σ k : Dv k )ϕ = lim i,k→∞ t 0 Ω (σ k : D i )ϕ.
But from (C6) and (C7) we compute the limit on the right-hand side above, which is equal to t 0 Ω (σ : Dv)ϕ. Thereby, we prove that

σ k : Dv k σ : Dv ∈ L 1 L 1 .
From result of Step 2 (a) and (b), by uniqueness of the limit we have the equality σ : Dv = q |Dv| a.e.

This proves the second relation of ( 17) so (σ, Dv) ∈ G[q], which concludes the proof.

Remark 3.2. As for test-function ψ = v k , it is theoretically not possible to choose ψ = v as a test-function in [START_REF] Chupin | A bi-projection method for Bingham type flows[END_REF]. But in the case of bi-dimensional flows, we have

∂ t v ∈ L 2 V .
So if we regularize the velocity v with respect to the time, say v ε = η ε (t) v and lim ε→0 v ε = v in L 2 V, then we can use it instead of v and take the limit as ε tends to 0. The result is the one we obtain by formally using v as a test-function.

Note that this method does not work in a three-dimensional case, because we only have

∂ t v ∈ L 4 3 V .

An example of flow with variable plasticity yield

In this section, we provide a numerical illustration of the previous model which highlights the effect of fluidization on the flow. We consider the following model:

           ∂ t v + v • ∇v + ∇p -∆v = 0.2 div (1 -p f ) + Dv |Dv| , div v = 0, ∂ t p f + v • ∇p f -0.1 ∆p f = 0. (27)
In order to illustrate the influence of the diffusion of the fluid pressure p f , we look at the behaviour of a fluid with such a rheology in a 2D-channel. The fluid enters the channel with a Poiseuille-type profile for velocity and flows out freely. Its rheology depends on the degree of fluidization, which is just given by (2). When testing the behaviour of a simple Bingham fluid in this configuration, we set p f = 0 everywhere, so that (1 -p f ) + = 0. But when we compute the same flow with a defluidization process, the fluid pressure equals the lithostatic pressure everywhere inside at the beginning and at the inlet, so that the fluid is initially totally fluidized and when entering the channel. The corresponding dimensionless boundary conditions for the interstitial pore pressure p f are given by 1 at the inlet and 0 (i.e. atmospheric pressure) on the other boundaries. Initial velocity is given by the steady state solution of the equivalent Poiseuille-type problem for a pure viscous fluid and initial fluid pressure depends on the experimental set-up.

The numerical scheme used to compute the solution we present here is described and analyzed in [START_REF] Chupin | A bi-projection method for Bingham type flows[END_REF]. It is a second-order projection method with respect to the time based on the combination of the BDF2 (second-order Backward Differentiation Formula) and the AB2 (Adams-Bashforth).

The non-linearity due to the Bingham threshold is treated using a projection method (see [START_REF] Chupin | A bi-projection method for Bingham type flows[END_REF] for details on the numerical analysis). A Cartesian uniform mesh is used for the spatial discretisation, and a classical MAC1 scheme is introduced in order to treat the incompressibility.

The parameters of the momentum equation for these numerical tests are usual ones (see [START_REF] Muravleva | Uzawa method on semi-staggered grids for unsteady Bingham media flows[END_REF]).

Other concrete examples of this type of model applied to real physical problems represent a work in progress. We illustrate the influence of the presence of the fluid pressure through the geometry of the rigid zones. In the case of a pure Bingham fluid, in the steady state the rigid zone corresponds to the plug in the middle of the channel. We can observe that in the Figure 1, describing the flow corresponding to an established Bingham flow. We can observe that the Poiseuille flow imposed on the left rapidly becomes a plug flow characterizing the Bingham flow. In a second simulation, we are looking at in the same quantities (plug zones and velocity profile) for a fluid taking into account the gas pressure. The rigid zone is thinner where the parameter q does not vanish, whereas it has quite the same behaviour where q = 0 (see Figure 2 and3).

Clearly the plug flow is being established further from the source corresponding to x = 0.

Conclusions and prospects

We have established existence of a weak solution to bi-dimensional homogeneous Bingham fluids with variable shear stress, the main new feature of the model being the coupling of the rheology with another partial differential equation through the definition of the order-parameter. This work is an extension in the field of existence theorems for homogeneous fluids (Navier-Stokes [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF], Bingham, Herschel-Bulkley [START_REF] Ček | On unsteady flows of implicitly constituted incompressible fluids[END_REF]). In order to compute some free surface-like flows, it would be interesting to complete the study by taking into account a variable density. Some results already exist for Bingham [START_REF] Amirat | Nonhomogeneous incompressible Herschel-Bulkley fluid flows between two eccentric cylinders[END_REF] and Herschel-Bulkley [START_REF] Basov | Nonhomogeneous incompressible Bingham viscoplastic as a limit of nonlinear fluids[END_REF] fluids, but only with constant plasticity yield. Also, following the case treated in [START_REF] Ček | On unsteady flows of implicitly constituted incompressible fluids[END_REF], it seems to be possible to extend the result to the three-dimensional case and thus to include more physical flows.

Figure 1 :

 1 Figure 1: The non-fluidized case (corresponding to the Bingham model) at t = 30. Top: Rigid zone (in the middle of the channel) developed in the flow. Bottom: Horizontal velocity profile at different positions.

Figure 2 :

 2 Figure 2: The fluidized case (variable plasticity yield) at t = 30. Top: Rigid zone (in the middle of the channel) developed in the flow. Bottom: Horizontal velocity profile at different positions.

Figure 3 :

 3 Figure 3: Distribution of the interstitial gas pressure p f in the flow of a Bingham fluid at t = 30.

Marker And Cell, see[START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF]

Acknowledgments

We would like to thank Karim Kelfoun and Olivier Roche (L.M.V.) for their active participation to the development of the ideas of this article, especially in the creation of the model. This research was partially supported by the French Government Laboratory of Excellence initiative noANR-10-LABX-0006, the R égion Auvergne and the European Regional Development Fund. This is Laboratory of Excellence ClerVolc contribution number 218.


Francis Bloch
email: francis.bloch@univ-paris1.fr.

Gabrielle Demange
email: demange@pse.ens.fr.

M Bacache

P J Benghozi

M Bourreau

B Caillaud

J Cremer

S Gauthier

L Gille

J Hamelin

L Janin

J M Lozachmeur

Taxation and Privacy Protection on Internet Platforms *

Keywords: JEL classification numbers: H23, L86, L50 Digital services, Privacy protection, Taxation, Opt-out and opt-in options

This paper studies data collection by a monopolistic internet platform We show that the optimal strategy of the platform is either to cover the market or to choose the highest data exploitation level, excluding users with high privacy costs from the platform. For likely parameter values, the platform chooses an excessive level of data exploitation from the point of view of users. We study how different tax instruments can be used to reduce the level of data collection and analyze the effect of an opting-out option, letting users access the platform with no data collection. We show that a differentiated tax, taxing access revenues and data revenues at different rates is the most effective instrument and that the introduction of an opting-out option may harm users as it induces the platform to raise the level of data exploitation.

Introduction

The precipitous decline in the cost of data collection and storage linked to the development of information technologies has transformed business models in advertising and commerce. While records on customers and sales histories have always existed, the digital economy now enables firms to exploit data at a much larger scale, opening up new opportunities for profit as well as new concerns about privacy and exploitation of personal data. Large sales platforms can now use detailed records of past sales histories to target users and engage in discriminatory dynamic pricing. Other platforms, like search engines or online social networks, use data on immediate search to auction off advertising spaces to clients, or sell search histories to intermediaries who accumulate data to better target users with ads. Business models of all giant internet platforms rely at different degrees on the collection and exploitation of personal data. The use of personal data is clearly one of the main specificities of the digital sector in modern industrial economies.

The development of "big data" and its potential exploitation raises two separate questions.

First, data are a valuable input for internet platforms, but users voluntarily upload their data without any payment. One can argue that internet platforms are engaged in a barter agreement, where platforms deliver a valuable service (targeted proposals for products, targeted ads, outcomes of search, access to friends) in exchange for the uploading of data. But absent any price and financial transaction, it is difficult to assess whether this barter is "fair" and if users receive a fair share of the surplus. The immense profits of (some not all) internet platforms suggests that it may not be the case and that platforms benefit from a "free" input which is not paid at its true value. Furthermore, in the absence of financial transactions, governments cannot properly tax the benefit of personal data, creating a distortion with respect to other sectors, clouding the territoriality principle for the taxation of profits, and leading to extremely low levels of taxation of internet platforms through a clever use of transfer prices and the absence of records of financial transaction in countries where users reside. Second, users are rightly afraid that the collection of personal data infringes on their privacy. In addition, the resale of data to unknown intermediaries through opaque arrangements results in a loss of control on the dissemination of personal data to third parties. The exploitation of data, while it provides a valuable service to users by improving targeting, also necessarily involves a cost in privacy loss.

Even though the two problems of the absence of fair payment of data and privacy loss seem unrelated at first glance, they are in fact closely connected. In this paper, we study how regulatory instruments, and in particular various forms of taxation, can be used to solve both problems at once. We construct a model where users are differentiated only along their privacy cost. Their (identical) benefit of accessing the platform is made of two components, one derived from the access to the platform (say from the use of the search engine or the purchase of a good) and one which depends in an increasing way on the collection of data. Data collection is represented by a scalar, which might be interpreted as the duration of time during which personal histories are stored by the platform or the extent to which geo-localization is used. The collection of data enables the platform to better target offers to users and users to products or advertisers, resulting in an increase in the benefit to users as well as the value of users to advertisers or to the firm. Initially, we suppose, following current usage in the digital industry, that platforms do not charge users for their service and that their entire revenue comes from the other side of the market (advertisers or sales of future goods). A monopolistic platform chooses (and commits to) a degree of data exploitation balancing two effects: on the one hand, an increase in data collection increases revenues by increasing the value of the user to advertisers or for targeted pricing, on the other hand, an increase in data collection may deter users with high privacy cost to access the platform.

In this model, we first compute the optimal level of data exploitation chosen by the platform and the users. We show that the platform either chooses to cover the market making sure that all users access the platform, or chooses the maximal level of data exploitation, thereby excluding some users from the platform. The welfare of users is also maximized at one of two values: either at a low value in which all users access the platform (hence the value is lower than level of market coverage) or at the maximal degree of data exploitation. The optimal levels of data collection from the point of view of the platform and users are characterized as a function of the particular shape of the revenue and benefit functions. In the likely situation where the access benefit is large relative to the benefit of data collection or when the marginal benefit of data collection decreases sharply with the amount of data, the optimal level of data collection of the platform is excessive from the point of view of users. Hence the objective of the regulator is to select instruments to reduce the level of data collection and exploitation of the platform.

We then study how different forms of taxation affect data collection. We first observe that a tax on profits (or equivalently a tax on revenues because variable costs are negligible) does not affect the choice of the platform. A tax paid by the platform per user does not affect the marginal benefit of data exploitation, but it reduces the profit made on the marginal user accessing the platform, thereby reducing the cost of data collection. Hence, a tax per user (or per flow of data as users do not choose the level of data they upload), results in an increase in data exploitation. A specific tax paid by users (like a tax on internet service providers) reduces the level of data exploitation when the market is covered, but also increases the region of parameters for which exclusion occurs. Hence, it produces ambiguous effects on the welfare of users. The only tax which allows to correct for excessive data collection is a tax on revenues which treats differentially platform's revenues accruing from one-time use (like auction revenues based on current keywords) and revenues linked to data collection (like resale of data to intermediaries).

If fiscal authorities charge a higher tax level on resale of data than on auction revenues, taxes deter the platform from exploiting the data, playing the classical role of a Pigovian tax correcting for externalities.

We then explore the effect of the introduction of an option for the users to access the platform with no data collection. This allows the platform to collect access revenue from those users who choose to opt-out, segmenting the market into two groups with different revenue levels. Users will now all access the platform -some with data collection and others without. We show that this changes the level of data collection chosen by the platform, resulting in a decrease in the maximal level of data exploitation for which the market is covered, but in an increase in the region of parameters for which the platform chooses the maximal degree of data exploitation. We show that for the most likely values of the parameters, the platform's profit decreases with the introduction of the opt-out option, as this creates competition with the regular service with data collection and hence reduces the platform's ability to generate data revenues. More surprisingly, we show that users may also be harmed, because the introduction of the opt-out option may lead the platform to switch from a r/'egime of market coverage to the maximal level of data exploitation, hurting users with low privacy costs who choose to access the platform with data collection.

Our analysis of the effect of taxation and regulation on data collection is related to two strands of the literature.

First, it is related to the literature on the economics of media, which considers a media (television, newspaper) as a platform in a two-sided market connecting readers with advertisers and collecting revenues from the advertising market. (See Gabsewicz, Laussel and Sonnac (2001) and ( 2004) for early contributions to the literature and the survey by [START_REF] Anderson | The media and advertising: A tale of two-sided markets[END_REF]). Advertisement in these models play the same role as data collection in ours. As in our model, users are assumed to suffer a linear cost from advertisements. Though there are two main differences between our analysis and this literature. One stems from the questions raised, the other on the modelling assumptions. The recent literature on media as two-sided platforms mostly focuses on program differentiation and competition, while we are interested in regulatory régimes and taxation of a monopolistic platform to improve privacy protection. For instance [START_REF] Peitz | Content and advertising in the media: Pay-tv vs freeto-air[END_REF], [START_REF] Choi | Broadcast competition and advertising with free entry: Subscription vs. free-to-air[END_REF], [START_REF] Crampes | Advertising, competition and entry in media industries[END_REF]) allow for subscription prices charged to viewers, and compare regimes of "free-to-air" with "pay-for-view" televisions. [START_REF] Reisinger | Two-sided markets with pecuniary and participation externatlities[END_REF] and [START_REF] Reisinger | Platform competition for advertisers and users in media markets[END_REF] pay close attention to competition among advertisers on different platforms. Casadesus-Masanell and Hervas-Drane (2015) analyze a model where consumers choose the amount of data they upload and platforms choose a two-pronged policy to extract access and data revenues. They show that platforms typically specialize in access or data collection revenues. The papers that consider a monopolistic platform rely on different modeling assumptions, which precludes a direct comparison between our results and theirs (see e.g. [START_REF] Anderson | Platform siphoning: Ad-avoidance and media content[END_REF] and [START_REF] Johnson | Targeted advertising and advertising avoidance[END_REF] who examine the reaction of a platform to ad-blocking technologies). For example [START_REF] Anderson | Market provision of broadcasting: A welfare analysis[END_REF] analyze the behavior of a monopolistic platform when the market is not covered, in their model, users are differentiated by their intrinsic benefit from the good and not their aversion to advertising, resulting in very different demand functions and different conclusions from here. In particular, they find that platforms typically choose too little advertising whereas we observe that the level of data collected by the platform is likely to be excessive. [START_REF] Tag | Paying to remove advertisements[END_REF] introduces the possibility for consumers to pay to pay to remove advertisements. Almost all models assume that users suffer a linearly increasing utility loss for ads, whereas we assume that, in addition to a linearly decreasing loss, they obtain a concave utility gain due to improved service. [START_REF] Crampes | Advertising, competition and entry in media industries[END_REF] however allow users to have a positive value for low levels of ads, which is similar but not equivalent to our assumption. De Cornière and de Nijs (2016) also discuss privacy in a monopolistic two-sided platform, but with a different angle. They study how the platform's choice between privacy and disclosure affect competition among advertisers.

Second, our paper is more distantly related to the literature on taxation on two-sided markets (see [START_REF] Kind | Efficiency enhancing taxation in two-sided markets[END_REF], (2010a), (2010b), [START_REF] Kind | Newspaper Differentiation and Investments in Journalism: The Role of Tax Policy[END_REF] and [START_REF] Kotsogiannis | Public goods and tax competition in a two-sided market[END_REF]), but again the focus of the analyses are different, as we focus on the effect of taxes on privacy protection rather than revenues and distortion. Finally, we note that our paper is closely related to the two contributions by [START_REF] Crémer | Taxing network externalities[END_REF] and [START_REF] Bourreau | Digital platforms, advertising and taxation[END_REF] as all three papers consider the effect of specific taxation on the behavior of internet platforms, even though the three papers rely on different models of internet platforms and focus on different choice variables.

The rest of the paper is organized as follows. We introduce our model in the next Section.

Section 3 is devoted to the baseline model. Section 4 analyzes the effect of different tax instruments. Section 5 considers the introduction of an option to access the platform with no data collection. Section 6 concludes. All proofs not given in the texts are collected in Section 7.

The Model

Platform and users

The platform We consider an internet platform which provides services to users. The platform collects revenues either directly (as in the case of e-commerce) or from third parties like advertisers in the case of search engines or digital social networks. The platform also collects data from users, recording their history of activity on the platform. We distinguish between two sources of revenues: some of the revenues, like sales revenues from e-commerce or instantaneous search-base advertising revenues are collected immediately, whether the platform records data on the customer or not. Other revenues, like the sale of personal data to aggregators or the expected revenue from future targeted advertising, only accrue if personal data are collected and stored. We suppose that the platform commits to the degree of exploitation of personal data, denoted x ∈ [0, 1]. The degree x can be interpreted along different dimensions. It can represent the duration of time during which personal histories are stored by the platform, the fraction of personal data which are sold by the platform to third parties or kept for direct exploitation, or any specified limitation on the use of personal data. We denote by v 0 the fixed value generated by a user independently of data collection (the access revenue) and by v(x) the value generated by data exploitation (data revenue). The function v(•) is supposed to be iso-elastic so that the total value of a user to the platform is

v 0 + v(x) where v(x) = bx β for some b > 0, 0 < β < 1.
Finally, we assume the marginal cost of providing digital services is negligible.

Users Users get a benefit from using the platform, which is also decomposed into a fixed positive component u 0 (the access benefit), and a component which depends on the collection of data x (the data benefit). The fixed component u 0 corresponds to the benefit from accessing the platform, i.e. buying a good, obtaining results from a query on a search engine, contacting friends on social networks. Data collection allows the platform to better match the user with a product and hence results in an improvement in the service. We let u(x) denote the benefit from the improvement in match quality, where u(•) is supposed to be iso-elastic. In addition, we suppose that users are aware that they may be harmed by the collection of storage and personal data, and suffer a privacy cost which we assume proportional to the degree of data exploitation.

While users are identical in all other characteristics (in particular their value to advertisers and the benefit they obtain from the platform), we suppose that they are heterogeneous in their sensitivity to privacy loss. We let θ denote the characteristic of the user with privacy cost θx, called a θ-user. We assume the distribution of privacy costs in the population of users, θ, to be uniform over [0,1]. The utility of a θ-user can be written as

u 0 + u(x) -θx where u(x) = ax α for some a > 0, 0 < α < 1.
We normalize the payoff of a user who does not access the platform to zero.

In the baseline scenario, users access the platform letting the platform use their data. Given the positive level of data exploitation x, a θ-user chooses to access the platform if and only if

u 0 + u(x) -θx ≥ 0.
For x small enough,1 u 0 + u(x) -x ≥ 0 so that all users access the platform. Otherwise, u 0 + u(x) -x < 0, and the characteristic of the marginal user accessing the platform is given by the solution to the equation: u 0 + u(x) -θx = 0. So letting T (x) defined by

T (x) = max( u 0 + u(x) x , 1) (1)
all users with characteristic in [0, T (x)] access the platform and the demand is given by T (x).

The assumptions of iso-elasticity on the data revenue and user's benefit functions can be relaxed as discussed at the end of Sections 3 and 4.

Our model can be compared to models of interaction on media markets when users are averse to advertisement. In these models, x is interpreted as the level of advertising, v(x) the revenues from advertising and θx the cost of ads to users. Models of advertising in media markets typically assume that v 0 = 0 (the media does not make any direct revenue from users) and that u(x) = 0 (users have no positive value for advertising).2 A more important difference stems from the dimension of heterogeneity of users. We assume that users have the same value for the service but different privacy costs, whereas models of media markets suppose that users have the same cost of advertising but differ in their value of the service or in their ideal point in models of horizontal differentiation across media. This difference between our model and other models of media with advertising avoidance results in non-trivial consequences. When users differ in their value for the service or ideal points, demand is linearly decreasing in x ; when users differ in their advertising or privacy cost, demand is no longer linear and in fact includes a part u 0 x which is decreasing and convex in x. Hence profit is typically concave in models with heterogeneity in values, but not in models with heterogeneity in sensitivity to advertising or privacy. This leads to a major difference in the characterization of optimal levels of advertising or data exploitation.

Regulatory instruments

The objective of the paper is to assess how different regulatory instruments affect the level of data collection chosen by the platform and the welfare of users. We distinguish between tax instruments and a regulation imposing an opt-out option. We investigate the incidence of four different tax instruments:

• A tax τ levied on the revenue of the platform • A tax t P levied on the platform per user

• A differentiated revenues tax system, with two different tax rates τ 1 and τ 2 applied to the revenues v 0 and v(x)

• A tax t U levied on each user for accessing the platform.

As an alternative regulation, we consider a policy mandating that the platform offer a choice to the user between access with data collection and an opt-out option without data collection.

We focus on the case where the opt-out option is offered at no charge to the user.

The baseline model

In the baseline model, there is no public intervention and the platform freely chooses the degree of data exploitation x. Our aim in this Section is to compare the optimal level of data exploitation chosen by the platform and the users. The profit of the platform is given by

Π(x) = [v 0 + v(x)]T (x).
(2)

The user's welfare is defined as the sum of the utility levels of all users. As the benefit of users who do not access the platform is normalized to zero, the user's welfare with data collection

x is given by

W (x) = T (x) 0 [u 0 + u(x) -θx]dθ. (3)
For x low enough, all users access the platform (for u 0 > 0). As x increases, users with high privacy cost stop accessing the platform, and the demand becomes decreasing in x. We define the coverage level to be the maximal value of data exploitation for which all users access the platform.

The coverage level The coverage level will be used to characterize the optimal behavior of the platform and discuss the effect of regulatory instruments. It is convenient to emphasize the dependance of the coverage level with respect to the access benefit u 0 . As a preliminary Lemma, we derive some comparative statics properties of the coverage level ξ.

Property 1 Let ξ(u 0 ) be the coverage level: T (x) = 1 if and only if x ≤ ξ(u 0 ).

If u 0 + a > 1, the coverage level ξ(u 0 ) is 1: all users access the platform at the maximal degree of data exploitation x = 1.

If u 0 + a ≤ 1, the coverage level ξ(u 0 ) is the unique solution in (0, 1) to the equation:

u 0 + ax α -x = 0. ( 4
)
ξ is increasing in u 0 ; when u 0 tends to 0, ξ(u 0 ) tends to ξ(0

) = a 1 1-α ; ξ(1 -a) = 1. ξ is
increasing in a and decreasing in α.

Not surprisingly, an increase in the access benefit and a uniform increase in the data benefit increase the utility of users, and raises the coverage level. The effect of a change in α is less

obvious. An increase in α makes the benefit less concave in data and hence reduces the marginal benefit of data for low values of x, lowering the coverage level.

In the sequel of the paper, we consider the interesting case where the coverage level is interior and suppose that u 0 + a < 1.

The platform's choice

To compute the optimal level of data collection of the platform, we observe that the expression of the platform's profit differs in the régime where the market is covered, when x ≤ ξ(u 0 ) and where some users do not access the platform, when x > ξ(u 0 ):

Π(x) = v 0 + v(x) if x ≤ ξ(u 0 ), = [v 0 + v(x)] u 0 + u(x) x if x ≥ ξ(u 0 ).
When the level of data exploitation is below the coverage level, the profit is strictly increasing in

x because the demand stays at the maximum level. When the level of data exploitation is above the coverage level, the profit might no longer be increasing because the demand T (x) = u 0 +u(x)

x is decreasing in x. With our iso-elastic specification, the elasticity of the data benefit function, u(•), is constant and smaller than 1. The marginal data benefit thus u (x) decreases less rapidly than the average benefit u(x)

x , so that the demand T (x) decreases slowly with an increase in x. Similarly, the marginal value of data exploitation v (x) decreases less rapidly than the average value v(x) x . This implies a weak form of convexity for the profit, namely that the elasticity of profit with respect to data exploitation, xΠ (x) Π(x) is increasing in x. As a result, the platform will never find it optimal to choose an intermediate value and the optimal choice is either equal to ξ(u 0 ) or 1.

There are thus two possible régimes depending on the values of the parameters: a régime of market coverage where the platform chooses x = ξ(u 0 ) and serves all users and a régime of exclusion where the platform selects x = 1 and serves users with a privacy cost smaller than u 0 + a. The values of the parameters determining each régime are given in next Proposition.

Proposition 1 The platform optimally chooses a degree of data exploitation equal either to the coverage level ξ(u 0 ) or to the maximal level 1. It chooses ξ(u 0 ) if and only if

v 0 b ≥ ν(u 0 ) ≡ u 0 +a-ξ(u 0 ) β 1-u 0 -a .
The threshold ν(u 0 ) is increasing in α and β and decreasing in u 0 .

Proposition 1 establishes the existence of a threshold ν(u 0 ) such that the platform chooses market coverage when the market when the fraction v 0 b is larger than ν(u 0 ) and exclusion otherwise. This result is easily understood. The fraction v 0 b measures the relative importance of the access revenue over the data revenue due. When the access revenue is large relative to the data revenue, the platform has an incentive to serve all users and hence will choose x low enough so that it covers the market. If the value of data exploitation is large relative to the value of access, the opposing intuition holds and the platform optimally chooses to restrict access and set a high degree of data exploitation. When the access benefit u 0 increases, the coverage level ξ(u 0 ) goes up, making market coverage more attractive to the platform. By contrast, an increase in α makes the data benefit less concave and reduces the coverage level, making exclusion more attractive. When β increases, the data revenue becomes less concave, increasing the effect of a change in x on the platform's profit, and thereby making exclusion more attractive.

These comparative statics effects of α and β can be illustrated in the simple case where u 0 = 0. We compute ν(0) = a(1-a α+β-1 ) 1-a . As a < 1, the sign of ν(0) is equal to the opposite of the sign of α + β -1, so that ν(0) < 0 if α + β < 1. As the threshold is decreasing in u 0 , we obtain the following corollary, characterizing conditions under which market coverage and exclusion are chosen by the platform:

Corollary 1 For α + β ≤ 1, the platform always chooses market coverage. For α + β > 1, there exists a threshold of the access benefit, u 0 such that the platform chooses market coverage for u 0 ≥ u 0 and market exclusion for u 0 < u 0 .

Users' welfare

The value of users' welfare also assumes two different expressions, depending on whether market is covered or not:

W (x) = u 0 + u(x) - x 2 if x ≤ ξ(u 0 ) = = 1 2x [u 0 + u(x)] 2 if x ≥ ξ(u 0 ).
When the level of data exploitation is below the coverage level, all users access the platform, and welfare is strictly concave in x. When the level of data exploitation is above the coverage level, the expression for welfare captures the fact that some users are excluded from the platform, and welfare is no longer necessarily concave in the level of data exploitation x. We note that, at the boundary value ξ(u 0 ), the utility of a marginal user is identical in the two régimes so that the welfare function W (x) is differentiable. Thus, contrary to the platform's profit, the welfare function does not present a kink at the coverage level and the optimal level of data collection will generically not be equal to the boundary value ξ(u 0 ).

Let x denote the value which maximizes the welfare of the average user, u 0 + u(x) -x 2 . With our specification, x = (2aα) 1 1-α . When x is above the coverage level, welfare increases with x up to the coverage level ξ(u 0 ): covering the market is not optimal. When the level of data exploitation is above the coverage level, the market is not covered, and the demand decreases slowly with x so that the welfare function displays the same weak form of convexity as the profit: the elasticity of welfare with respect to the level of data exploitation, xW (x) W (x) is increasing in x, which shows that the optimal choice cannot be an intermediate value between ξ(u 0 ) and 1.

We conclude that the optimal choice of users will either be equal to x (only if this value is lower than ξ(u 0 )) or to 1.

Proposition 2 The level of data collection maximizing users' welfare is either x or 1. If α ≤ 1 2 , users prefer market coverage at x. If α ≥ 1 2 , there exists a threshold value of the access benefit, u 0 such that users prefer market coverage at x if u 0 ≥ u 0 and prefer exclusion if u 0 < u 0 .

Proposition 2 characterizes the optimal level of data collection from the users point of view.

We first observe that when u 0 increases, the difference W ( x) -W (1) increases: when the access benefit is higher, exclusion becomes less attractive and users prefer market coverage to exclusion.

We now focus attention to the case u 0 = 0, and observe that (i) x ≤ ξ(u 0 ) if and only if α ≤ 1 2 and (ii) for any x > ξ(u 0 ), W (x) ≤ 0 if and only if α ≤ 1 2 . The value α = 1 2 plays a special role in the computations: when the data revenue function is a square root function, the welfare W (x) is increasing up to ξ(u 0 ) = x = a 2 , and equal to a 2 2 afterwards. When the data benefit function is more concave than the square root function (α < 1 2 ), the maximum is obtained for a value lower than ξ(u 0 ), and the welfare function is decreasing after ξ(u 0 ). By contrast, when the data benefit function is less concave than the square root function, (α > 1 2 ), the maximum is achieved at a value x > ξ(u 0 ) and the welfare function is increasing after ξ(u 0 ). We conclude that, when u 0 = 0, if α ≤ 1 2 , users prefer market coverage to exclusion, but if α ≥ 1 2 , users prefer exclusion to market coverage. As an increase in u 0 makes market coverage more attractive, we know that even when α ≥ 1 2 , market coverage will be preferred to exclusion when the access benefit is sufficiently large.

Propositions 1 and 2 show that the optimal level of data collections of the platform and of users may differ and that their ranking depends on the parameters. These Propositions cannot rule out situations where the platform chooses market coverage and the users prefer exclusion (this may happen for low values of u 0 when α > 1 2 and α+β < 1). However, we believe that such situations are unlikely to arise in practice as it requires access benefits to be small with respect to data benefit, and the data benefit function to be more elastic than the data revenue function, implying that advertisers are less sensitive to incremental increases in data than users. If α ≤ 1 2 or u 0 ≥ u 0 , users always favor market coverage at a level of data collection x ≤ ξ(u 0 ) so that users always favor a level of data exploitation which is lower than the platform's. We conjecture that this is the case which is empirically relevant, as it corresponds to an access benefit which is high with respect to the data benefit, and a data benefit function which achieves most gains for low levels of data exploitation. We state this result in the following corollary.

Corollary 2 If α ≤ 1 2 or α ≥ 1 2 and u 0 ≥ u 0 users favor a level of data exploitation x which is lower than ξ(u 0 ), hence surely lower than the platform's choice. In that case welfare decreases when x increases from x to ξ(u 0 ).

Robustness We now argue that Propositions 1 and 2 can be extended to data benefit and value functions which are more general than the iso-elastic functions. Our observation that the optimal level of data exploitation of the platform and of users cannot be an interior point in the segment [ξ(u 0 ), 1] relies on the fact that the elasticity of profit and welfare is non-decreasing in the region where T (x) = u 0 +u(x)

x < 1. A sufficient condition for the elasticity of profit to be non-decreasing is that both the elasticities of the u(•) and v(•) functions be non-decreasing, as

xΠ (x) Π(x) = xv (x) v 0 + v(x) + xu (x) u 0 + u(x) -1.
Similarly, a sufficient condition for the users' welfare to be non-decreasing is that the elasticity of the data benefit function u(•) be non-decreasing as

xW (x) W (x) = 2 xu (x) u 0 + u(x) -1.
These conditions are clearly satisfied for iso-elastic functions as xv (x) v(x) = β and xu (x) u(x) = α. We also note that the observation that the welfare function is differentiable at ξ(u 0 ) does not depend on the specification of the data benefit function. Hence, the characterization of the optimal level of data collection of the platform as either the coverage level ξ(u 0 ) or the maximal level 1 and of the optimal level of data collection of users as either the optimal level x for the average consumer or 1 does not rely on the specification of the functions u(•) and v(•).

Taxation

In this Section, we analyze the effect of the imposition of a tax on the strategy of the platform.

The imposition of different taxes can be interpreted as changes in the parameters of the model.

A tax τ on the revenues only scales down the profit and has no effect on the strategy of the platform. A tax per user paid by the platform reduces the access revenue from v 0 to v 0 -τ P . A differentiated revenues tax reduces the access revenue from v 0 to v 0 -τ 1 and the data revenue from v(x) to v(x)(1 -τ 2 ). A tax per user paid by users reduces the access benefit from u 0 to u 0 -τ U . The effect of taxes on the optimal choice of the platform can thus be computed by using the comparative statics results of the previous Section. They are summarized in the following Proposition.

Proposition 3 The imposition of a tax has the following effects on the degree of data exploitation by the platform:

• An ad valorem tax τ on the profits or revenues of the platform has no effect on the degree of data exploitation

• A tax paid by the platform per user τ P results in an increase in the degree of data exploitation • In a differentiated revenues tax system, a tax τ 1 on the access revenues results in an increase in the degree of data exploitation, whereas a tax τ 2 on the revenues generated by data exploitation results in a decrease in the degree of data exploitation • A tax paid by users for accessing the platform τ U results in a decrease in the degree of data exploitation when the platform chooses market coverage, but may increase the probability that the platform chooses exclusion.

Proposition 3 analyzes the incidence of the different taxes on the optimal choice of the platform. Clearly, a uniform tax on profits, or on revenues given that marginal cost are negligible, is neutral, and does not affect the platform's choice of data exploitation. A tax per user paid by the platform has no effect on the marginal data revenue of users accessing the platform, but it lowers the loss of not serving the marginal user who chooses not to access on the platform.

Hence a tax per user induces the platform to increase its level of data exploitation. Similarly, in the context of differentiated taxation on access and data revenues, a tax τ 1 on the access revenue leaves the marginal benefit of data exploitation unchanged but reduces the loss of not serving the marginal user and hence raises data exploitation. By contrast, a tax τ 2 targeted at data revenues leads to a reduction in the elasticity xv (x) v 0 +v(x) while leaving the effect on demand unchanged. It thus reduces the platform's incentive to collect data and results in a lower level of data exploitation. Finally, a tax supported by users lowers the level of data exploitation when the platform chooses market coverage. However, it also makes market coverage less attractive and hence increases the parameter configurations for which the platform chooses exclusion.

However, recall that when the data revenue and benefit functions are highly inelastic, α + β < 1, the platform always chooses to cover the market. In that case, a tax on users unambiguously leads to a reduction in data exploitation.

User's welfare Proposition 3 enables us to study the effect of taxation on user's welfare through the change in the level of data exploitation in the likely situation identified in Corollary 2

where users prefer a level of data exploitation x, which is lower than ξ(u 0 ).

Corollary 3 If α ≤ 1
2 or α ≥ 1 2 and u 0 ≥ u 0 , any taxation which raises the level of data exploitation hurts users and any taxation which lowers the level of data exploitation, keeping it above 3 x, benefits users. This result does not compare the different tax instruments. Different taxation schemes may generate different fiscal revenues that could be incorporated in the measure of social welfare.

Robustness Finally we extend the tax incidence results of Proposition 3 to general utility functions with non-decreasing elasticity (for which the the same types of results obtain, see the discussion at the end of Section 3). Proposition 3 only depends on the comparative statics effects of changes in u 0 , v 0 and b on the optimal level of data exploitation of the platform. Now note 1)) is negative or any functions v(•) (still assuming that T (1) < 1, i.e. not everyone access the platform if the maximum level is chosen). In addition, the coverage level ξ(•) is an increasing function, by the 3 Welfare increases when x decreases down to x. When α ≤ 1 2 , then x < ξ(0), so that the platform's choice stays always larger than x: users surely benefit from a decrease in the level of data exploitation. When α ≥ 1 2 and u0 ≥ u0 the decrease in x might be harmful if the platform chooses a level below x. But this can happen only if the tax is high enough to have a large effect. concavity of u4 These observations are sufficient to guarantee that Proposition 3 carries over to general utility functions, as the marginal effects of an increase in u 0 and v 0 on the behavior of the platform do not depend on the iso-elastic specification.

that Π(ξ(u 0 )) -Π(1) = v 0 (1 -T (1)) + b(v(ξ(u 0 )) -T (1)v(1)) is increasing in v 0 ; it is decreasing in b if x = 1 is preferred by the platform since then surely v(ξ(u 0 )) -T (1)v(

Opt-out option

In this section, we suppose that the platform is mandated to propose an opt-out option, where users are guaranteed that data will not be collected but suffer from the resulting loss in service quality. For example, users may refuse to be geo-localized or refuse to let cookies monitor their activity, but will in consequence obtain a lower quality of service on the platform. Users now freely choose between two options: one with no data collection and one with a single positive level x of data collection. 5 As each user prefers the opt-out option to the outside option (u 0 > 0) all users access the platform. We examine the optimal level of data collection with an opt-out option, compare the profit of the platform and the welfare of users in the baseline model and with the opt-out option.

A θ-user chooses the positive level of data collection x instead of the opt-out option if and only if

u(x) + u 0 -θx ≥ u 0 , or θ < u(x)
x . With the opt-out option, the coverage level is the solution to ax α -x = 0, or ξ(0) = a 1 1-α , the coverage level defined in the baseline model when u 0 = 0. The profit of the platform is

Π(x) = v 0 + v(x) if x ≤ ξ(0), Π(x) = v 0 + v(x) u(x) x if x ≥ ξ(0).
Notice that the platform now collects the access revenue v 0 on all users, and that the demand for data collection is independent of the access benefit u 0 . The optimal level of data collection of the platform is thus easier to characterize than in the baseline model, as it corresponds to the level chosen by the platform when u 0 = 0. The profit Π(x) is increasing in x for x ≤ ξ(0)

and either increasing or decreasing in x for x ≥ ξ(0) depending on whether α + β exceeds 1. We immediately obtain the following characterization:

Proposition 4 With an opt-out option, the platform optimally chooses a degree of data exploitation x ∈ {ξ(0), 1}. It chooses ξ(0) if and only if α + β ≤ 1.

Proposition 4 shows that, as in the baseline case, the platform either chooses to cover the market or sets the highest possible degree of data exploitation. It covers the market when the data revenue and benefit functions are highly concave, α + β ≤ 1 and chooses maximal data exploitation otherwise. The comparison between the level of data collection of the platform with and without the opt-out option produces ambiguous results. On the one hand the coverage level ξ(0) is always lower than the coverage level in the baseline case, ξ(u 0 ). The existence of the opt-out option forces the platform to reduce x in order to attract all users. On the other hand, the region of parameters for which the platform chooses maximal data exploitation may increase. The platform now collects the access revenue v 0 on all users and does not face the risk of non-participation as in the baseline case. When α + β > 1 and u 0 is large, this effect leads the platform to choose maximal exploitation when the opt-out option is present, whereas it chooses to cover the market in the baseline case.

We next characterize the optimal level of data exploitation from the users point of view. An easy adaptation of Proposition 2 when u 0 = 0 yields the following result.

Proposition 5 With an opt-out option, the welfare maximizing level of data collection is x if

α ≤ 1 2 and 1 if α ≥ 1 2 .
With an opt-out option, all users access the platform and obtain the access benefit u 0 .

Thus the optimal value of data collection is not affected by u 0 , and is the same as the optimal level in the baseline case when u 0 = 0. Not surprisingly, exclusion is optimal more often than under a uniform policy. Comparing Propositions 2 and 5, we see that when α > 1 2 and u 0 is large, users favor the maximal level of data exploitation when the outside option is present whereas they prefer market coverage in the baseline mode. Hence the presence of the outside option unambiguously increases the optimal level of data exploitation of the users, because this benefits those who do not value privacy much while leaving the access benefit to others.

We next compare the profit of the platform in the baseline case and with the opt-out option.

Proposition 6 The profit of the platform is higher in the baseline case than with the outside option whenever α + β < 1. If α + β ≥ 1, there exists a threshold value ũ0 ∈ (0, 1 -a) of the access benefit such that the profit of the platform is higher in the baseline case than with the outside option if and only if u 0 > ũ0 .

Proposition 6 shows that the only situation where the platform prefers to introduce the opt-out option is the unlikely situation where the access benefit is low and the data revenue and benefit functions are elastic enough. In all other circumstances, the platform is hurt by the introduction of the opt-out policy. The intuition underlying Proposition 6 is easy to grasp.

When α + β < 1, the platform covers the market at level ξ(0), which is lower than ξ(u 0 ).

Given that, profit is increasing in x under market coverage, this policy is always dominated by a market coverage policy at ξ(u 0 ) which can be chosen by the platform in the baseline case. When α + β ≥ 1, we observe that the profit of the platform with the outside option is independent of u 0 . By contrast, the profit of the platform is strictly increasing in u 0 in the baseline scenario.

At u 0 = 0, the platform chooses exclusion in both régimes and the profit of the platform is strictly higher with the opt-out option as it collects access revenue v 0 on all users. On the other hand at u 0 = 1 -a, the profit of the platform is higher in the baseline model, as it attracts all users by setting the maximal level of data collection. Hence there exists a threshold value ũ0 ∈ (0, 1 -a) under which profit is higher with the outside option and over which profit is higher in the baseline case.

We now ask whether users benefit from the switch from the introduction of the outside option. The answer to this question is not easy. On the one hand, if the baseline policy involves exclusion, excluded users benefit from the introduction of the opt-out option. On the other hand, the introduction of the opt-out option may lead to a deviation of the level of data collection away from the optimum, hurting users who access the platform. The following Proposition details situations under which the introduction of the outside option benefits users.

Proposition 7 Suppose that α ≤ 1 2 . If α + β ≤ 1 or α + β ≥ 1 and u 0 < u 0 , users benefit from the introduction of the opt-out option. If α + β ≥ 1 and u 0 > u 0 , users are harmed by the introduction of the opt-out option if and only if ξ(u 0 ) < a 2 . Proposition 7 focuses on the situation α ≤ 1 2 , where users' optimal level of data collection is lower than the platform's choice both in the baseline model and with the outside option.

If the platform chooses market coverage in both régimes (i.e. when α + β < 1), users prefer the situation with the opt-out option as the coverage level is lower than in the baseline model.

Similarly, users prefer the situation with outside option when the platform chooses the maximal level of data exploitation under the two régimes (i.e. when α + β > 1 and u 0 > u 0 ) as excluded users obtain a positive benefit with the opt-out option. However, when the platform chooses to cover the market in the baseline model but to set the maximal level of data collection in the presence of the outside option (i.e. when α + β > 1 and u 0 > u 0 ), users with low privacy cost who choose data collection are harmed and users' welfare may be higher in the régime without outside option. Hence, for the most likely values of the parameters, the introduction of an optout option reduces the profit of the platform, but does not necessarily have a positive impact on users, as it may lead the platform to switch from market coverage to the maximal level of data exploitation.

Conclusion

This paper studies data collection by a monopolistic internet platform We show that the optimal strategy of the platform is either to cover the market or to choose the highest data exploitation level, excluding users with high privacy costs from the platform. For likely parameter values, the platform chooses an excessive level of data exploitation from the point of view of users. We study how different tax instruments can be used to reduce the level of data collection and show that user-based taxes lead to an increase in data collection. Taxation with different rates according to the source of revenues, with higher tax level for revenues generated by data exploitation can reduce data collection as can a tax paid by users on access. We also analyze the effect of an opting-out option, letting users access the platform with no data collection and show that the platform is hurt by the introduction of the outside option and that, more surprisingly, users may also be harmed as the level of data exploitation increases.

Our analysis provides a first step in understanding how different regulatory instruments affect the optimal degree of data collection of an internet platform (or equivalently the optimal level of advertising of a media platform). Our analysis is based on a simple model of behavior of platforms and users and three directions seem particularly fruitful to follow in future research.

First, we wish to study how another dimension of heterogeneity among users (for example demographic characteristics and the frequency of access) affects our results. Second, we would like to open the black box of the value of data to the platform, and construct a full-blown model of advertising or dynamic pricing to derive the revenues generated by the data. Third, taxes may have other welfare effects which are not captured in our model. Taxes may affect the behavior of the platform, possibly leading the platform to exclude some users and advertisers or to start charging a subscription fee. This calls for extending the analysis. We plan to expand the analysis by allowing the platform to choose a larger menu of policies with different degrees of data exploitation in order to discriminate more finely among users.

Proofs

Proof of Lemma 1: By implicit differentiation of equation 4 ∂ξ ∂u 0 = 1 1 -αaξ α-1 > 0, ∂ξ ∂a = ξ α 1 -αaξ α-1 > 0, ∂ξ ∂α = aξ α log ξ 1 -αaξ α-1 < 0.

Proof of Proposition 1: The function Π(x) is strictly increasing in x for x ≤ ξ(u 0 ) and for x > ξ(u 0 ), we show that xΠ (x) Π(x) = βv(x) v 0 + v(x) + αu(x) u 0 + u(x)

-1, is non-decreasing in x and strictly increasing if u 0 > 0 or v 0 > 0. But this implies that there cannot be an interior maximum of Π(x) in (ξ(u 0 ), 1) as this would imply that Π switches from a positive to a negative value, contradicting the fact that xΠ (x) Π(x) is non-decreasing in x. We now compare Π(ξ(u 0 )) and Π(1). We have Π(ξ(u 0 )) = v 0 + bξ(u 0 ) β and Π(1) = (v 0 + b)(u 0 + a)

(5)

showing that ξ(u 0 ) is chosen when v 0 b ≤ ν. We compute the comparative statics effects of changes in the parameters on ν. To compute the effect of changes in u 0 , we compute the sign of ∂ν ∂u 0 = 1 -βξ(u 0 ) β-1 ∂ξ ∂u 0

(1 -u 0 -a) -ξ(u 0 ) β .

When u 0 converges to 1 -a, ∂ν ∂u 0 converges to zero. We also compute the second derivative of ν with respect to u 0 :

∂ 2 ν ∂u 2 0 = βξ(u 0 ) β-1 ∂ξ ∂u 0 -β(β -1)ξ(u 0 ) β-2 [ ∂ξ ∂u 0 ] 2 (1 -u 0 -a) -βξ(u 0 ) β-1 ∂ 2 ξ ∂u 2 0 (1 -u(0) -a) -βξ(u 0 ) β-1 ∂ξ ∂u 0 = -β(β -1)ξ(u 0 ) β-2 [ ∂ξ ∂u 0 ] 2 (1 -u 0 -a) -βξ(u 0 ) β-1 ∂ 2 ξ ∂u 2 0 (1 -u(0) -a)
Next note that

  xx β > 0.

The function u 0 +u(x)x is decreasing in x.

One exception is[START_REF] Crampes | Advertising, competition and entry in media industries[END_REF] who allow users to have a positive value for low levels of advertising as we do.

( ξ(u0) satisfies the equation. Thus surely u(x) ≤ x at x = ξ(u0), which implies u (x) < 1. Applying the implicit function theorem thus yields that ξ(•) is an increasing function..

For reasons of tractability and technical implementability, we do not consider more complex policies where the platform discriminates among users by proposing several positive degrees of data collection.

(1 -aαx α-1 ) 2 ∂ξ ∂u 0 < 0, so that we finally obtain ∂ 2 ν ∂u 2 0 > 0 which together with the fact that ∂ν ∂u 0 = 0 at u 0 = 1 -a guarantees that ∂ν ∂u 0 < 0.

Proof of Proposition 2: The welfare admits two different expressions for x ≤ ξ(u 0 ) and

x ≥ ξ(u 0 ). We observe that the derivative W (x) is continuous at ξ(u 0 ), is continuous, by comouting the left-hand and right-hand derivatives, W -(ξ(u 0 )) and W + (ξ(u 0 )):

where the last equality stems from the fact that ξ(u 0 ) = u 0 + u(ξ(u 0 )). This implies that generically, ξ(u 0 ) will never be the optimal level of data exploitation. When x ≤ ξ(u 0 ), all users access the platform, and welfare is strictly concave in x. The maximum in this region is attained

) and at ξ(u 0 ) otherwise. In the latter case, the derivative is positive at ξ(u 0 ) and ξ(u 0 ) cannot be the global optimum.

When x|geqξ(u 0 ), we compute

The elasticity of welfare is non-decreasing in x, so that the maximum in the exclusion region is either attained at ξ(u 0 ) (if the derivative is negative at that point) or at 1. We conclude that the global optimum is either achieved at x or at 1.

To determine more precisely the optimum, we consider the simple case where the access benefit is zero, u 0 = 0. Since x = (2aα)

Furthermore, the elasticity of welfare is constant equal to 2α -1 in the exclusion region, so that welfare decreases if α < 1 2 and increases if α > 1 2 . We conclude that, when u 0 = 0, welfare is

As u 0 increases, the welfare under market coverage increases faster than under exclusion, as every marginal increase in u 0 raises welfare by 1 under market coverage but by a + u 0 < 1 under exclusion. Hence, when u 0 increases, users are more likely to prefer market coverage at x. When u 0 is equal to 1 -a, one checks that W ( x) > W (1) when α > 1 2 , completing the proof of the Proposition.

Proof of Proposition 3:

The Proposition is a simple consequence of the comparative statics results of Lemma 1 and Proposition 1, we observe that a reduction in v 0 increases the region of parameters for which exclusion is chosen while not affecting the coverage level ξ(u 0 ). A reduction of v(x), which can be interpreted as a reduction in b, leaves ξ(u 0 ) unaffected and reduces the region of parameters for which exclusion is chosen. A reduction in u 0 increases the bound ν, thereby increasing the region of parameters for which exclusion occurs. However, a reduction in u 0 also reduces the coverage level, thereby reducing the degree of data exploitation when the platform chooses to cover the market.

Proof of Proposition 7: If α + β ≤ 1, the platform chooses coverage at ξ(0) < ξ(u 0 ) under the binary policy. As α < 1 2 , x ≤ ξ(0) < ξ(u 0 ), and hence users favor a lower value of data collection and prefer the régime with the opt-out option. If now α + β > 1 and u 0 < u 0 , the platform selects the maximal level of data collection under both régimes and users prefer the opt-out option as it allows excluded users to collect the access benefit u 0 . Finally, consider the case where α + β > 1 and u 0 ≥ u 0 . In the baseline scenario, the platform covers the market and welfare is

With the outside option, the platform chooses the maximal degree of data exploitation and welfare becomes

. The conclusion follows.