G Clair

B Despres

E Labourasse

A new method to introduce constraints in cell-centered Lagrangian schemes

Keywords: Lagrangian Hydrodynamic, Compressible Gas dynamics, Constraints, Minimization method

We describe a new method to introduce constraints in cell-centered Lagrangian schemes in the framework of compressible hydrodynamics. In this paper we apply it to the modeling of contact and sliding on solid wall boundaries. We illustrate our method, which is based on the minimization of a specific objective function, by several basic problems.

Introduction

In this work we describe a new method to solve the Euler's equations of Lagrangian hydrodynamics [START_REF] Lifschitz | Fluid Mechanics (Course of Theoretical physics[END_REF][START_REF] Liepmann | Elements of Gasdynamics[END_REF] on general unstructured meshes when several constraints apply on the considered mesh. In integral form, the fundamental equations are: (

                         d dt V (t) dV = V (t) ∇ • u dV volume conservation,
) 1
Here V(t) is a volume moving with the fluid and D Dt = ∂ ∂t + u • ∇ is the material derivative. The physical variables are density ρ (and the specific volume τ = 1/ρ), velocity u, total energy e and pressure p.

Constraints we have in mind are encountered in contact and impact problems in the framework of classical mechanics [START_REF] Wriggers | Computational Contact Mechanics[END_REF][START_REF] Stronge | Impact Mechanics[END_REF], even though the formalism we describe here is more general. Such problems deal with interfacial boundaries between non-mixing media (solid, liquid or gas). To analyze specifically interfacial boundaries, numerical simulations are often used. Several contact algorithms have been proposed in the literature: a survey has been recently proposed by N.G. Bourago in the reference [START_REF] Burago | [END_REF]. According to the author, contact algorithms can be separated into two major groups, regarding the Lagrangian and Eulerian Motion. The method described in this paper matches with the first category. It is proposed in the framework of Lagrangian cell-centered numerical schemes, for which there is, to our knowledge, no equivalent method that has been published yet (let mention however recent work of Bertoluzza et al. in the context of pure sliding algorithms [4]).

Indeed, the uniqueness of our method is due to the fact that historically, cell-centered Lagrangian schemes are much more recent than staggered schemes. That's why most of contact algorithms have been developed for staggered codes: we cite for instance the contact algorithms in LS-DYNA [6,[START_REF] Hallquist | Report UCRL 52066[END_REF] and PRONTO [START_REF] Attaway | [END_REF]9,10] codes, as well as several contact algorithms based on Lagrange multipliers [12,13,14,15] and penalty function techniques [16,17]. This is also the case of the pioneer work of Wilkins about the treatment of sliding lines [START_REF] Wilkins | Computer Simulation of Dynamic Phenomena[END_REF] and most of the sliding line treatments that have followed [START_REF] Caramana | [END_REF]20]. In cell-centered schemes, two different discrete velocities are used: the first one is cell-centered and satisfies the conservation of momentum. The second one is vertex-centered and is obtained by the solution of a well-posed problem. This structure makes a large difference with staggered methods which have only one instance of the velocity staggered at the vertex, and is one fundamental reason why cell-centered schemes offer potential advantages over staggered methods for the discretization of contact problems.

In this paper, we apply our method to cell-centered Lagrangian schemes of Godunov type (that is fluxes are based on a generalized notion of Riemann invariant), based upon the knowledge of some corner vectors at each time step. The velocities of the N nodes in the mesh are computed using N independent d×d linear systems (d is the dimension of the considered problem) whose unknowns are the N nodal velocities of all nodes in the mesh. Such basic formulation is inappropriate to take into account constraints, such as impact and contact, that may apply on several nodes in the mesh. For this purpose, we propose a new formulation of the nodal solver, which is based on a minimization procedure commonly encountered in the modeling of constraints in the solid mechanics community. That is we adopt the framework of constraint optimization which is very natural way of formulating problems with contacts, see [START_REF] Wriggers | Computational Contact Mechanics[END_REF][START_REF] Stronge | Impact Mechanics[END_REF] and references therein. What is new is the coupling with the Riemann solver. Indeed the solution of the multidimensionnal Riemann solver is then seen as the minimum of an objective function within a set of admissible velocities expressing all constraints applying on nodes in the mesh. We prove that this new formulation preserves the main expected properties in the cell-centered Lagrangian schemes that are mass, volume, momentum and total energy conservations. If all constraints are inactive, the new formulation naturally degenerates to the usual method. We stress this paper is dedicated to the abstract framework developed in order to take into account contact and impact, and to first applications to impact and contact between a fluid and a wall. Let emphasize that this method is free from computing the time of impact between the nodes of the mesh and the wall: this feature seems to us very pleasant for multidimensional problems for which the exact computation of this time of impact for all nodes and the associated restriction on the time step could generate an important computational burden. This formulation can be extended to the sliding between two fluids, but is beyond the scope of this paper and will be the subject of another paper. While theory is well-suited for 3D implementation, we restrict here to 1D and 2D numerical examples.

An outline of this work is as follows. In a first section, we present an overview of traditional nodal solver that are used in cell-centered Lagrangian numerical schemes. We take the example of the GLACE scheme [21], but extension to the EUCCLHYD scheme [5] is easy. Then, we describe the new formulation of the nodal solver. Finally, the efficiency of the method is illustrated with several numerical examples.

Overview of nodal solvers in traditional cell-centered Lagrangian solvers

Introduction of geometrical vector C j,r

Before going further, we remind the notations found in the reference [21] which are needed to define the objective function J. We consider a computational domain Ω ⊂ R d (d is the dimension of the problem) composed of control volumes (or cells) indexed by j and points (or nodes) indexed by r. The volume of the j-th cell at time t n is denoted V n j such that:

j V n j = Ω and dim V n i ∩ V n j < d if i = j.
The set of cells which the r-th node belongs to is denoted C(r). Similarly, the set of nodes that belong to the j-th cell is denoted N (j). In addition, we denote respectively x r and u r position and velocity vectors of the r-th node within the mesh. In the text, we will often refer to nodal position and nodal velocity.

Let's introduce the corner vectors C j,r defined as the gradient of the volume V j with respect to the nodal positions x r :

C j,r = ∇ xr V j .
Using this notation, the semi-discretization of the system (1) is the following:

             M j τ ′ j (t) = r∈C(r) (C j,r , u r ) , M j u ′ j (t) = - r∈C(r) C j,r p j,r , M j e ′ j (t) = - r∈C(r) (C j,r , u r ) p j,r . (2)
where u r is the velocity of the r-th node, and p j,r is a nodal pressure. The latter is computed from a linearized-Riemann-invariant relation in the direction of the C j,r :

p jr -p j + ρ j c j u r -u j , C j,r |C j,r | = 0. ( 3
)
This relation ensures the entropy property of this family of numerical methods. In the following, we describe the usual method used to compute the nodal velocity u r .

Nodal solver

The nodal velocity is computed by expressing that the sum of all forces around the vertex x r is zero:

j C jr p jr = 0. ( 4
)
Substituting the expression of the pressure (3) in the relation (4) leads to solve the system of equations:

A r u r = b r . (5)
where A r and B r are defined as follow:

A r = j ρ j c j C j,r ⊗ C jr |C j,r | , ( 6
) b r = j C j,r p j + j ρ j c j C j,r ⊗ C j,r |C j,r | u j . (7)
The matrix A r is symmetric positive-definite on general meshes, so that the unique solution is:

u r = A -1 r b r . (8)

Boundary conditions for nodal solver

Usual boundary conditions treatment for cell-centered compressible Lagrangian schemes, as sliding on a planar wall, pressure prescribed on a free surface, given velocity and finally mixed of these conditions at corners, are described in [5,21,[START_REF] Després | Lagrangian gas dynamics in 2D and Lagrangian systems[END_REF]. There is no optimal solutions in the literature for sliding on non planar walls. The method that we propose hereafter to treat more general unilateral boundary conditions can also be used to impose the previous more classical boundary conditions.

Formalization of the problem

Constraints influence the motion of several nodes in the mesh. Therefore, they should be taken into account in the method of computation of the nodal velocities u r . In this sense, solving the system (5) is unadapted. We propose therefore a new formulation based on the minimization of an objective function J under constraints. Before giving more details, we give the following proposition: Proposition 2.1 (local formulation). Solving the linear system A r u r = b r , ∀r ∈ [1 : N ] is equivalent to find within R d the minimum of the convex quadratic functions J r defined as :

J r : R d → R u r → J r (u r ) = 1 2 (A r u r , u r ) -(b r , u r ) .
Proof. J r is well defined over R d . Since A r is a symmetric and positive definite matrix, J r is α-convex. Both existence and uniqueness of the global minimum of J r within R d are ensured by the α-convexity of J r (refer for instance to [START_REF] Allaire | Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation[END_REF]). This minimum is found at the zero of the derivative of J r : ∇J r (u r ) = A r u rb r = 0.

We now describe the new formulation for nodal solvers. Our idea is to minimize an objective function J under constraints, that is looking for the minimum of J in a set of admissible velocities that expresses the constraints. Function J must be a global function of all nodal velocities in the mesh, since, in general, constraints globally apply on the mesh and may couple several nodes. Denoting U = (u 1 , u 2 , . . . , u N ) T a new velocity vector gathering all nodal velocities in the mesh, our first idea is to define J from the J r functions:

J : R N ×d → R (9)
U → J(U) = N r=1 α r J r (u r ). ( 10
)
We suppose α r > 0, ∀r ∈ [1 : N ], which ensures that J is a strictly convex quadratic function of U. An important question is to determine the optimal value of the coefficients α r .

In the section 2.5, we will show that the property of conservation of momentum is preserved with specific values for α r . Obviously, we have the following proposition.

Proposition 2.2 (global formulation). For unconstrained problems, solving the local linear systems A r u r = b r , ∀r ∈ [1 : N ] is equivalent to find within R N ×d the minimum of the convex quadratic function J defined by (10).

Proof. J is well defined over R N ×d . J is strictly convex and the convexity of the set R N ×d then ensures the existence and the uniqueness of a minimum of J within this set. Such minimum is found at the zero of the gradient of J:

U min = (u 1 , u 2 , . . . , u N ) T s.t. ∀r ∈ [1 : N ] , ∂J ∂u r (U) = α r ∇ ur J r (u r ) = α r (A r u r -b r ) = 0.
Constraints are taken into account by looking for the minimum of J r within a set of admissible velocities K ⊂ R N ×d . In this work, we admit that K = ∅ and can be written as:

K = {U ∈ R N ×d , F (U) ≤ 0}. ( 11
)
where

F = (F 1 (U), F 2 (U), . . . , F M (U))
T are real functions expressing M constraints applying on all nodes in the mesh. Here F(U) ≤ 0 means that F j (U) ≤ 0, 1 ≤ j ≤ M . We will show that this formalism is well adapted to the modeling of contact and impact problems.

For a specific constraint problem, the constrained solution is unique. It means that the minimum of J in K must exist and be unique. This implies that K should have the two following properties: K is closed, ensuring the existence of the minimum.

K is convex, ensuring its uniqueness. This gives the definition of U min in the case of constrained problem:

U min = argmin U∈K J(U) (12)
In theory, the method of global minimization must preserve momentum and total energy. For this purpose, K should have the two additional properties:

Momentum is preserved if translations are admissible solution of any constrained problems. Translations are elements in the form U min + W a , where W a = (a, a, . . . , a) T , ∀a ∈ R d . This property means that the translation of all nodes in the mesh by any vector a ∈ R d naturally preserves the solution of the minimization procedure.

Total energy is preserved if K is a cone, that is:

U ∈ K ⇒ ∀λ > 0, λU ∈ K
The exact definition of K will be given for several cases of interest in the section 3. In particular, the validity of the properties , , and mentioned above will be discussed for each numerical test case proposed in this paper. We'll see that some of them are sufficient but not necessary. In the next section, we describe the properties of the method.

Properties of the method

This new formulation ensures the main properties that are expected for the numerical schemes developed in the framework of fluid dynamics: conservation of mass, momentum and total energy. The following propositions give the proof of the two last properties. Proposition 2.3. If K has the property and α r = 1 for all r ∈ [1 : N ], the minimization of the objective function J within K preserves momentum.

Proof. The total momentum variation over a time step writes:

j M j u ′ j (t) = - r j C jr p jr . ( 13
)
The minimum of the objective function J satisfies the Euler inequality [START_REF] Allaire | Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation[END_REF]:

(∇J(U min ), V -U min ) ≥ 0, ∀ V ∈ K. (14)
The r-th component of ∇J(U min ) writes:

∂J ∂u r (U min ) = α r ∇ ur J r (u r ).
The expression of the derivative of J r with respect to u r is:

∇ ur J r (u r ) = A r u r -b r = j ρ j c j C j,r (C j,r , u r ) |C j,r | - j C j,r p j - j ρ j c j C j,r (C j,r , u j ) |C j,r | = j C jr -p j + ρ j c j u r -u j , C j,r |C j,r | = - j C j,r p j,r . (15)
Proof ends by recalling the property of K: translations are admissible solutions of the constrained problem. This means that, ∀a ∈ R d , elements V = U min + W a where W a = (a, a, . . . , a) T , and V = U min + W -a , where W -a = (-a, -a, . . . , -a) T , are in K. Thus, inequality (14) becomes the equality:

(∇J(U min ), W a ) = 0 ∀a ∈ R 2 , (16)
⇔ - r α r j C j,r p j,r a = 0 ∀a ∈ R 2 , ⇔ r α r j C j,r p j,r = 0.
From (13), conservation of momentum is ensured by choosing α r = 1, ∀r ∈ [1 : N ]. This ends the proof.

From now on, we take α r = 1, ∀r, so that J may be written in the more general form:

J : R N ×d → R U → J(U) = 1 2 (AU, U) -(B, U) . ( 17
)
where

A =         A 1 . . . 0 . . . . . . A r . . . . . . 0 . . . A N         and B =        b 1 . . . b r . . . b N       
.

Next, we show that total energy is preserved.

Proposition 2.4. Assuming moreover that K has the property , then total energy is preserved Proof. Total energy variation over a time step writes:

j M j e ′ j (t) = - r j (C j,r , u r ) p j,r = (∇J(U min ), U min ) . (18)
Again, the proof is based on the Euler inequality:

(∇J(U min ), V -U min ) ≥ 0 ∀ V ∈ K.
K is a cone, that means we can find two elements V 1 and V 2 that are written

V 1 = (1 + µ) U min and V 2 = (1 -µ) U min with µ ∈ ]0; 1]
. Thus, directions µU min and -µU min are feasible. From this point, Euler inequality (2.5) turns into equality:

(∇J(U min ), µU min ) = 0 ⇒ µ - r j (C j,r , u r ) p j,r = 0
and, referring to eq. ( 18), it shows that total energy is preserved.

Numerical implementation

Minimization under constraints is a subject that has been largely described in the literature [START_REF] Glowinski | Computing Methods in Applied Sciences and Engineering[END_REF][START_REF] Allaire | Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation[END_REF]. It amounts to introduce a Lagrangian which, as usual, is the sum of the objective function J and of the M constraints applied on the node multiplied by suitable Lagrange multipliers λ = (λ 1 , . . . , λ M ) T :

L(U, λ) = J(U) + λ • F (U). (19)
In equality-constrained problems , finding U corresponding to the minimum of J is equivalent to cancel the gradient of the Lagrangian. Assuming now that functions F are now continuously differentiable, we are looking U ∈ K and λ ∈ R M such as:

min U∈K J(U) ⇐⇒    ∂L ∂U = ∇J(U) + F ′ (U) T • λ = 0, ∂L ∂λ = F (U) ≤ 0. ( 20
)
where

F (U) ′ = F ′ 1 (U), . . . , F ′ M (U)
T is the gradient of the function F with respect to the variable U.

In inequality-constrained problems, U and λ are respectively searched in K and R M + . Additional conditions, called constraint qualifications, must be respected in consequence:

min U∈K J(U) ⇐⇒    ∂L ∂U = ∇J(U) + F ′ (U) T • λ = 0, λ • F (U) ≤ 0, with λ i ≥ 0, ∀i ∈ [1 : M ] . (21)
To solve the system (20) or the system (21), we use the Uzawa method [START_REF] Arrow | Studies in nonlinear programming[END_REF] which can be reformulated and implemented with the usual matricial and variational formulations described for example in [START_REF] Bacuta | A unified approach for uzawa algorithms[END_REF][START_REF] Wriggers | Computational Contact Mechanics[END_REF].

Numerical examples

We use explicit schemes with a convenient control of the global time step through the usual CFL constraint [START_REF] Després | Lagrangian gas dynamics in 2D and Lagrangian systems[END_REF]5,21]. The computation of the time of impact is not necessary. That is why the implementation the method described in this paper is easy. We remark that our algorithm is, in dimension one, exactly the usual one before and after the time of impact.

1D impact between a fluid and a wall

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 The most obvious example is the 1D contact between a fluid and a wall. Such situation is depicted on figure 1. The surface of the wall is taken as the origin of positions x = 0, so the fluid moves in the region x ≤ 0. The mesh used for the computation is a regular grid containing R = 100 nodes. We define the global velocity vector U = (u 1 , u 2 , . . . , u N ) T . Initially, the fluid of length L=1 is detached from the wall and moves with an initial velocity u 0 = 1. Its density is 1. We choose to work with an ambient pressure equal to 0, by using a stiffened gas pressure law:

Wall u 0 u R
p = (γ -1) ρ e - v 2 2 -γΠ 0
where γ = 5/3 and Π 0 is arbitrarily set to 1/γ.

Let us describe in this example the form of the objective function J. We use the general form given by relation (17). A is a diagonal matrix whose diagonal coefficients are written using the relation (6) in the 1D framework:

     A 1 = ρ 1 c 1 , A i = (ρ i-1 c i-1 + ρ i c i ) ∀i ∈ [2 : N -1] , A N = ρ N -1 c N -1 .
Similarly, the vector B is written using the relation [START_REF] Hallquist | Report UCRL 52066[END_REF]:

     B 1 = -p 1 + ρ 1 c 1 u 1 , B i = (p i-1 -p i ) + ρ i-1 c i-1 u i-1 + ρ i c i u i ∀i ∈ [2 : N -1] , B N = p N -1 + ρ N -1 c N -1 u N -1 .
The fluid must remain in the subset R -, so every point must respect the condition:

x(t) ≤ 0, ∀t > 0.
The discrete form of the previous relation writes:

x n+1 r ≤ 0 ∀(r, n) ∈ [1 : N ] × N, ⇔ x n r + ∆t u n r ≤ 0 ∀(r, n) ∈ [1 : N ] × N, ⇔ u n r ≤ - x n r ∆t , ∀(r, n) [1 : N ] × ∈ N. (22)
In practice, constraint will only be active for the node that impacts the wall, that is the boundary node at the extreme right of the mesh (velocity u N ). Thus, relation [START_REF] Glowinski | Computing Methods in Applied Sciences and Engineering[END_REF] always remains a strict inequality for j ∈ [1 : N -1], and the constraint may be simply written:

u n N ≤ - x n N ∆t , ∀n ∈ N.
The set of admissible velocities then writes and the function F writes (M = 1)

K n = {U ∈ R N , u N ≤ - x n N ∆t }, (23)
F (U) = u n N + x n N ∆t . ( 24
)
During the impact, x n N = 0 and the constraint simplifies to u N ≤ 0.

Figure 2 presents results about the velocity of the fluid at four different times of computation. Figure 2(a) illustrates the time t = 0.1 at which impact has already occurred. The constraint is active on the node impacting the wall, forcing the fluid to stop at the wall. The latter enters in a compression phase: the kinetic energy is turned into internal energy, inducing a decrease in the velocity value from 1 to 0 in each cell, and an increase in the pressure. This induces the formation of a left-running shock wave, that may be observed at x ≈ -0.1 on the part (a) of the velocity profile. On Figure 2(b), shock wave is going to reach the left boundary. After that, the fluids enter in an expansion phase, during which the internal energy turns into kinetic energy. The velocity decreases from 0 to a negative value in each cell. The drop in pressure results in the formation of a rarefaction fan that moves to the right on Figure 2(c). When the velocity of the constrained node becomes negative (the constraint becomes inactive for u N < 0), the fluid detaches from the wall, as seen on Figure 2(d).

Let us check the properties of K n in this numerical test case:

1. K is closed. 2. K is convex.
3. When impact occurs, and as long as the fluid remains attached to the wall, translations in the x-positive direction are not admissible solution of the constrained problem. In others words, we can't find any element in K n in the form U min +W a ∈ K n , where W a = (a, a, . . . , a) T , a ∈ R * + . Indeed, during the impact, the solution for the N-th component of U min is u N = -x n N ∆t , and we can't find any element a > 0 that satisfies the constraint:

u N + a ≤ -x n N ∆t .
As a consequence, momentum is not preserved. We can show that it decreases as long as impact occurs. Since directions W -a = (-a, -a, . . . , -a) T , a ∈ R * + are always feasible, , we can deduce from ( 14) that:

(∇J(U min ), W -a ) ≥ 0 ∀a ∈ R * + , ⇔ - r j
C j,r p j,r , a ≤ 0.

From (13), it shows that momentum decreases during the impact, as expected from mechanical considerations.

On the contrary, when the constraint is inactive, U min is the solution of the unconstrained problem, for which ∇ U J(U min ) = 0. Recalling the expression of the gradient of J from relation (15), and using (13), momentum is then automatically preserved.

4. K n is a cone when the fluid is attached to the wall. Indeed, in this case, x n N = 0. The solution of the constraint problem is u N = 0 and any λ > 0 satisfies the relation λ u N ≤ 0. Thus, total energy is preserved. However, a change in total energy arises during the time step over which the fluid impacts the surface. In this case, the solution of the constrained problem for the Ncomponent of the velocity is u N = -x n N ∆t > 0. There is no λ > 0 that satisfies the constraint λu N ≤ -x n N ∆t , involving that K n is not a cone. Going further, there are no elements in the form (1 + µ) U min , ∀µ > 0 in K n . However, elements in the form (1µ) U min , ∀µ > 0 can be found in K n . Recalling the proof of the proposition 2.4, the Euler can't turn into an equality, and can only be written for the feasible direction -µU min :

(∇J(U min ), -µU min ) ≥ 0, ⇔ -µ r j (C j,r , u r ) p j,r ≤ 0. ( 25
)
From the relation (18), we have j M j e ′ j (t c ) ≤ 0, explaining the drop in total energy at the time of impact. We illustrate this loss on the figure 3. A simple expression of the lost amount of energy can be given by writing explicitly the relation [START_REF] Wriggers | Computational Contact Mechanics[END_REF] :

-

r j (C j,r , u r ) p j,r = -(p N -1,N u N ) .
Before the impact and when the fluid detaches from the wall, p N -1,N = 0 since we work with a zero ambient pressure. Total energy is conserved. This is also the case when the fluid is attached to wall, because the constraint imposes u N = 0. However, over the time step of impact, u N = -x N ∆t > 0 and the relation (3) gives a positive value for p N -1,N . Then there is a loss of total energy over this time step. This is illustrated on figure 3 for several time steps.

As long as the fluid is attached to the wall, the situation may be seen as a 1D Noh problem [START_REF] Gehmeyr | [END_REF], which admits a simple analytical solution. We present therefore a comparison between the latter solution and the numerical solution given by our method. For that purpose, we suppose that the fluid is initially attached to the wall at x = 0. The fluid lies in the subset R -. Initial conditions are uniform in density, ρ = 1, internal energy e = 0, and gas velocity u = 1. In our case, the fluid moves to the right (u = 1), while it moves to the left in the original Noh test case (u = -1). The pressure law is a perfect gas law with γ = 5 3 . The exact solution of the Noh problem and the solution given by our method at the time of computation t = 0.6 s are presented on figure 4. The comparison, made for the density and the pressure, shows a good agreement between the numerical and the exact solution.

The shock front is well captured. On the density profile, the "wall-heating" effect may be evidenced above the wall, which is a common drawback of the Lagrangian schemes.

1D Impact between two fluids

We describe in this part the impact of two fluids with different initial velocity and density. Each fluid has a length L = 1 and is meshed with a regular grid with N = 100 nodes. Nodes are continuously indexed from 1 to 2N: nodes indexed from 1 to N belong to the fluid on the left, while the ones from N+1 to 2N belong to the fluid on the right. Again, we define a global velocity vector U = (u 1 , u 2 , . . . , u 2N ) T . Initial velocities of the fluids are respectively 1 and -1. Initial positions of each fluid are chosen such as the impact occurs at x = 0. Initial densities are 1. Again the pressure law is a stiffened gas law and the initial ambient pressure is zero. Velocity constraint applies on both N-th and (N+1)-th nodes, expressing that the distance between them must be greater than or equal to 0 at every time:

x n+1 N -x n+1 N +1 ≤ 0, ( 26
) ⇔ x n N + ∆t u n N ≤ x n N +1 + ∆t u n N +1 , (27)
⇔ u n N -u n N +1 - x n N +1 -x n N ∆t ≤ 0. ( 28
)
The set of admissible velocities can be written:

K n = {U ∈ R 2N , u N -u N +1 - x n N +1 -x n N ∆t ≤ 0}. ( 29
)
and the function F writes (M = 1)

F (U) = u N -u N +1 - x n N +1 -x n N ∆t . ( 30
) -0,8 -0,4 0,0 0,4 0,8 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Velocity (m.s -1 ) Distance (m) t = 0,1 (a)
-0,8 -0,4 0,0 0,4

-0,6 -0,4 -0,2 0,0 0,2 0,4

-1,2 -0,8 -0,4 0,0 0,4 0,8 Results about velocity are presented on figure 5 at four different times of computation. One can see that the behavior of both fluids is very similar to the behavior of the fluid in the 1D problem of the impact on a wall. At t = 0.1 (a), both fluids have met, inducing in each one a shock wave that propagates symmetrically to the unconstrained boundary. In the intermediate region, velocity is logically zero by symmetry. When each shock wave reaches the boundary (just after (b)), both fluids have stopped and then expand in the opposite direction of their initial motion. This is observed on figure (c) by the presence of two rarefaction waves propagating in each fluid towards the constrained nodes. The velocity of the constrained node u N becomes negative, while u N +1 becomes positive. It leads to the detachment of both fluids, finally moving in opposite directions.

Again, let us examine the validity of properties of K n in this case. Properties and hold. On the contrary to the previous case, it is obvious that property always holds, since translations by any real a ∈ R * are in K n . Thus, momentum is preserved at any time. Let us now discuss about the validity of the property . It is clear that the impact between two fluids can be regarded as two separated impact problems for each fluid. Therefore, the same drawbacks that have been described in the case of the impact with a wall will be found in this one. We can show that that total energy is not preserved at the time of impact, due to the fact that K n is not a cone and there's no element in the form (1 + µ) U min in K n , with µ ∈ ]0; 1]. Total energy decreases at the time of impact, which is shown on figure 6. Again, this decrease is O(∆t) for the usual first-order discretization in time of the relation [START_REF] Wilkins | Computer Simulation of Dynamic Phenomena[END_REF]. Let us describe in this part the extension of the 1D impact problem to the 2D framework. The situation is depicted on figure 7. We admit that the equation of the wall surface has the general form f (x, y) = 0, so that the fluid initially moves in the subset f (x, y) ≤ 0. Length and height of the fluid are respectively L = 1 and H = 1. It is meshed with triangular cells for simplicity, and the number of nodes will be denoted N. Again, we define the global velocity vector U = (u 1 , u 2 , . . . , u N ) T that gathers the velocity vectors of all nodes in the mesh. The fluid initially moves along the x-direction with a constant velocity equal to 1. Its density is 1, the ambient pressure is 0 and we still work with a stiffened gas pressure law. We denote t c the time of impact.

Constraints apply on every node in the mesh, expressing that their position must remain in the subset Ω defined as:

Ω = {(x, y) ∈ R 2 , f (x(t), y(t)) ≤ 0 ∀t ≥ 0}. (31)
Obviously, constraint for a specific node will be active only if the node impacts the wall. Let us give three examples of 2D impact problems with different surface geometries. In each case, the properties of K will be discussed.

Plane wall

Let us consider a plan wall orthogonal to the x-direction, whose surface position is located at x surf = 0. We denote x r = (x r , y r ) T and u r = (u r , v r ) T respectively the position and the velocity vectors of the r-th node in the mesh. Recalling the expression (31), the position of each node must remain in the subset f (x r (t), y r (t)) ≤ 0, ∀t ≥ 0. This leads in this particular case to the following discrete formulation of the constraint:

x n+1 r ≤ 0 ∀(r, n) ∈ [1 : N ] × N, ⇔ x n r + ∆t u n r ≤ 0 ∀r ∈ [1 : N ] × N, ⇔ u n r ≤ -x n r ∆t ∀(r, n) ∈ [1 : N ] × N.
The set of admissible velocity K n is therefore defined as

K n = {U ∈ R 2N , u r ≤ -x n r ∆t , ∀r ∈ [1 : N ]}.
and the functions F write (M = N )

F r (U) = u r + x n r ∆t , ∀r ∈ [1 : N ] . (32)
In this case, K n is obviously closed and convex (properties and holds). When the fluid is attached to the wall, property is violated since translations inducing a motion in the x-positive direction are not allowed. The x-component of the momentum (see eq. ( 13)) is therefore not preserved. On the contrary, translations inducing a motion in the y-component are admissible, so the y-component of the momentum is preserved. Finally, we can show that the norm of the momentum decreases as long as impact occurs, as well as in the case of the 1D impact between a fluid and a wall.

As long as the fluid is attached to the wall, K is a cone, and total energy is preserved. Over the time step of impact, the 2D plane case can been seen as P independent 1D impact problems, where P is the number of nodes impacting the wall. As a consequence, total energy will decrease as soon as a node impacts the wall. In the plane case, the P nodes impact the wall at the same time, so that total energy will decrease just over the time step of impact.

Results of our computation are presented on figure 8 at three different times. As expected, we denote that, once the impact has occurred, the x-component u r of the velocity vectors of each node impacting the wall must cancel, while the y-component v r remains non zero. The fluid then slides on the wall.

Concave wall

Let us now consider a concave wall whose surface equation is given by the relation:

f (x, y) = x + y 2 = 0.
In this case this is not only the impact which is challenging but also the treatment of the sliding on such a curve because to our knowledge it is not addressed in the literature in the

Properties and hold. When the fluid has impacted the wall, properties and do not hold. Property does not hold for the same reason described in the plane case: translations in any direction that induces a penetration of the fluid into the wall are not allowed. Thus, momentum is not preserved and we can show that it decreases as in the plane case. Property does not hold when impact has occurred. On the contrary to the plane case, nodes impact the wall at different times, inducing for each impact a drop in total energy. Between two consecutive impact, total energy is not preserved since we can't find in K any element in the form (1 + µ) U min with µ > 0. To illustrate this argument, we propose the figure 10. Let us consider the r-th node has already impacted the wall and is sliding over it. Let x n r and x n+1 r be the position of the r-th node at time t n and t n+1 . u r is the r-th component of the vector U min . Choose any λ ∈ ]0; 1]. According to the proposed figure, the velocity (1µ) u r is clearly an element of K n , since it satisfies the constraint (with such velocity, the node would not enter the wall). On the contrary, the velocity (1 + µ) u r can't be in K for any µ > 0. This argument can be extended to all the nodes impacting the wall, so that there are elements in the form (1µ) U min in K n but no elements in the form (1 + µ) U min , for µ > 0. Recalling the proof of the proposition (2.4), the consequence is that total energy decreases between two consecutive impacts. We propose the result on figure 11.

x n r x n r + ∆t (1 -µ) u r x n r + ∆t (1 + µ) u r x n+1 r = x n r + ∆t u r
Figure 9 presents results at three different times of computation. As expected, fluid stops at the wall and then slides over it as long at impact occurs. This means that the component of the nodal velocities normal to the wall cancels as long as impact occurs.

Convex wall

We consider here a convex wall whose surface equation is given by the relation:

f (x, y) = x -y 2 = 0
The form of the set K n is The main difference with the previous case is that the property of convexity of K n does not hold, which questions the uniqueness of the minimum of the objective function J within K n . Results are proposed on figure 12. The behavior of the fluid is yet the one expected : while impacting the wall, the fluid slides over it. In this test case, the radius of curvature is sufficiently gentle to ensure that the minimum found in the minimization procedure is the one that corresponds to a correct physical behavior of the fluid. However, if the radius of curvature is much smaller, the minimum may be the wrong one and might lead a priori to the crash of the computation. We illustrate this situation by using the figure 13. We consider a single point impacting a concave wall with a steep radius of curvature. The constraint expresses that the point must not enter the wall. In this case, the position of the point X may be found within the set ]-∞; X A ] ∪ [X B ; +∞[, giving therefore two admissible values for the velocity. The uniqueness of the minimum is then not ensured. However, we think that it is always possible to capture the correct minimum of the objective function J in the case of non convex wall. This is due to the fact that we use an algorithm of minimization (in our case the Uzawa method) that requires an initial estimation U 0 to compute the solution U min . Efficiently, U 0 is close to U min . In our case, we always choose U 0 = U n min (the solution at the time of computation t n ) to compute the minimum of the objective function U n+1 min at the time t n+1 . U n min is all the closer to U n+1 min as the time step ∆t is small. Thus, by choosing a certain small CFL condition, our method will capture the correct solution even in the case of non convex sets for which several minima of J may exist. This is equivalent to say that there exists an implicit CFL condition that ensures the efficiency of the method. That's what we observe on the figure 12 in the case of a non convex wall.

K n = {U ∈ R 2N , x n r + ∆t u r -(y n r + ∆tv r ) 2 ≤ 0 ∀r ∈ [1 : N ]} and the functions F write (M = N ) F r (U) = x n r + ∆t u r -(y n r + ∆tv r ) 2 ∀r ∈ [1 : N ] . (34)

Conclusions

We proposed a new formulation of nodal solvers in cell-centered Lagrangian schemes allowing to take into account several types of constraints. Such formulation is based on the minimization of a strictly convex objective function J, depending on velocities of the constrained nodes, over a set of admissible velocities K which expresses constraints applying on the mesh. Four fundamental properties ensure the efficiency of the method. The existence and the uniqueness of the constrained solution U min is ensured if K is a closed set (property ), and convex (property ). The property of convexity is not necessary. We show thanks to an example that the physical solution can be captured for a non-convex set by restricting the CFL condition. Momentum is preserved if K contains translations (property ). Finally, total energy is preserved if K is a cone (property ), but again, such assumption is not necessary : the presence in K of elements in the form (1 + µ) U min and (1µ) U min for a µ > 0, leads to the preservation of total energy. In our numerical tests, the possible loss of energy over the time step of impact, as well as during the impact in the case of curved walls, vanishes for ∆t → 0, so that it is not a major problem according to us. A perspective of our work is the application of the method to the numerical modeling of slide lines in the interior of the computation domain. Slide lines can be expressed as equality constraints, and can be considered as a sub-class of contact problems formulated as inequality or unilateral constraints. Therefore the theory described in this work is convenient to treat slide lines. However, the difficulty results in the discretization of the slip condition to construct K n . This major difficulty will be the subject of a forthcoming paper. Future works will also focus on the 3D extension of the method. While theoretically easy, such issues are practically tedious to implement.

d

  dt V (t) ρ dV = 0 mass conservation,d dt V (t) ρ u dV + V (t) ∇p dV = 0 momentum conservation, d dt V (t) ρ e dV + V (t) ∇ • (pu) dV = 0 total energy conservation.

Figure 1 :

 1 Figure 1: schematics of the fluid-wall 1D impact

Figure 2 :

 2 Figure 2: Computed velocity at t = 0.1 (a), t = 0.4 (b), t = 0.6 (c) and t = 1.3 (d)

Figure 3 :

 3 Figure 3: Numerical loss of total energy at the time of impact in the case of 1D impact problem between a fluid and a wall. Results are given for decreasing values of the time step.

Figure 4 :

 4 Figure 4: Comparison between the analytical solution of the 1D Noh Problem (line) with our numerical solution for the 1D impact problem (symbol + line)

Figure 5 :

 5 Figure 5: Computed velocity at t = 0.1 (a), t = 0.4 (b), t = 0.6 (c) and t = 1.3 (d)

Figure 6 :

 6 Figure 6: Loss of total energy at the time of impact in the case of 1D Impact Problem between two fluids. Results are given for decreasing values of the time step

Figure 7 :

 7 Figure 7: schematics of the fluid-wall impact

Figure 8 :

 8 Figure 8: Fluid impacting on a plane wall. Results refers to three different time of computation (a) t = 0, (b) = t = 0.59 and (c) t = 0.67.

Figure 9 :

 9 Figure 9: Fluid impacting on a concave wall whose equation is x + y 2 = 0. Results refers to three different time of computation (a) t = 0, (b) = t = 0.59 and (c) t = 0.67.

Figure 10 :

 10 Figure 10: schematics of the fluid-wall 1D impact

Figure 11 :Figure 12 :

 1112 Figure 11: Loss of total energy for 2D impact of a fluid on a wall

Figure 13 :

 13 Figure 13: Loss of uniqueness of the minimum of J within K n


Boris Andreianov
email: boris.andreianov@univ-fcomte.fr

Frédéric Lagoutière
email: frederic.lagoutiere@math.u-psud.fr

Nicolas Seguin
email: nicolas.seguin@upmc.fr

Takéo Takahashi
email: takeo.takahashi@inria.fr

Well-posedness for a one-dimensional fluid-particle interaction model

Keywords: Fluid-particle interaction, Burgers equation, Non-conservative coupling, Well-posedness, BV estimates, Wave-front tracking, Splitting, Fixed-Point 2010 Mathematics Subject Classification: primary 35L65, secondary 35L81, 35R06, 65M12

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L ∞ data is also given.

Introduction

The aim of this paper is to prove the existence (in the BV setting and in the L ∞ setting) and uniqueness (in the BV setting) for the following system that models fluid-particle interaction via drag force:

∂ t u + ∂ x (u 2 /2) = λ (h ′ (t) -u) δ 0 (x -h(t)) (t > 0, x ∈ R), (1)
mh ′′ (t) = λ (u(t, h(t)) -h ′ (t)) (t > 0) [START_REF] Andreianov | Small solids in an inviscid fluid[END_REF] with the initial conditions

u(0, x) = u 0 (x) (x ∈ R), h(0) = h 0 , h ′ (0) = V 0 . (3)
In the above one-dimensional system, u(t, x) is the velocity of the fluid and we assume it satisfies the Burgers equation with a source term corresponding to the action of the particle on the fluid. The position of the particle is denoted by h(t) and its motion is governed by the Newton law where we have modeled the force of the fluid on the particle by λ (u(t, h(t)) -h ′ (t)), with λ > 0. For more details on this model, we refer to [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF] and also to [START_REF] Andreianov | Small solids in an inviscid fluid[END_REF] (see also [START_REF] Borsche | Mixed systems: ODEs -balance laws[END_REF] for related models). As emphasized in [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF], in the Burgers equation, the source term has to be defined: the product uδ 0 has a priori no sense since we are looking for a solution which is not continuous in variable x. We also need to interpret the right-hand side of the ODE (2) since u(t, •) may be discontinuous at x = h(t). Following classical approaches ( [START_REF] Isaacson | Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law[END_REF], [START_REF] Yves | Riemann solvers for some hyperbolic problems with a source term[END_REF], [START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF], [START_REF] Vasseur | Well-posedness of scalar conservation laws with singular sources[END_REF]), the definition of the source term in [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF] was obtained by considering a regularization process. This approach led to a rule of interface coupling that was formalized in our preceding works [START_REF] Andreianov | Small solids in an inviscid fluid[END_REF] and [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF] using the germ terminology introduced in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF].

Definition 1. The admissibility germ G λ ⊂ R 2 associated with [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] is defined by

G λ = (c -, c + ) ∈ R 2 ; c --c + = λ ∪ (c -, c + ) ∈ R + × R -; -λ ≤ c -+ c + ≤ λ . (4)
In what follows, we use the notation

G λ (V ) = (V, V ) + G λ (V ∈ R).
The key feature for the subsequent analysis of the problem is the following dissipativity and maximality property of the germ G λ (V ) (see [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF]Prop. 2.3]):

(c L , c R ) ∈ G λ (V ) ⇐⇒ ∀(b L , b R ) ∈ G λ (V ) Φ(c L , b L ) -V |c L -b L | ≥ Φ(c R , b R ) -V |c R -b R | ; (5)
here and in the whole paper, Φ is the function defined by

Φ(a, b) = sgn(a -b)(a 2 /2 -b 2 /2) (a, b ∈ R). (6)
Now let us give precise definitions of solutions, separately for (1) and for [START_REF] Andreianov | Small solids in an inviscid fluid[END_REF]. For the sake of simplicity, consider a finite time horizon T > 0 (global in time solutions are defined via localization to [0, T ] for all T > 0). Definition 2. (i) Given h ∈ W1,∞ ([0, T ]; R), a function u is a solution of (1) with initial datum u 0 if u ∈ L ∞ ((0, T ) × R) ∩ C([0, T ]; L 1 loc (R)), if u is a Kruzhkov entropy solution of the Burgers equation with initial datum u 0 in {(t, x) ; t ∈ (0, T ), x = h(t)}, and if for a.e. t ∈ (0, T ) the one-sided traces of u at the particle position satisfy u(t, h(t) -) , u(t, h(t)

+ ) ∈ G λ (h ′ (t)). (7)
(ii) A function h is a solution of (2) with the initial data h 0 , V 0 if h ∈ W 2,∞ ([0, T ]), if h(0) = h 0 , h ′ (0) = V 0 and if, given u a Kruzhkov entropy solution of the Burgers equation in {(t, x) ; t ∈ (0, T ), x = h(t)}, we have for a.e. t ∈ (0, T ) mh ′′ (t) = u(t, h(t) -) 2 2 -h ′ (t) u(t, h(t) -) -u(t, h(t) + ) 2 2 -h ′ (t) u(t, h(t) + ) .

(iii) A pair (u, h) satisfying the previous two items is said to be an entropy solution of (1), ( 2), [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF].

Remark 3. The above interpretation [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF] of the ODE (2) stems from the principle of conservation of the total momentum Q(t) := R u(t, •) dx + mh ′ (t).

Remark 4. It is worth noticing that, although in the above definition, a Kruzhkov entropy solution u of the Burgers equation is a priori only in L ∞ ((0, T ) × R) ∩ C([0, T ]; L 1 loc (R)) formulas [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF] and ( 8) make sense: indeed, according to Panov in [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] (see also Kwon,Vasseur [12]), the traces t → u(t, h(t) ± ) on the path {x = h(t)} exist in the strong L 1 sense whenever u is an entropy solution away from the path (due to the regularity assumption on the path h(t)) 1 .

Yet, the explicit use of traces of u in the above definition makes it delicate to discuss approximation of solutions; fortunately, an equivalent "traceless" formulation can be provided:

Definition 5. (i) Given h ∈ W 1,∞ ([0, T ], R), a function u is a solution of (1) with initial datum u 0 if u ∈ L ∞ ((0, T ) × R) ∩ C([0, T ]; L 1 loc (R)), if and if for all (κ L , κ R ) ∈ R 2 there exists M > 0 such that ∂ t |u -κ| + ∂ x Φ(u, κ) ≤ M dist 1 (κ L , κ R ) , G λ (h ′ (t)) δ 0 (x -h(t)) in D ′ (R + × R) with initial datum u 0 , (9)
where dist 1 is the distance on R 2 induced by the norm (x, y) 1 = |x| + |y|, and where

κ(t, x) := κ L 1 1 R-(x -h(t)) + κ R 1 1 R+ (x -h(t)). (ii) A function h is a solution of (2) with the initial data h 0 , V 0 if h ∈ W 2,∞ ([0, T ]), if h(0) = h 0 , h ′ (0) = V 0
and if, given u a Kruzhkov entropy solution of the Burgers equation in {(t, x) ; t ∈ (0, T ), x = h(t)}, for all ξ ∈ D([0, T )), for all ψ ∈ D(R) such that ψ ≡ 1 on the set h([0, T ]), there holds

-m T 0 h ′ (t)ξ ′ (t)dt = mV 0 ξ(0)+ T 0 R uψ(x)ξ ′ (t)+ u 2 2 (t, x)ξ(t)ψ ′ (x) dx dt+ R u 0 (x)ψ(x)ξ(0) dx. ( 10
)
(iii) A pair (u, h) satisfying the previous two items is said to be an entropy solution of (1),( 2), [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF].

Proof of the equivalence between Def. 5 et Def. 2. The proof of equivalence of Definition 2(i) and Definition 5(i) follows the guidelines of [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF]. Namely, one can perform the change of variable y = x -h(t) in order to rectify the particle path, and then use [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]. The equivalence of Definition 2(ii) and Definition 5(ii) comes from an application of the Green-Gauss formula, see [2, Lemma 2.1].

Remark 6. In the above definition, let us stress that we can choose the constant M in [START_REF] Isaacson | Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law[END_REF] such that it depends only on (κ L , κ R ), and on u L ∞ ((0,T )×R) , h ′ L ∞ (0,T ) .

We are now in position to state the first main result of this paper:

Theorem 7 (L ∞ case). Assume u 0 ∈ L ∞ (R) and h 0 , V 0 ∈ R.
Then for all T > 0 there exists at least one solution (u, h) of (1), ( 2) and (3) in the sense of Definition 2 (or equivalently of Definition 5).

The second main result of this paper is the following Theorem 8 (BV case). Assume u 0 ∈ BV (R) and h 0 , V 0 ∈ R. Then any solution (u, h) of (1), ( 2) and (3) in the sense of Definition 2 (or equivalently of Definition 5) satisfies, for all T > 0, u ∈ L ∞ (0, T ; BV (R)).

Moreover, there exists at most one solution of (1), ( 2) and (3).

Actually, the uniqueness result of Theorem 8 is a "weak-strong" type uniqueness result (see Theorem 16 for a precise statement). Let us note that the uniqueness in the L ∞ case (hypothesis of Theorem 7) remains an open problem. Some ingredients of the proof of the theorems are the results of the work [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF] of the first and the third authors, where the Burgers equation with singular source term supported by the path h(t) ≡ 0 of a stationary particle is considered; it is not difficult to extend them to the case of a general "frozen" particle path (we assume that this path is given and it is C 1 or piecewise C 1 ). The other ingredient of Theorem 8 is the new BV estimate that we prove in this paper. These results are presented in Section 2. Using the technique of continuous dependence on the flux of solutions of conservation laws (see Bouchut and Perthame [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF], Karlsen and Risebro [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF], Lécureux-Mercier [START_REF] Lécureux-Mercier | Improved stability estimates for general scalar conservation laws[END_REF] and references therein), we evaluate the gap between two solutions of (1) corresponding to different particle paths. Based on this estimate, in Section 3 we derive uniqueness of a solution to the coupled problem (1),( 2), [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF]. Then in Section 4, we describe a splitting approximation scheme for the coupled problem and prove its convergence. Section 5 contains the proof of existence in the L ∞ case, that is based upon a fixed-point argument and a tedious check of the continuity of the map that associates the solution u in the sense of Definitions 2(i),5(i) to a given particle path h. The Appendix contains the calculations needed to establish the BV estimate; they are based upon the wave-front tracking algorithm of Dafermos [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF] as presented in Holden and Risebro [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF].

2 Burgers equation driven by a particle with a given path: known results and new BV estimates

In this section, we consider a given path h ∈ W 1,∞ ([0, T ]). We study well-posedness for the Cauchy problem for (1) in the sense of Definition 2(i) (or 5(i)).

Well-posedness for the Cauchy problem

With the results and techniques of [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF], one can deduce several consequences. Therefore, in the sequel of this subsection, we skip some of the proofs or we only give the main steps. First, we focus on the case of a continuous and piecewise affine path h ℓ . Let 0 = t 0 < t 1 < • • • < t N ℓ = T be the sequence of times where the derivative of h ℓ is discontinuous and note (V n ℓ ) n the associated sequence of velocities. Then, we can write h ′ ℓ (t) = N ℓ n=1 V n ℓ 1 1 (t n-1 ,t n ] (t) and we assume without loss of generality that h ℓ (0) = 0. A direct consequence of [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF] is the well-posedness result for the Cauchy problem and the following L ∞ bound. Lemma 9. Assume that u 0 ∈ L ∞ (R) and that h ℓ is a given piecewise affine path. Then there exists a unique solution u ℓ of (1) in the sense of Definition 2(i) (or 5(i)). This solution obeys the bound u ℓ (t, •) ∞ ≤ u 0 ∞ +λ.

Proof. The existence and uniqueness proofs were already discussed in [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF]Sect. 4.2]; our main focus is on the L ∞ estimate. Performing the change of variables v(t, y)

:= u ℓ (t, y + V 1 ℓ t) -V 1 ℓ , we notice that in the time interval (0, t 1 ], v is a solution of ∂ t v + ∂ y (v 2 /2) = -λ v δ 0 , v(0, •) = u 0 -V 1 ℓ . Let us define c R := min ess inf R- u 0 -λ , ess inf R+ u 0 and c L := c R + λ. Since (c L -V 1 ℓ , c R -V 1 ℓ ) ∈ G λ , the function c(y) := c L 1 1 R-(y) + c R 1 1 R+ (y) -V 1 ℓ (y ∈ R)
is a stationary entropy solution of ∂ t u + ∂ x (u 2 /2) = -λ u δ 0 (x) for t > 0 and x ∈ R (see Definition 2, (i), with h(t) = 0 for any t). Moreover, v(0, •) ≥ c(•) and thus, using [3, Thm. 2.6] we deduce that v(t, y) ≥ c(y) for all y ∈ R and all relevant t. Consequently,

u ℓ (t, x) ≥ c L 1 1 R-(x -h ℓ (t)) + c R 1 1 R+ (x -h ℓ (t)) (x ∈ R, t ∈ (0, t 1 ]).
By induction, we can prove that for all n,

u ℓ (t, x) ≥ c L 1 1 R-(x -h ℓ (t)) + c R 1 1 R+ (x -h ℓ (t)) (x ∈ R, t ∈ (t n-1 , t n ]).
More precisely, we perform the change of variables v(t, y)

:= u ℓ (t+t n , y+h ℓ (t n )+V n+1 ℓ t)-V n+1 ℓ
for t ∈ (t n-1 , t n ] and we follow the above proof. Using the same technique to get the upper bound, we finally get

min{ess inf R -u 0 -λ , ess inf R+ u 0 } ≤ u ℓ (t, x) ≤ max{ess sup R - u 0 , ess sup R+ u 0 + λ}. (11)
Hence the claim follows.

Let us turn now to the case of a particle path h ∈ W 1,∞ ([0, T ]). The uniqueness result can be adapted from [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF] (it is actually independent of the previous lemma): Proposition 10. Assume that u 0 ∈ L ∞ (R) and that h ∈ W 1,∞ ([0, T ]) is given. There is at most one solution u of equation (1) in the sense of Definition 2(i) (or 5(i)).

Proof. We take a sequence of classical test functions ψ R n n∈N (see [START_REF] Stanislav | First order quasilinear equations with several independent variables[END_REF]) approximating (as n tends to +∞) the characteristic function of the trapezoid {(t, x) ; |x| ≤ R + L(T -t), 0 ≤ t ≤ T } with L ≥ u ∞ and we multiply them by a truncation function

ξ n ∈ D([0, T ] × R * ) around the particle path approximating (t, x) → min{ n|x -h(t)| , 1 }.
Let us consider u, û two entropy solutions in the sense of Definition 2(i) with the same initial condition u 0 ∈ L ∞ (R). Using the Kruzhkov doubling of variables method with the test functions ψ R n ξ n , which vanish on {x = h(t)}, we get the so-called Kato inequality

- T 0 R |u -û| (∂ t ψ R n ) ξ n dx dt - T 0 R Φ(u, û) (∂ x ψ R n ) ξ n dx dt ≤ T 0 R (|u -û|, Φ(u, û)) • (∂ t ξ n , ∂ x ξ n ) ψ R n dx dt =: I n (12)
where • denotes the scalar product in R 2 ; now, showing that the limit, as n → ∞, of the right-hand side I n is non-positive, taking R as large as desired, one concludes to the uniqueness of a solution, as in [START_REF] Stanislav | First order quasilinear equations with several independent variables[END_REF]. To study the sign of lim n→∞ I n , we introduce the notation

Φ(h ′ (t); u, û) := Φ(u, û) -h ′ (t)|u -û|.
The right-hand side in ( 12) is an approximation of

T 0 R n sign(x -h(t)) Φ(h ′ (t); u, û) 1 1 [0, 1 n ] (|x -h(t)|)ψ R
n , so that using the strong traces of u at x = h(t) ± at the limit n → ∞ we can write this term as

- T 0 Φ(h ′ (t); u(t, h(t) -), û(t, h(t) -)) -Φ(h ′ (t); u(t, h(t) + ), û(t, h(t) + )) ψ R n (h(t)) dt.
Due to the trace condition in Definition 2(i) and to the germ dissipativity property (5), the above term is non-positive. Then, the non-positivity of the left-hand side of ( 12) implies the uniqueness, following classical arguments. Details can be found in [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF], for the particular case h(•) ≡ 0.

Using Proposition 10 and Lemma 9, we deduce the following result.

Theorem 11. Assume h ∈ W 1,∞ ([0, T ]) and u 0 ∈ L ∞ (R)
, there exists a unique solution u of equation (1) in the sense of Definition 2(i) (or 5(i)). Moreover,

∀t ∈ (0, T ) u(t, •) ∞ ≤ u 0 ∞ + λ. ( 13
)
Proof. The uniqueness is proved in Proposition 10. In order to prove existence of a solution to (1) we can approximate in W 1,1 ([0, T ]) the function h by a sequence (h ℓ ) ℓ of continuous piecewise affine paths. Applying Lemma 9, we deduce that for each ℓ, there exists a unique solution u ℓ of (1) with initial datum u 0 ∈ L ∞ (R) and with path h ℓ . Moreover, the following estimate holds:

u ℓ (t, •) ∞ ≤ u 0 ∞ + λ. ( 14
)
Thus we can use the strong compactness result of [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF], [START_REF] Lions | Kinetic formulation of the isentropic gas dynamics and p-systems[END_REF] to deduce that u ℓ → u a.e. (0, T ) × R. To deduce that u is solution of (1), we write for each ℓ the global integral formulation of Definition 5(i) (it is done in Section 4.2 for the coupled problem) with a positive ψ ∈ D((0,

T ) × R): for all (κ L , κ R ) ∈ R 2 there exists M > 0 such that T 0 R |u ℓ -κ| (∂ t ψ) dx dt + T 0 R Φ(u ℓ , κ) (∂ x ψ) dx dt + M T 0 dist 1 ((κ L , κ R ), G λ (h ′ ℓ (t))ψ(h ℓ (t)) dt ≥ 0. ( 15
)
Note that the constant M above does not depend on ℓ since we have the uniform estimate ( 14) and since Remark 6). The result of weak compactness allows to pass to the limit in the first two integrals. For the last one, it is sufficient to notice that for almost every t ∈ (0, T ),

h ′ ℓ ∞ ≤ h ′ ∞ (see
dist 1 ((κ L , κ R ), G λ (h ′ ℓ (t))) ≤ 2|h ′ ℓ (t) -h ′ (t)| + dist 1 ((κ L , κ R ), G λ (h ′ (t))), |ψ(h ℓ (t)) -ψ(h(t))| ≤ C h ℓ -h ∞ .
In particular, since h ℓ → h in W 1,1 (0, T ), we can pass to the limit in [START_REF] Lécureux-Mercier | Improved stability estimates for general scalar conservation laws[END_REF] and deduce that u := lim N ℓ →∞ u ℓ ∈ L ∞ ((0, T ) × R) satisfies the global integral formulation of Definition 5(i). To end the proof, we invoke a classical result (see [START_REF] Cancès | On the time continuity of entropy solutions[END_REF]) to deduce that u ∈ C([0, T ]; L 1 loc (R)). Remark 12. Notice that another way to prove the convergence of (u ℓ ) ℓ is to use the continuity estimate (23) below, first with u 0 ∈ BV (R) and different paths h ℓ , then with the fixed path h and a sequence of BV initial data approximating a general L ∞ initial datum u 0 .

Accurate BV estimates on the solutions

This paragraph is devoted to careful estimation of the space variation for the unique entropy solution for (1) with h ≡ 0. More precisely, we re-consider the following particular case that was the object of [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF]:

∂ t u + ∂ x (u 2 /2) = -λ u δ 0 (x) (t > 0, x ∈ R), (16)
with the initial condition

u(0, x) = u 0 (x) (x ∈ R). (17)
Even in that case, the problem of existence, in the sense of Definitions 2(i),5(i) remains delicate. Specifically, in order to get fine estimates on the solutions, wave interactions with the source path have to be handled carefully. While existence was shown in [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF], the method employed was not based on a uniform BV control of approximate solutions (BV loc control away from the interface was used instead). Here, constructing the solutions of ( 16), ( 17) by the wave-front tracking algorithm (see [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF]), we control wave interactions quite precisely and justify the following bound. Proposition 13. Assume u 0 ∈ BV (R). The unique solution of ( 16), [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] in the sense of Definitions 2(i) (or 5(i)) belongs to L ∞ ([0, T ]; BV (R)) and satisfies

sup t∈(0,T ] TotVar u(t, •) ≤ TotVar u 0 + 2 dist 1 (u 0 (0 -), u 0 (0 + )) , G λ . (18)
A proof of this proposition, based on a wave front tracking, lies in the Appendix. Notice that for approximations u δ obtained with the wave-front tracking algorithm, we obtain an additional remarkable property: namely, TotVar u δ is a non-decreasing function of t ∈ (0, T ]. At the present stage, we are unable to justify this property for u = lim δ→0 u δ solution of ( 16), [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF].

As a consequence, we can improve the result of Theorem 11, giving a precise BV control of solutions corresponding to BV data. In order to state this result we introduce the space

P BV ([0, T ]) := {h ∈ W 1,∞ ([0, T ]) ; h ′ ∈ BV ([0, T ])}.
Corollary 14. Let u be the solution of (1) with given h ∈ P BV ([0, T ]) and with initial datum u 0 ∈ BV (R). Then for all t ∈ (0, T ) we have

TotVar u(t, •) ≤ TotVar u 0 + 2 dist 1 (u 0 (0 -), u 0 (0 + )) , G λ (h ′ (0)) + 4TotVar [0,t] h ′ . ( 19
)
Proof. It is easy to see that Proposition 13 can be extended to the case of affine path h by using a change of variable (the important property being h ′′ = 0). We choose a sequence of piecewise affine functions h ℓ with variation on [0, T ] not exceeding the variation of h ′ (•) on [0, T ]. It is enough to apply the estimate of Proposition 13(ii) on each interval [t n-1 ℓ , t n ℓ ] where the approximate piecewise affine path h ℓ is straight. At the initial time, the total variation of u 0 may increase by 2 dist 1 (u 0 (0 -), u 0 (0 + )) , G λ (h ′ (0)) due to the interaction that replaces (u 0 (0 -), u 0 (0 + )) by a state (c L , c R ) at the interface arising from the solution of the corresponding Riemann problem. At every subsequent step t = (t n ℓ ) -the pair of traces belongs to G λ (h ′ ℓ ((t n ℓ ) -)) and the first interaction replaces it by a pair of states in G λ (h ′ ℓ ((t n ℓ ) + )), therefore, following Proposition 13, we have

TotVar u ℓ ((t n ℓ , •) + ) ≤ TotVar u ℓ ((t n ℓ ) -, •) + 2 dist 1 G λ (h ′ ℓ ((t n ℓ ) -)) , G λ (h ′ ℓ ((t n ℓ ) + )) .
Hence, recalling that

G λ (V ) = G λ + (V, V ), we have dist 1 G λ (h ′ ℓ ((t n ℓ ) -)) , G λ (h ′ ℓ ((t n ℓ ) + )) ≤ 2 h ′ ℓ ((t n ℓ ) + ) -h ′ ℓ ((t n ℓ ) -)
and we deduce

TotVar u ℓ (t, •) ≤ TotVar u 0 + 2 dist 1 (u 0 (0 -), u 0 (0 + )) , G λ (h ′ (0)) + 4 TotVar [0,t] h ′ ℓ
which yields [START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF] passing to the limit as in the proof of Theorem 11.

Lipschitz continuous dependence of BV solutions with respect to h

Assume h ∈ W 1,∞ ([0, T ]) and u 0 ∈ L ∞ (R), and consider the unique solution u of equation ( 1) in the sense of Definition 2(i) (or 5(i)) (see Theorem 11). We make the coordinate transformations that rectify the path of the particle, and consider

w(t, x) = u(t, x + h(t)). Then w ∈ L ∞ ((0, T ) × R) ∩ C([0, T ]; L 1 loc (R))
is the entropy solution of the problem

∂ t w + ∂ x (w 2 /2 -V w) = λ (V -u) δ 0 (x) (t ∈ (0, T ), x ∈ R), (20) w(0, x) = w 0 (x) (x ∈ R), (21)
where

V = h ′ ∈ L ∞ (0, T ). In particular, setting Φ(V ; a, b) := |a -b| a+b 2 -V , we deduce from Definition 2 that for all k ∈ R ∂ t |w -k| + ∂ x Φ(V ; w, k) ≤ 0 in D ′ ((0, T ) × R * ) (22)
(where R * stands for R \ {0}) and w(t, 0 -) , w(t, 0 + ) ∈ G λ (V (t)) for a.e. t ∈ (0, T ).

In this section, we compare two entropy solutions w and ŵ of ( 20)-(21) for two different particle velocities V and V and two different initial conditions w 0 and ŵ0 . The corresponding result applies in the case where at least one of the two solutions enjoys a uniform in time BV in space estimate. Its proof is similar to the arguments of [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] and [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF].

Lemma 15. Assume V, V ∈ L ∞ (0, T ), w 0 , ŵ0 ∈ L ∞ (R) and w 0 -ŵ0 ∈ L 1 (R). Let w, ŵ ∈ L ∞ ((0, T ) × R) ∩ C([0, T ]; L 1 loc (R))
be the entropy solutions of system (20)-( 21) associated respectively with (V, w 0 ) and ( V , ŵ0 ). Assume that there exists C T such that for all t ∈ (0, T ), TotVar w(t, •) ≤ C T . Then there exists

C ∈ R such that for all t < T R | ŵ -w|(t, x) dx ≤ R | ŵ0 -w 0 | dx + (C T + C) t 0 | V -V |(s) ds ∈ R (23) (C depends on w ∞ , ŵ ∞ and V -V ∞ ).
Proof. We notice that for all k ∈ R

∂ x Φ( V ; w, k) -Φ(V ; w, k) = (V -V )∂ x |w -k| ≤ |V -V | |∂ x w|.
Using (22), we see that w satisfies

∂ t |w -k| + ∂ x Φ( V ; w, k) ≤ ∂ x Φ( V ; w, k) -Φ(V ; w, k) ≤ µ in D ′ ((0, T ) × R * ) ( 24
)
where µ ∈ L ∞ (0, T ; M b (R)) is a non-negative measure with

µ L 1 (0,t;M b (R)) ≤ C T t 0 | V -V | ds. (25)
Now we can use ( 22) and ( 24) perform the doubling of variables of Kruzhkov, with a test function ψ R n ξ n with, as in the proof of Proposition 10, ψ R n approximating the characteristic function of the trapezoid {(t, x)

; |x| ≤ R + L(T -t), 0 ≤ t ≤ T } with L ≥ w ∞ + ŵ ∞ + V ∞ and ξ n ∈ D(R * ) approximating x → min{ n|x| , 1 }.
Using (25), we deduce the following Kato inequality for all t ∈ (0, T )

R | ŵ -w|(t, •)ψ R n (t, •)ξ n dx ≤ t 0 R Φ( V ; ŵ, w) ∂ x (ψ R n ξ n ) dx ds + t 0 R | ŵ -w|∂ t ψ R n ξ n dx ds + R | ŵ0 -w 0 |ψ R n (0, •)ξ n dx + ψ R n ξ n ∞ C T t 0 | V -V | ds. ( 26
)
Using the properties of Φ( V ; ŵ, w) with the choice of L, we deduce

R | ŵ -w|(t, •)ψ R n (t, •)ξ n dx ≤ t 0 R Φ( V ; ŵ, w) ψ R n ∂ x ξ n dx ds + R | ŵ0 -w 0 |ψ R n (0, •)ξ n dx + C T t 0 | V -V | ds. ( 27
)
Proceeding as in the proof of Theorem 11, taking n → ∞, we find

R | ŵ -w|(t, •)ψ R n (t, •) dx ≤ R | ŵ0 -w 0 |ψ R n (0, •) dx + C T t 0 | V -V | ds + t 0 Φ( V (t); ŵ(t, 0 + ), w(t, 0 + )) -Φ( V (t); ŵ(t, 0 -), w(t, 0 -)) ds. ( 28
)
It remains to estimate the last term of the above inequality. In order to do this, we recall that, for a.e. t > 0, (w(t, 0 -), w(t,

0 + )) ∈ G λ (V (t)) = G λ ( V (t)) -(∆(t), ∆(t)), ( ŵ(t, 0 -), ŵ(t, 0 + )) ∈ G λ ( V (t)), (29)
with ∆ := V -V . This suggests the following decomposition (removing the time variable for readability):

Φ( V ; ŵ(0 + ), w(0 + )) -Φ( V ; ŵ(0 -), w(0 -)) = Φ( V ; ŵ(0 + ), w(0 + )) -Φ( V ; ŵ(0 + ), w(0 + ) + ∆) + Φ( V ; ŵ(0 + ), w(0 + ) + ∆) -Φ( V ; ŵ(0 -), w(0 -) + ∆) + Φ( V ; ŵ(0 -), w(0 -) + ∆) -Φ( V ; ŵ(0 -), w(0 -)) =: E r + E o + E l
respectively. By the dissipativity property (5) of the germs and from (29), we deduce E o ≤ 0 (because (w(0 -) + ∆, w(0 + ) + ∆) ∈ G λ ( V (t))). Moreover, using that Φ is a locally Lipschitz continuous function, we obtain

|E l | + |E r | ≤ |∆|C( w ∞ , ŵ ∞ , V ∞ )
for ∆ sufficiently small, where C is the (local) Lipschitz modulus of Φ. Therefore

Φ( V ; ŵ(0 + ), w(0 + )) -Φ( V ; ŵ(0 -), w(0 -)) ≤ |V -V |C( w ∞ , ŵ ∞ , V ∞ ).
Combining the above estimate with (28) and taking R → ∞, we conclude the proof of the lemma.

3 Uniqueness for the coupled problem with BV initial data

Based on the result of Section 2.3 and on the BV estimate of Section 2.2, we are able to prove the uniqueness of an entropy solution to the Burgers equation driven by a particle with a given path in P BV ([0, T ]), in the sense of Definition 2 (iii) (or equivalently Definition 5 (iii)).

Theorem 16. Assume u 0 ∈ BV (R) and assume there exists a solution (u, h) of problem (1),( 2),(3) in the sense of Definition 2 and such that u ∈ L ∞ (0, T ; BV (R)). Then this solution is unique within the whole class of (not necessarily BV in space) solutions of (1),(

Proof. Assume that (u, h) is solution of ( 1),( 2),(3) in the sense of Definition 2 with u ∈ L ∞ (0, T ; BV (R)). We consider C T such that TotVar u(t,

•) ≤ C T for almost every t ∈ [0, T ]. Assume that (û, ĥ) ∈ L ∞ ((0, T ) × R) ∩ C([0, T ]; L 1 loc (R)) × W 1,∞ ([0, T ]
) is another solution with the same initial data. Making the change of variables of Section 2.3 w(t, y) = u(t, y + h(t)), ŵ(t, y) = û(t, y + ĥ(t)).

we find for all t < T R |û(t, x) -u(t, x)| dx = R |û(t, x + ĥ(t)) -u(t, x + ĥ(t))| dx ≤ R | ŵ(t, y) -w(t, y)| dy + |h(t) -ĥ(t)| TotVar u(t, •), and finally thanks to Lemma 15 we have

R |û(t, x) -u(t, x)| dx ≤ 2C T + C( u ∞ , û ∞ , h ′ -ĥ′ ∞ ) t 0 |h ′ (s) -ĥ′ (s)| ds, (30)
where we have bounded |h(t) -ĥ(t)| by t 0 |h ′ (s) -ĥ′ (s)| ds by using the fact that h(0) = ĥ(0). Now, we use formulation [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF] to estimate the right hand side of (30). Taking ψ suitable for both paths h and ĥ and using an approximation of ξ = 1 1 [0,s] , we obtain

m(h ′ (s) -h ′ (0)) = - R u(s, •)ψ + R u 0 ψ + mh ′ (0)ξ(0) + s 0 R u 2 2 ψ ′ + R u 0 ψξ(0), (31)
and a similar formula for ĥ′ (s). Making the difference of the two expressions yields

m |h ′ (s) -ĥ′ (s)| ≤ C( u ∞ , û ∞ ) R |u(s, x) -û(s, x)| dx + s 0 u(τ, •) -û(τ, •) L 1 (R) dτ
(actually C may depend on ψ). Inserting this estimate in (30) and using the Gronwall inequality allows to conclude (û, ĥ) = (u, h).

A splitting scheme and existence for the coupled problem

In this section, starting from BV data u 0 we construct approximate solutions and prove their convergence to a solution of the coupled problem; together with the uniqueness result of the previous section, this establishes Theorem 8.

Splitting algorithm

Take N ∈ N * = N \ {0} and set ∆t = T /N > 0. We set t N n = n∆t for n ≤ N and we define the approximate solutions (u N , h N ) by the following algorithm.

• Define u N 0 := u 0 ∈ BV (R), h N 0 = h 0 and V N 0 = V 0 .
• Assume that we know u N n , h N n and V N n . To compute u N n+1 , h N n+1 and V N n+1 , we solve the following two steps.

we define h

N on (t N n , t N n+1 ] by h N (t) = h N n + V N n (t -t N n
), and u N on (t N n , t N n+1 ] as the solution of the problem

∂ t u + ∂ x u 2 /2 = -λ u -V N n δ 0 x -h N (t) , u| t=t N n = u N n ;
in the sense of Definitions 2(i) with the straight path h N .

Then, we write u

N n+1 := u N (t N n+1 ), h N n+1 := h N (t N n+1
), and

V N n+1 := V N n - 1 m t N n+1 t N n (u N (t)) 2 2 -V N n u N (t) x=h N (t) dt (32)
where α(t) = α(t, h N (t) + )-α(t, h N (t) -) denotes the jump across the path h N of the quantity α(t).

Remark 17. In the spirit of [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF], fixing some function ψ ∈ D(R) satisfying ψ ≡ 1 on the subset h N ([0, T ]) of R and ξ(t) = 1 1 [t N n ,t N n+1 ] (t), and using the Green-Gauss formula, we can propose the alternative equation for

V N n+1 : m(V N n+1 -V N n ) = t N n+1 t N n R (u N (t)) 2 2 ψ ′ dx dt + R (u N n -u N n+1 )ψ dx . (33)
From Theorem 11, it is clear that the algorithm makes sense, that is to say that it yields a unique solution (h N , u N ) defined in [0, T ] for any N ∈ N * . In the next paragraph, we prove compactness and convergence of the splitting approximation procedure as N → +∞.

Compactness and passage to the limit

Assume for simplicity that the initial datum u 0 is compactly supported (the general case follows using the property of finite speed of propagation, see [START_REF] Stanislav | First order quasilinear equations with several independent variables[END_REF]). Notice that in this case, u N ∈ L ∞ (0, T ; L 1 (R)).

Using Theorem 11, we obtain that for all N , the sequence (u N ) satisfies

u N ∞ ≤ u 0 ∞ + λ. (34)
Using this estimate and (32), we deduce that there exists a positive constant C such that

|V N n+1 -V N n | ≤ C∆t + C∆t|V N n | and |V N n+1 | ≤ (1 + C∆t)|V N n | + C∆t.
Thanks to a discrete Gronwall lemma, the above estimate yields that (V N n ) is bounded independently of n ∈ {0, . . . , N } and N , and thus that there exists a constant C independent of N such that

h N ∞ + V N ∞ + TotVar (0,T ) V N ≤ C. (35)
Using Helly's theorem, we deduce that (up to a subsequence) V N → V in L 1 (0, T ) and that h N → h in W 1,1 (0, T ), with V ′ = h. Moreover, using Corollary 14, we deduce that there exists a constant C independent of N such that TotVar R u N ≤ C (t ∈ (0, T ), N ≥ 0).

Setting w N (t, x) = u N (t, x + h N (t)) and applying Lemma 15, we see that

w N → w in L ∞ (0, T ; L 1 (R)).
We then set u(t, x) = w(t, x -h(t)) and we deduce from the above results that

u ∈ L ∞ (0, T ; L 1 (R) ∩ BV (R)) ∩ L ∞ ((0, T ) × R). ( 36
)
Proceeding as in the proof of Theorem 16 (see ( 30)), we deduce that u N → u in L ∞ (0, T ; L 1 (R)) up to a subsequence.

It remains to show that (u, h) is a solution to problem (1),( 2), [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF]. First, we write for each N the global integral formulation of Definition 5(i) with a non-negative ψ ∈ D((0, T ) × R): for all (κ L , κ R ) ∈ R 2 and for all κ as in Definition 5 [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], there exists M > 0 such that

T 0 R |u N -κ| (∂ t ψ) dx dt+ T 0 R Φ(u N , κ) (∂ x ψ) dx dt+M T 0 dist 1 ((κ L , κ R ), G λ (V N ))ψ(t, h N (t)) dt ≥ 0. ( 37
)
Note that the constant M above does not depend on N because of the above uniform estimates (34) on u N and (35) on V N (see Remark 6). From the convergence of (u N ), we can pass to the limit in the above formulation, taking into account that

dist 1 (κ L , κ R ), G λ (V N (t)) ≤ dist 1 (κ L , κ R ), G λ (V (t)) + 2|V N (t) -V (t)|.
Finally, to obtain [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF], we first derive its discrete version from (33). To this end, we take ξ ∈ D([0, T )), we set ξ N n = ξ(t N n ) for n ∈ {0, . . . , N }, and we multiply (33) by ξ N n and sum over n ≥ 0. The Abel transform yields

-m N -1 n=0 ∆t V N n ξ N n+1 -ξ N n ∆t -mV 0 ξ 0 = N -1 n=0 ξ N n t N n+1 t N n R (u N ) 2 2 ∂ x ψ + N -1 n=0 ∆t R u N n ξ N n+1 -ξ N n ∆t ψ + R u 0 ψξ 0 .
Passing to the limit as N → ∞ gives [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF]. Since h ∈ P BV ([0, T ]) and using that u is regular enough (see (36)), we conclude that h satisfies (8). Consequently, h ∈ W 2,∞ ([0, T ]). This concludes the proof of Theorem 8.

Note that thanks to the uniqueness Theorem 16, the whole sequence u N , h N N ∈N * actually converges.

5 Proof of Theorem 7: existence for L ∞ initial data

The goal of this section is to prove the existence result in L ∞ setting, i.e. Theorem 7. We use the fixed-point theory in the following way. Fix u 0 ∈ L ∞ (R), h 0 , V 0 ∈ R, fix T > 0 and define

E := {h ∈ C 1 ([0, T ]) : h(0) = h 0 , h ′ (0) = V 0 } endowed with the C 1 ([0, T ]) norm.
Then the map A is defined as follows: let us consider h ∈ E. Then

• We first associate the solution u of (1) in the sense of Definition 2(i) (or 5(i)) which is given by Theorem 11; note in particular that the traces t → u(t, h(t) ± ) on the path {x = h(t)} exist in the strong L 1 sense.

• Second, we define by

h * ∈ W 2,∞ ([0, T ]) the solution of h * (0) = h 0 , (h * ) ′ (0) = V 0 and m(h * ) ′′ (t) = u(t, h(t) -) 2 2 -h ′ (t) u(t, h(t) -) - u(t, h(t) + ) 2 2 -h ′ (t) u(t, h(t) + ) , ( 38
)
where u is obtained from h in the first step.

This solution is expressed as h * = A(h).

Remark 18. Let us notice that a fixed point h to A and the corresponding solution u of (1) in the sense of Definition 2(i) (or 5(i)) which is given by Theorem 11 provide a solution (h, u) to our problem. We notice that an application of the Green-Gauss formula yields that, whenever u solves the Burgers equation away from the path {x = h(t)}, equation ( 38) is equivalent to the fact that for all ξ ∈ D([0, T )), for all ψ ∈ D(R) such that ψ ≡ 1 on the set h([0, T ]), there holds

m T 0 (h * ) ′′ (t)ξ(t)dt = T 0 R uψξ ′ + u 2 2 ξψ ′ dx dt + R u 0 ψξ(0) dx (39)
(see [START_REF] Hvistendahl | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF] in Definition 5). Therefore, in the second step of the definition of A, (38) can be replaced by (39).

Theorem 19. The mapping A : E → E is continuous.

Proof. We split the proof into two parts corresponding to the above construction of A. The most delicate part of this proof is to show that the solution u of (1) depends continuously on h. Due to the finite speed of propagation property for the Burgers equation and to the boundedness of the set of relevant paths ĥ, we can assume that u 0 is an L 1 ∩ L ∞ function.

Step 1 Consider h ∈ E and u the associated solution of (1) with initial datum u 0 given by Theorem 11. Fix ε > 0. Given δ > 0, consider another path ĥ ∈ E such that h -ĥ E ≤ δ; let û be the associated solution of (1) (given by Theorem 11). Using the same kind of change of variables as in Sections 2.3 and 4.2, we set w(t, x) = u(t, x + h(t)), ŵ(t, x) = û(t, x + h(t)) and h(t) = ĥ(t) -h(t). Then

w 0 (x) := w(0, x) = u 0 (x + h 0 ), ŵ0 (x) := ŵ(0, x) = u 0 (x + h 0 ), (40)
|| h|| E = | h(t)| + |( h) ′ (t)| ≤ δ, (41)
and w and ŵ are Kruzhkov entropy solutions of

∂ t w + ∂ x w 2 2 -h ′ (t)w = 0 for t ∈ R + , x ∈ R * and ∂ t ŵ + ∂ x ŵ2 2 -h ′ (t) ŵ = 0 for t ∈ R + , x -h(t) ∈ R *
with interface coupling prescribed by the germs G λ (( ĥ) ′ (t)) -(∆, ∆) and G λ (( ĥ) ′ (t)), respectively, where ∆ = ( h) ′ (t). Notice that, unlike in Subsection 2.3, we use the same change of variable on the two equations; consequently, we have the same differential operator on the left, but the source terms are localized on two different interfaces.

Fix T > 0 and introduce the traces w(t, 0 ± ) of w on {x = 0}; in particular, for all η, α > 0, there exists r such that for all y ∈ (0, r) meas{t ∈ (0, T ) ; |w(t, ±y) -w(t, 0 ± )| > η/2} < α (42) because L 1 convergence implies convergence in measure. Further, we can approximate the measurable vector map t ∈ (0, T ) → w(t, 0 -), w(t, 0 + ) ∈ G λ (h ′ (t)) a.e. by a piecewise constant map as follows: for all η, α > 0, there exist N and (c

± n ) N n=1 such that meas{t ∈ (0, T ) ; |w(t, 0 ± ) -C ± N (t)| > η/2} < α (43)
where

C ± N (t) := N n=1 c ± n 1 1 (tn-1,tn] (t) and t n := nT /N ; since h ′ is continuous, we can pick (c - n , c + n ) ∈ G λ (h ′ (t n ))
; moreover, we can assume that C ± N satisfies the same L ∞ bound as w. Note that both r and N depend on u and thus on h, but they are independent of δ. Now, assuming that δ < 1/4, let us consider

ψ δ 0 ∈ D(- √ δ, √ δ) such that 0 ≤ ψ δ 0 ≤ 1, ψ δ 0 ≡ 1 in (-2δ, 2δ), ∂ x ψ δ 0 L ∞ ≤ 2 √ δ . We set ψ δ ∞ = 1 -ψ δ 0 (defined on R). In particular, supp(ψ δ ∞ ) ⊂ R \ (-2δ, 2δ
) and we can write

R |w -ŵ|(T, •) = R |w -ŵ|(T, •) ψ δ ∞ + R |w -ŵ|(T, •) ψ δ 0 =: I δ ∞ + I δ 0 . ( 44
) Since u, û ∈ L ∞ ((0, T ) × R), |I δ 0 | ≤ C √ δ (45)
with a constant C that only depends on T and on the L ∞ bounds on u, û. To estimate the first term of (44), notice that by (41) and by construction of ψ δ ∞ , w and ŵ satisfy the same conservation law without source in supp ψ δ ∞ . Thus we can deal with the first term of (44) by using the Kato inequality with the test function ψ δ ∞ (y):

I δ ∞ ≤ R |w 0 -ŵ0 | ψ δ ∞ + T 0 R Φ(h ′ (t); w, ŵ) ∂ x ψ δ ∞ = - T 0 R Φ(h ′ (t); w, ŵ) ∂ x ψ δ 0 (46)
(see (40)), where Φ is the Kruzhkov entropy flux Φ(h

′ (t); a, b) := (a+b)|a-b| 2 -h ′ (t)|a -b| associated with the hyperbolic operator ∂ t w + ∂ x ( w 2 2 -h ′ (t)w). Then we write Φ(h ′ (t); w, ŵ) = Φ(h ′ (t); ĈN , ŵ) + Φ(h ′ (t); C N , ŵ) -Φ(h ′ (t); ĈN , ŵ) + Φ(h ′ (t); w, ŵ) -Φ(h ′ (t); C N , ŵ) =: T 1 + T 2 + T 3 ,
(respectively) where

C N (t, x) = N n=1 c - n 1 1 R-(x) + c + n 1 1 R+ (x) 1 1 (tn-1,tn] (t) = C - N (t)1 1 R-(x) + C + N (t)1 1 R+ (x), ĈN (t, x) = N n=1 c - n 1 1 R-(x -h(t)) + c + n 1 1 R+ (x -h(t)) 1 1 (tn-1,tn] (t) = C - N (t)1 1 R-(x -h(t)) + C + N (t)1 1 R+ (x -h(t)) = C N (t, x -h(t)).
Recall that Φ is a locally Lipschitz continuous function and that w, C N , ĈN and ŵ enjoy a uniform L ∞ bound.

Because the functions C N and ĈN only differ on the set {(t, x) ; x is between 0 and h(t)} which measure is at most T δ, there exist A, C such that

T 0 R T 2 ∂ x ψ δ 0 ≤ A √ δ T 0 R |C N -ĈN | ≤ C √ δ. (47)
As far as T 3 is concerned, for η, α > 0, let us choose δ < √ r with r small enough as in (42), and N large enough as in (43). We obtain that there exist A 1 , A 2 , and C such that

T 0 R T 3 ∂ x ψ δ 0 ≤ A 1 √ δ √ δ - √ δ T 0 |w -C N | ≤ A 1 √ δ 2 √ δ(A 2 α + T η) ≤ C(α + η). (48)
Here, we have split, for y ∈ [-√ δ, √ δ], the integral on [0, T ] into an integral on the measurable set {t ∈ (0, T ) ; |w(t, y) -C N (t)| > η} and an integral on its complement in [0, T ]. We then have used (42) and the fact that w -C N is bounded.

Finally, to estimate the contribution of T 1 to the right-hand side of (46) we use the global entropy inequalities for ŵ in each interval [t n-1 , t n ] (notice that ĈN is constant on each side of the interface {y = h(t)}, as required in the global entropy inequalities

∂ t |w -κ| + ∂ x Φ(h ′ (t), w, κ) ≤ M dist 1 (κ L , κ R ), G λ (h ′ (t)) δ 0
which can be deduced from [START_REF] Isaacson | Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law[END_REF]. We have, for n = 1, . . . , N ,

R | ŵ(t n ) -ĉn | ψ δ 0 ≤ R | ŵ(t n-1 ) -ĉn | ψ δ 0 + tn tn-1 R Φ(h ′ (t); ĈN , ŵ) ∂ x ψ δ 0 + M tn tn-1 dist 1 (c - n , c + n ), G λ (( ĥ) ′ (t)) .
with ĉn (t, x)

:= c - n 1 1 R-(x -h(t)) + c + n 1 1 R+ (x -h(t)) defined for t ∈ [t n-1 , t n ]
. Summing up the above inequalities in n, we find

- T 0 R T 1 ∂ x ψ r 0 ≤ R | ŵ(t 0 )-ĉ 1 | ψ δ 0 - R | ŵ(T )-ĉ N | ψ δ 0 + N n=2 R |c n -c n-1 | ψ δ 0 +2M T δ+2M tn tn-1 |h ′ (t n )-h ′ (t)|, because (c - n , c + n ) ∈ G λ (h ′ (t n )) = G λ ( ĥ′ (t)) + (h ′ (t n ) -h ′ (t), h ′ (t n ) -h ′ (t)) -(( h) ′ (t), ( h) ′ (t)
). The right-hand side of the above expression is therefore estimated by

C(1 + TotVar C - N + TotVar C + N ) √ δ + Cδ + o N , (49)
where o N denotes a function of N that converges to 0 as N tends to +∞. This inequality comes from the fact that the rectangle method applied to a continuous function is convergent. We see that this last expression can be bounded by ε, by taking N large enough and then δ small enough.

In conclusion: given ε > 0,

• we first pick η and α so that C(η + α) in ( 48) is less than ε/4;

• we choose r small enough, δ < √ r and N large enough in order that (48) hold true; • actually we pick δ possibly smaller and N possibly greater, to ensure that quantity (49) is less than ε/4; • at last, δ small enough is chosen to ensure that C √ δ in both (45) and ( 47) is smaller than ε/4.

Therefore,for any given ε > 0 there exists δ > 0 such that w -ŵ L ∞ (0,T ;L 1 (R)) < ε as soon as h -ĥ E < δ.

It remains to notice that u(t, •) -û(t, •) L 1 (R) = w(t, •) -ŵ(t, •) L 1 (R) .
Since | h(t)| ≤ δ and u ∈ L 1 ((0, T ) × R) admits a finite modulus of continuity for translations, we conclude that û tends to u in L 1 (0, T ; L 1 (R)) as ĥ → h in E. This ends the first step of the proof.

Step 2

We use the reformulation (39) for (h, u) and ( ĥ, û). Due to the L ∞ estimate (13), we can choose ψ such that ψ ≡ 1 on ĥ([0, T ]) for all relevant ĥ belonging to a neighborhood of h. Using that û tends to u in L 1 (0, T ; L 1 (R)) (previous step) and having in mind the L ∞ estimate [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF], from (39) we see that ( ĥ * ) ′′ → (h * ) ′′ weakly in L 1 (0, T ) as ĥ → h in E; therefore ĥ * converges strongly to h * in E. Now we are in a position to prove the existence claim, i.e. Theorem 7. We consider the nonempty closed convex of C 1 ([0, T ])

Ẽ := h ∈ E : |h ′ (t)| ≤ |V 0 | + T m ( u 0 ∞ + λ) 2 exp 2t m ( u 0 ∞ + λ) ∀t ∈ [0, T ] .
Theorem 19 yields that A : Ẽ → E is continuous. Moreover, a standard calculation and the a priori L ∞ bound [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF] show that if h ∈ Ẽ, then h * = A(h) ∈ Ẽ and moreover,

h * W 2,∞ (0,T ) ≤ C = C(m, λ, T, |V 0 |, u 0 ∞ ).
Thus we can apply the Schauder fixed-point theorem in Ẽ and eventually get the claim of Theorem 7.

Appendix: BV estimate with wave-front tracking algorithm

The goal of this appendix is to prove Proposition 13. Unfortunately, the method of proof used in [START_REF] Andreianov | Well-posedness of a singular balance law[END_REF] makes it difficult to control carefully the increase of variation of the approximate solutions at the moment of interaction of incoming waves with the interface located at the particle position. Yet we have uniqueness of a solution, therefore we can establish the BV estimate with the help of any other convergent approximation method. Here, in order to prove the Proposition 13, we use the method of wave front tracking. More precisely, let us take M λ ∈ N * and let us set δ := λ/M λ . We choose M sufficiently large to ensure

0 ≤ u 0 ∞ + λ ≤ M δ,
We approximate u 0 in L 1 loc (R) by a piecewise constant function u δ 0 taking value in δZ, with u δ 0 ∞ ≤ u 0 ∞ and with TotVar u δ 0 converging to TotVar u 0 as δ → 0. Finally, we approximate the map u → u 2 /2 by a continuous piecewise affine function f δ , which breakpoints are kδ, k ∈ {-M, . . . , M }. We take f δ (kδ) = (kδ) 2 /2 (this yields an even and convex function f δ strictly increasing on R * + ). We also consider the discrete germ

G λ,δ = G λ ∩ (δZ) 2 .
Due to the assumption that λ/δ ∈ N * , every point of G λ is at a distance dist 1 of at most δ from the set G λ,δ (see Fig. 5; pay particular attention to the straight line G 1 λ that is a part of G λ .) Then we consider the following approximation of ( 16), [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] (the problem should be interpreted in the way analogous to Definition 2(i)):

∂ t u + ∂ x (f δ (u)) = 0 in R * , (50)
(u(t, 0 -), u(t, 0 + )) ∈ G λ,δ for almost every t ∈ R + , (51)
u(0) = u δ 0 . ( 52
) (0, 0) u - u + λ -λ G 1 λ G 2 λ G 3 λ Figure 1: Representation of the admissibility germ G λ = G 1 λ ∪ G 2 λ ∪ G 3 λ (with the notation of [3]).
The solution u δ is obtained by juxtaposition of piecewise constant solutions to Riemann problems originated from every point of discontinuity of u δ (0, x) = u δ 0 (x) and from every point of interaction of shocks already present in the solution. These problems are solved in a classical way if the interaction point lies away from the interface {x = 0}; and on the interface, one uses the Riemann solver entirely analogous to the solver described in [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF]. Because it is instrumental for the below proof of Proposition 13, we recall the form of this Riemann solver at the end of this section. The only difference between the Riemann solver for the nonlinearity f δ from the one of [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF], f (u) = u 2 /2, is that the rarefaction waves present in solutions of the cases (I), (II), (III), (V I) and (V II) are partitioned into a sequence of shocks of amplitude equal to δ. For instance, in case (I) of Theorem 23, if u L = -aδ, u R = bδ, with a, b ∈ N * , u -= u + = 0 and we insert at the left of the particle the states u

(a-1) L * = -(a -1)δ, . . . , u (1)
L * = -δ and the right of the particle the states u

(1) R * = δ, . . . , u (b-1) R * = (b -1)δ.
With the above construction, the solution u = u δ remains a step function with values in {-M δ, . . . , M δ} (in particular, notice that the technique of proof of Lemma 9 allows to prove the same L ∞ bounds for (50), (51), (52) as for problem [START_REF] Lions | Kinetic formulation of the isentropic gas dynamics and p-systems[END_REF], [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF]).

We need to check that this wave front tracking algorithm gives a global in time solution. To ensure this, we can show that there are only a finite number of interactions between discontinuities. In passing, we will show a uniform in δ bound on the total variation of these solutions. We consider the two functionals Σ(t) := 2N (t) + N a (t), and TotVar u δ (t, •), where N (t) is the number of fronts at time t and where N a (t) is the number of fronts approaching the particle. We call a front a discontinuity of u at some point x = 0 and we say it approaches the particle if this discontinuity is at the left of 0 with a positive velocity or at the right of 0 with a negative velocity. Remark that since the initial condition u 0 has a bounded total variation and since u δ 0 takes values in δZ, N (0) is finite. By convention, we assume that N (•) and N a (•) are defined except at the front collision times.

Let us start the analysis with some important remarks: with the above construction of f δ , we notice that a front separating the states u (1) , u (2) is moving with the velocity (u (1) + u (2) )/2. We deduce from this remark that any collision in the fluid of several fronts results in at most one front. In particular, in a collision between fronts, N is decreasing by at least one, and N a is increasing at most by one. Consequently, Σ is decreasing for any collision in the fluid. It is also simple to show that TotVar u(t, •) is non-increasing during the collision. It remains to analyze the evolution of Σ and TotVar u(t, •) for a collision between one or several fronts and the particle (at x = 0).

More precisely, let us consider an interaction that will occur at time t between the particle and fronts at the left of the particle separating, before t, the states u L =: u

Here above, it is assumed that the position of the discontinuity separating u is an increasing function of k. Note that since we consider the case of collision between at least one front and the particle, max{n, m} ≥ 2. After the collision, we are reduced to solve a Riemann problem with the states (u L , u R ).

Remark 20. After the initial time, for any front separating the states u (1) on its left and u (2) on its right, we have either u (1) > u (2) or u (1) = u (2) -δ. As a consequence, if u L < 0, then there are no fronts satisfying (53) (with m ≥ 2) and colliding the particle: we would have u L ≤ -δ and u

(2) * ≤ 0 which is impossible, because in this case the wave speed of the front separating u L and u

(2) * would be negative. Symmetrically, if u R > 0, then there are no fronts satisfying (54) with n ≥ 2 and colliding the particle.

Remark 21. Note that by the very definition of the solutions, (u -, u + ) ∈ G λ,δ .

At the instant of such a collision, one has to solve the Riemann problem with the states u L and u R . The solution being given by Theorem 23, let us analyze each of the eight cases (according to the values of u L , u R and λ) of this theorem. It is sufficient to consider only the evolution of Σ and of TotVar u in a neighborhood of the particle, which means that we can assume that the only fronts in the fluid are the fronts in (53) and (54) and the fronts produced after the collision of these fronts with the particle. For simplicity, we write N -, N - a , Σ -and T V -(respectively N + , N + a , Σ + and T V + ) for the value of N , N a , Σ and TotVar u at t -(respectively at t + ). Our aim is to show that for every case of Theorem 23 we have Σ + < Σ -and T V + ≤ T V -.

(55)

The above relation yields that there is only a finite number of collisions between fronts and the particle and that the total variation of each solution u δ is bounded uniformly with respect to δ.

1. Case (I): u L ≤ 0 and u R ≥ 0. In virtue of Remark 20, if u L < 0 then u L = u -< 0. In that case, (u -, u + ) ∈ G λ,δ implies u + = u --λ < 0. This last equality and Remark 20 yield that u R = 0. We deduce that

N -≥ -u L + λ δ , N - a ≥ 1, T V -≥ 2λ -u L .
From Theorem 23, we have

N + = -u L δ , N + a = 0, T V + = -u L .
The above relations imply (55). The case u L = 0, u R > 0 can be done in a similar way and we skip it. In the case u L = u R = 0, Theorem 23 yields N + = 0, N + a = 0, T V + = 0, and thus (55) holds.

2. Case (II): u L < 0 and u R < 0 and u R > -λ. From Remark 20, we deduce u L = u -< 0. In that case, (u -, u + ) ∈ G λ,δ implies u + = u L -λ = -λ < u R . We deduce that

N -≥ u R -u L + λ δ , N - a ≥ 1, T V -≥ 2λ + u R -u L .
From Theorem 23, one has

N + = -u L δ = 0, N + a = 0, T V + = -u L -u R = T V -.
The above relations imply (55).

3. Case (III): u L ≤ u R + λ < 0. This yields u L ≤ 0. If u L = 0, then u R = -λ. In that case,

N -≥ 1, N - a ≥ 1, T V -≥ u L -u R = -u R ,
while Theorem 23 gives N + = 0, N + a = 0, T V + = -u R . This yields (55) in this case.

( 1 )

 1 * , . . . , u (m) * := u -,(53)and fronts at the right of the particle separating, before t, the statesu + =: u (m+1) * , u(m+2)* , . . . , u (m+n) * := u R .

In[START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF], the trace existence result is stated for C 1 boundary, and it is pointed out that generalization to W 1,∞ boundary is straightforward. Also notice that throughout the paper, W 1,∞ can be replaced by the space of continuous piecewise C 1 functions: in this case, the result of[START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] can be used piecewise on the interface {x = h(t)}.

Acknowledgments The work of B.A. was partially supported by ANR Grant CoToCoLa. The work of F.L., N.S. and T.T. was partially supported by ANR Grant CISIFS (09-BLAN-0213-02). The work of F.L. was partially supported by ANR Grants INTOCS (08-JCJC-0132-01)

If u L < 0, then u L = u -and u + = u L -λ. In that case,

while Theorem 23 gives

This yields (55) in this case. Note that it is possible to have N -= N + and T V -= T V + . In this case, the presence of N a in Σ is essential in our analysis. [START_REF] Borsche | Mixed systems: ODEs -balance laws[END_REF]. Case (IV ): u R < u L -λ and u R < -u L -λ. In that case,

while Theorem 23 gives

This yields (55) in this case. As in the previous case, it is possible to have N -= N + and T V -= T V + and we need N a to obtain the inequality of Σ in (55).

5. Case (V ): u L ≥ 0 and u R ≤ 0 and u R ≥ -u L -λ and u R ≤ -u L + λ. In that case,

and Theorem 23 gives

The remaining cases are obtained in a symmetrical way.

Proof. (of Prop. 13) In the above construction, we obtained a globally defined sequence u δ δ∈R * + of solutions to (50)-(52) such that TotVar u δ (t, •) is non-increasing for t ∈ (0, T ]. Moreover, during the initial interaction in the wave-front tracking algorithm, the variation may increase by at most 2 dist 1 (u δ 0 (0 -),

Notice that the space-time compactness in the sense of pointwise a.e. convergence is ensured by the uniform BV bound in space: along with the uniform L ∞ bound and equation (50), it ensures a BV bound in time. By the lower semi-continuity of the total variation, there exists a limit u of (a subsequence of) (u δ ) δ that verifies the variation bound (18) of Proposition 13. Further, the functions u δ can be seen as approximate solutions of ( 16), [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF]. Writing the entropy formulation of the kind (9), we can pass to the limit in the entropy formulation and inherit property [START_REF] Isaacson | Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law[END_REF] for the accumulation point u of our sequence, as δ → 0.

In conclusion, we reproduce the result of [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF] used here above. We need only the particular result corresponding to the case when the velocity of the particle is 0. Notice that the solutions of the Riemann problem are entropy solutions in the sense of Definition 2 (or Definition 5).

Theorem 22. Consider the Riemann problem for [START_REF] Lions | Kinetic formulation of the isentropic gas dynamics and p-systems[END_REF]. For every pair (u L , u R ) ∈ R 2 , the solution with the Riemann datum u 0 (x) = u L 1 1 R -(x) + u R 1 1 R+ (x) is given by the formula u(t, x) = U(x/t) with U described by the expressions below, where Id is the identity map.

6. If u L > 0 and u L ≤ λ and u R > 0,

It is not difficult to see that for the discretized problem

we have a similar result that is stated below. Let us note that a crucial ingredient for this theorem is that λ/δ ∈ N * .

Theorem 23. Consider the Riemann problem for (56) ∂ t u + ∂ x f δ (u) = -λuδ 0 (x). For every pair (u L , u R ) ∈ (δZ) 2 , the solution with the Riemann datum u 0 (x) = u L 1 1 R -(x) + u R 1 1 R+ (x) is given by the formula u(t, x) = U(x/t) with U described by the same expressions as in Theorem 22 with Id replaced by the function I δ which is the discretized identity function: I δ (ξ) = k∈Z kδ1 1 [(k-1/2)δ,(k+1/2)δ) (ξ).