On the Riemann problem for the simplified Bouchut-Boyaval system

In this paper we study the one-dimensional Riemann problem for a new hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered $3 \times 3$ system is such that every characteristic field is linearly degenerate. Then, in despite of the fact that it is of Temple class, the analysis of the Cauchy problem is more involved since general results for such a systems are not yet available. We show a explicit solution for the Cauchy problem with initial data in $L^\infty$. We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.


Introduction
The modeling of viscoelastic materials and fluids it is important for many applications. In particular, a viscoelastic fluid is a material that exhibit both viscous and elastic characteristics upon deformation. Examples of viscoelastic fluids that are important for applications are: latex paint, gelatin, unset cement, liquid acrylic, asphalt and biological fluids such as synovial fluids, among others. In [4] the authors introduced a new system of conservation laws that models shallow viscoelastic fluids. This new system is motivated by Bouchut and Boyaval in [4, eq(5.6)] and is written as                ρ t + (ρu) x = 0, (ρu) t + (ρu 2 + π) x = 0, (ρ π s 2 ) t + (ρu π s 2 + u) x = 0, s t + us x = 0, c t + uc x = 0, where ρ denotes the layer depth of fluid, u is the horizontal velocity, s is related to the stress tensor and it is a conserved quantity, π is the relaxed pressure and c > 0 is introduced in order to parametrize the speeds. This system describes a simple model for a thin layer of non-Newtonian viscoelastic fluid over a given topography at the bottom when the movement is driven by gravitational forces such as geophysical flows (mud flows, landslides, debris avalanches).
In [11], since s is a conserved quantity, the author considers the case s = const. > 0. Additionally, we observe that the field c does not appear in the first four equations and, in order to simply even further, introduce the new variable v = π s 2 . After this observations, it is obtained the following simplified viscoelastic shallow fluid model, We refer to the system above as the simplified Bouchut-Boyaval system (sBB) and it was introduced by Lu in [11] as simplified version of a model proposed by Bouchut and Boyaval. We note that this system is of Temple class and therefore, it is also it is of Rich type. We note that sBB is of Rich type but it is not diagonal, so its analysis is not standard. In this paper we are concern with the Riemann and Cauchy problems for (1). The existence of global weak solutions including vacuum regions, was obtained in [11] using the vanishing viscosity method in conjunction with the compensated compactness argument. We also mention that, when dealing with the system (1), one of the main difficulties is to obtain existence and uniqueness of solutions of Cauchy problems in the presence of vacuum regions, that is, regions where the layer deep ρ = 0. We also note that, there are numerous studies on existence and uniqueness for general Temple class system [1,2,3,6,8,13]. However, some of these results do not apply to (1) since it has all fields being linearly degenerate and the initial data may have oscillations. In this paper we obtain explicit solutions for a Cauchy problem associated to the sBB system (1) with ρ 0 (x) ≥ ρ > 0 with a possibly oscillating initial data. Also, we construct the Riemann solution for the system focussing our attention on delta shock waves of certain type. The existence and uniqueness of solutions involving delta shock waves can be obtained by solving the generalized Rankine-Hugoniot relation under a entropy condition [7,9]. The paper is organized as follows, in Section 2 we present the problem and put conditions on the initial data for physical properties are maintained in ρ. In Section 3, we show the explicit solutions for the Cauchy problem associated with the sBB system (1) without the presence of vacuum regions. In Section 4, we solve the Riemann problem and we observe that the first and third contact discontinuity are asymptotic to the vacuum. In the last section, we study the existence and uniqueness of solutions delta shock waves type.
2 Properties of the simplified Bouchut-Boyaval system and some assumptions The eigenvalues associated to the system (1) are given by, where the corresponding Riemann invariants are From the expressions for the eigenvalues and the Riemann invariants we obtain From here we can see that system (1) This means that system (1) is of Rich type. We recall that this classifications if due to [14].
In this manuscript we focus on the study of the LBB system of conservation laws (1) with bounded initial data subject to the following conditions: H1: The functions ρ 0 , u 0 and v 0 satisfy where c i , i = 1, . . . , 5, are suitable constants satisfying H2: The total variations of u 0 (x) − sv 0 (x) and u 0 (x) + sv 0 (x) are bounded.
The conditions H1 and H2 are somehow natural to impose since they ensure that ρ is positive giving a physical meaning to the sBB system (1).
As we mentioned before, we note that in [11], the author shows existence of solutions for the Cauchy problem (1)-(5) for the case ρ 0 (x) ≥ 0. This is done using the vanishing viscosity method and a compensated complicatedness argument. In [11] it is also shown that all entropies associated to (1) are of the form, where F, G, H are arbitrary functions having entropy flux Moreover, if the functions F, G y H are convex, then, the entropy is also convex (see [11,Theorem 2]). Thus, from each convex pair (η, q) we have the following condition η t (ρ, u, v) + q x (ρ, u, v) = 0 (10) in the sense of distributions.

Explicit solutions
In this section we obtain explicit solutions for the Cauchy problem associated to sBB system with initial data (5) subject to H1-H2 conditions. For this purpose, we used the results given Wagner, Weinan and Kohn, Li, Peng and Ruiz (for example see [15,16,10,12]). We uses the Euler-Lagrange (E-L) transformation (t, x) → (t, y) = (t, Y (t, x)) defined by In Lagrangian coordinates, the system (1) becomes where ω denotes the quantity 1 ρ in Lagrangian coordinates, that is, ω(t, y) = 1 ρ(t,x) , and we also have ν(t, y) = u(t, x) and κ(t, y) = v(t, x).
The eigenvalues associated to (11) are given by and the the corresponding Riemann invariants are given by In Lagrangian coordinates the entropy condition (10) transforms into for each η with where F, G, H are (arbitrary) convex functions.
The initial conditions (5) becomes Due to the fact that system (11) is linear, the explicit solution of the corresponding Cauchy problem (11)- (15) is Moreover, by condition H1 we obtain that and since ρ 0 (x) ≥ ρ = const. > 0 by (5), we have that ω(t, y) ≥ ω > 0, ensuring that the function y → X(t, y) is invertible and bi-Lipschitzian from R to R for all t ≥ 0, and by [15,12], we also have uniqueness of the entropy solution of (1)- (5) if and only if we have uniqueness of the entropy solution of (11)- (15).
Therefore, we consider X 0 = Y −1 0 . Then, the unique function x = X(t, y) that satisfy X(0, y) = X 0 (y) is given by From the above, we obtain the following Theorem.
Then, the Cauchy problem (1)-(5) has an unique global solution (ρ, u, v) ∈ L ∞ (R + × R) that satisfy the entropy condition (10) for all pair (η, q) defined in (8)- (9). Moreover, this solution is given by and Now we apply our result to particular example, which behaves as the advection equation.
Example. We consider the initial data u 0 (x) = u and v 0 (x) = v as being constant functions, and ρ 0 (x) ≥ 0 as a bounded function. By the E-L transformation, Then, In this way the solution of the Cauchy problem is given by

Riemann problem
In this section we study the solution for the Riemann problem associated with the sBB system, in which the left and right constant states (ρ l , u l , v l ) and (ρ r , u r , v r ), respectively, satisfy the conditions H1-H2 and Consider the Riemann problem of the system (1) with initial data where (ρ 0 , u 0 , v 0 )(x) = (ρ, u, v)(0, x) satisfies the conditions H1 and H2. First, observe that system (1) is equivalent to with s = const. > 0, where m = ρu, n = ρv, and the initial data (19) is given by with m r = ρ r u r , n r = ρ r v r , m l = ρ l u l and n l = ρ l v l . The eigenvalues of the system (1), in the variables ρ, m, n, are given by the right eigenvectors become From (23) we get that the 1-rarefaction curve can be found as, That is, Therefore, the integral curves of the vector field r 1 are given by straight lines in the direction of the vector r 1 (ρ 0 , m 0 , n 0 ) and goes trough the point (0, s, −1), that is, Analogously, from (24) we can analyze the 2-rarefaction curve. In this case the integral curves corresponding to the vector field r 2 are given by straight lines going through the origin in the direction of the vector r 2 (ρ 0 , m 0 , n 0 ). Also, from (25) we can see that for the 3-rarefaction curve, the integral curves of the vector field r 3 , are give by straight lines trough (0, −s, −1) that are parallel to r 3 (ρ 0 , m 0 , n 0 ). Thus, the i-rarefaction curve R i (σ)(ρ, m, n) satisfy This also may be deduced from self-similar solution for which system (1) becomes and initial data (19) changes to the boundary condition This is a two-point boundary value problem of first-order ordinary differential equations with the boundary values in the infinity. For smooth solution, (29) is reduced to  It provides either the general solutions (constant states) or singular solutions Integrating (33) from (ρ l , u l , v l ) to (ρ, u, v), one can get that Oberve that (34) in the variables ρ, m, n is equivalent to (27). For a bounded discontinuity at ξ = ω, the Rankine-Hugoniot conditions hold. That is, where [q] = q l − q is the jump of q across the discontinuous line and ω is the velocity of the discontinuity. From (35), we have From (34) and (36), we conclude that the rarefaction waves and the shock waves are coincident, which correspond to contact discontinuities. Namely, for a given left state (ρ l , u l , v l ), the contact discontinuity curves, which are the sets of states that can be connected on the right by a 1-contact discontinuity J 1 , a 2-contact discontinuity J 2 or a 3-contact discontinuity J 3 , are as follows: In the space (ρ > 0, u ∈ R, u ∈ R), through the point (ρ l , u l , v l ), we draw curves (37) which are denoted by J 1 , J 2 and J 3 respectively. So, J 1 has asymptotes ρ = 0 and (ρ, u l −s/ρ l , v l +1/ρ l ) for ρ ≥ 0, and J 3 has asymptotes ρ = 0 and (ρ, u l + s/ρ l , v l + 1/ρ l ).
Lema 4.1. Given left and right constant states (ρ l , u l , v l ) and (ρ r , u r , v r ), respectively, such that they satisfy conditions H1,H2 and λ 1 (ρ l , u l , v l ) < λ 3 (ρ r , u r , v r ). Then, the following relation is satisfied The results of this section can be summarized in the following Theorem.
Then, there is a unique global solution to the Riemann problem (1)-(19). Moreover, this solution is given by

Delta shock solution
In this section, we discuss the solution for the Riemann problem associated with the sBB system, in which the left and right constant states (ρ l , u l , v l ) and (ρ r , u r , v r ), respectively, satisfy the conditions H1 and H2, but unlike previous section they satisfy λ 1 (ρ l , u l , v l ) ≥ λ 3 (ρ r , u r , v r ).
Denote by BM(R) the space of bounded Borel measures on R, and then the definition of a measure solution of sBB system in BM(R) can be given as follows.
Definition 5.1. A triple (ρ, u, v) constitutes a measure solution to the sBB system, if it holds that and v are measurable with respect to ρ at almost for all t ∈ (0, ∞), for all test function φ ∈ C ∞ 0 (R + × R).
and satisfies Definition 5.1, where (ρ l , u l , v l )(t, x) and (ρ r , u r , v r )(t, x) are piecewise smooth bounded solutions of the sBB system (1).
We set dx dt = u δ (t) since the concentration in ρ need to travel at the speed of discontinuity. Hence, we say that a delta shock wave (55) is a measure solution to the sBB system (1) if and only if the following relation holds, In fact, for any test function φ ∈ C ∞ 0 (R + × R), from (53), we obtain dt dt, and Relations (56) is called the generalized Rankine-Hugoniot relation. It reflects the exact relationship among the limit states on two sides of the discontinuity, the weight, propagation speed and the location of the discontinuity. In addition, to guarantee uniqueness, the delta shock wave should satisfy the admissibility (entropy) condition Now, the generalized Rankine-Hugoniot relation is applied to the Riemann problem (1)-(19) with left and right constant states U − = (ρ − , u − , v − ) and U + = (ρ + , u + , v + ), respectively, satisfying the conditions H1 and H2, the fact λ 3 (ρ + , u + , v + ) ≤ λ 1 (ρ − , u − , v − ) and Thereby, the Riemann problem is reduced to solving (56) with initial data under entropy condition From (56) and (59), it follows that Multiplying the first equation in (61) by u δ (t) and then subtracting it from the second one, we obtain that that is, d dt which is equivalent to From (64), one can find u δ (t) := u δ is a constant and x(t) = u δ t. Then, (64) can be rewritten When [ρ] = ρ − − ρ + = 0,the situation is very simple and one can easily calculate the solution which obviously satisfies the entropy condition (60), since by condition (58), and Similarly we can deduce that When [ρ] = ρ − − ρ + = 0, the discriminant of the quadratic equation (65) is and then we can find t, t. (69) Next, with the help of the entropy condition (60), we will choose the admissible solution from (68) and (69). Observe that by the entropy condition and since the system is strictly hyperbolic, we have that Observe that, then, for the solution given in (68), we have showing that the solution (69) does not satisfy the entropy condition (60). Thus we have proved the following result.