Advances in Mathematical Physics

Volume 2014 (2014), Article ID 354349, 11 pages

http://dx.doi.org/10.1155/2014/354349

Research Article

Delta Shock Wave for the Suliciu Relaxation System

Richard De la cruz,1,2 Juan Galvis,² Juan Carlos Juajibioy,² and Leonardo Rendón²

¹School of Mathematics and Statistics, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia

²Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia

Received 13 March 2014; Accepted 27 May 2014; Published 18 June 2014

Academic Editor: Manuel De León

Copyright © 2014 Richard De la cruz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered system is such that every characteristic field is linearly degenerate. We show an explicit solution for the Cauchy problem with initial data in . 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.

1. Introduction

The modeling of viscoelastic materials and fluids is important for many applications. In particular, a viscoelastic fluid is a material that exhibits 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 [1] the authors introduced a new system of conservation laws that models shallow viscoelastic fluids. This new system is motivated by Bouchut and Boyaval in the equation of [6]

where denotes the layer depth of fluid, is the horizontal velocity, is related to the stress tensor and it is a conserved quantity, is the relaxed pressure, and 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, and debris avalanches).

In [2], since is a conserved quantity, the author considers the case . Additionally, we observe that the field does not appear in the first four equations and, in order to simply notation, introduce the new variable . After this observation, the following simplified viscoelastic shallow fluid model is obtained:

We refer to the system above as the Suliciu relaxation system [3–5]. This system can be considered as a relaxation for the isentropic Chaplygin gas dynamics system

where and , respectively, stand for the density and the velocity of the gas, while the pressure is given by the state equation with .

We note that the Suliciu relaxation system (2) is of Temple class and, therefore, it is also of Rich type but it is not diagonal, so its analysis is not standard. We also mention that, when dealing with the system (2), 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 . The existence of global weak solutions, including vacuum regions, was obtained in [2] using the vanishing viscosity method in conjunction with the compensated compactness argument. We also note that there are numerous studies on existence and uniqueness for general Rich type and Temple class system [6–13]. However, some of these results do not apply to (2) since it has all fields being linearly degenerate and the initial data may have oscillations.

The Riemann problem for the Suliciu relaxation system has been extensively studied, for instance in [3, 14]. We think that there are other cases to investigate. Now, we propose delta wave solutions type for the Suliciu relaxation system.

In this paper we obtain explicit solutions for a Cauchy problem associated with the Suliciu relaxation system (2) with , a.e. , with a possibly oscillating initial data. Also, we construct the Riemann solution for the system focusing 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 an entropy condition [15, 16].

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 . In Section 3, we show the explicit solutions for the Cauchy problem associated with the Suliciu relaxation system (2) 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 Suliciu Relaxation System and Some Assumptions

The eigenvalues associated with the system (2) 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 (2) is linearly degenerate. On the other hand, we have that, for each with , , it holds that

This means that system (2) is of Rich type. We recall that this classifications is due to [17].

In this paper we focus on the study of the Suliciu relaxation system of conservation laws (2) with bounded initial data

subject to the following conditions.

(H1) The functions , , and satisfy

where , , are suitable constants satisfying

(H2) The total variations of and are bounded.

The conditions (H1) and (H2) are somehow natural to impose since they ensure that is positive giving a physical meaning to the Suliciu relaxation system (2).

As we mentioned before, we note that, in [2], the author shows existence of solutions for the Cauchy problem (2)–(8) for the case . This is done using the vanishing viscosity method and a compensated compactness argument. In [2] it is also shown that all entropies associated with (2) are of the form

where , and are arbitrary functions having entropy flux

Moreover, if the functions and are convex, then the entropy is also convex (see [2, Theorem 2)]. Thus, from each convex pair we have the following condition:

in the sense of distributions.

3. Explicit Solutions

In this section we obtain explicit solutions for the Cauchy problem associated with Suliciu relaxation system with initial data (8) subject to (H1)-(H2) conditions. For this purpose, we used the results given by Wagner, Weinan and Kohn, Li, Peng and Ruiz (for example see [12, 18–20]).

We use the Euler-Lagrange (E-L) transformation defined by

In Lagrangian coordinates, the system (2) becomes

where denotes the quantity in Lagrangian coordinates; that is, , and we also have and .

The eigenvalues associated with (15) are given by

and the corresponding Riemann invariants are given by

In Lagrangian coordinates the entropy condition (13) transforms into

for each with

where , and are (arbitrary) convex functions.

The initial condition (8) becomes

Due to the fact that system (15) is linear, the explicit solution of the corresponding Cauchy problem (15)–(20) is

Moreover, by condition (H1) we obtain that

and since by (8), we have that , ensuring that the function is invertible and bi-Lipschitzian from to , for all , and by [12, 18], we also have uniqueness of the entropy solution of (2)–(8) if and only if we have uniqueness of the entropy solution of (15)–(20).

Therefore, we consider . Then, the unique function that satisfies is given by

Now, we show the result based on the ideas proposed by Weinan and Kohn [19].

Lemma 1. Let be two weak solutions of Suliciu relation system (2). Let be such that , conditions (H1), (H2), and

are satisfied. Suppose that, for any function with compact support,

Then , , and almost everywhere.

Proof. Consider a solution of

A solution for (26) exists because the consistency condition is the first equation of (2). Let

then is unique and by (H1)-(H2) conditions the map is a bi-Lipschitz homeomorphism of onto itself. Observe that

is a weak solution of

Let be a continuous function with compact support. Then

thereby

Similarly,

Now, we can do a transformation for . If solve

and choosing

Also is a weak solution of

with

Since and have the same initial data, and from the dummy variables and , and solve the same linear hyperbolic equation with the same initial condition. Therefore, , , and for all .

To conclude that , , and we must show that .

The Jacobian of the map is the inverse of that of the map , so

A similar relation holds for . Since , , and , we see that , and it follows that .

From the above, we obtain the following theorem.

Theorem 2. Assume that with , a.e. , the conditions (H1), (H2) hold and

Then, the Cauchy problems (2)–(8) has a unique global solution that satisfies the entropy condition (13) for all pair defined in (11)-(12). Moreover, this solution is given by

where

Now we apply our result to particular example, which behaves as the advection equation.

Example 3. We consider the initial data and as being constant functions, and , a.e. , as a bounded function.

By the E-L transformation,

Then,

In this way the solution of the Cauchy problem is given by

4. Riemann Problem

In this section we study the solution for the Riemann problem associated with the Suliciu relaxation system, in which the left and right constant states and , respectively, satisfy the conditions (H1)-(H2) and .

Consider the Riemann problem of the system (2) with initial data

where satisfies the conditions (H1) and (H2).

First, observe that system (2) is equivalent to

with , where , , and the initial data (44) are given by

with , , , and .

The eigenvalues of the system (2), in the variables , are given by

and the right eigenvectors become

From (48) we get that the 1-rarefaction curve can be found as

That is,

Therefore, the integral curves of the vector field are given by straight lines in the direction of the vector and goes through the point ; that is,

Analogously, from (49) we can analyze the 2-rarefaction curve. In this case the integral curves corresponding to the vector field are given by straight lines going through the origin in the direction of the vector . Also, from (50) we can see that, for the 3-rarefaction curve, the integral curves of the vector field , are given by straight lines trough that are parallel to .

Thus, the -rarefaction curve satisfies

This also may be deduced from self-similar solution

for which system (2) becomes

and initial data (44) 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, (56) is reduced to

It provides either the general solutions (constant states)

or singular solutions

Integrating (60) from to , one can get that

Observe that (61) in the variables is equivalent to (54).

For a bounded discontinuity at , the Rankine-Hugoniot conditions hold. That is,

where is the jump of across the discontinuous line and is the velocity of the discontinuity. From (62), we have

From (61) and (63), we conclude that the rarefaction waves and the shock waves are coincident, which correspond to contact discontinuities. Namely, for a given left state , the contact discontinuity curves, which are the sets of states that can be connected on the right by a 1-contact discontinuity , a 2-contact discontinuity , or a 3-contact discontinuity , are as follows:

In the space , through the point , we draw curves (64) which are denoted by , , and , respectively. So, has asymptotes and for , and has asymptotes and .

In order to solve the Riemann problem (2)–(44), we consider constant left and right states and , respectively, such that the conditions (H1)-(H2) are satisfied and . Then there exists intermediate states, and , such that , , and , for some , , and   .

Furthermore, because of (54), the states and    should satisfy

From (65d), (65b), and (65f), we have

On the other hand, from (65c), (65a), and (65e), we have

From (66) and (67) we conclude that

Thus, (65a), (65b), (65c), (65d), (65c), (65e), (65f), and (68) give

Observe that, by conditions (H1) and (H2) on and , we have that and also satisfy (H1) and (H2). This guarantees that and are positive.

Note that implies

Remark 4. Observe that if , then

If we assume that and , then (71) reduces to . In this case the Riemann problem does not have a solution since the lines and are parallel.

Thus, by conditions (H1), (H2), and , the solution of the Riemann problem is given by

Additionally, as usual, since the system is linearly degenerate, , , and .

Lemma 5. Given left and right constant states and , respectively, such that they satisfy conditions (H1), (H2), and then, the following relation is satisfied

The results of this section can be summarized in the following theorem.

Theorem 6. Given left and right constant states and , respectively, such that they satisfy conditions (H1), (H2), and , then, there is a unique global solution to the Riemann problem (2)–(44). Moreover, this solution is given by

where

5. Delta Shock Solution

In this section, we discuss the solution for the Riemann problem associated with the Suliciu relaxation system, in which the left and right constant states and , respectively, satisfy the conditions (H1) and (H2), but unlike previous section they satisfy .

Denote by the space of bounded Borel measures on , and then the definition of a measure solution of Suliciu relaxation system in can be given as follows.

Definition 7. A triple constitutes a measure solution to the Suliciu relaxation system, if it holds that

(a),(b),(c), ,(d) and are measurable with respect to at almost for all ,

and

for all test function .

Definition 8. A two-dimensional weighted delta function supported on a smooth curve parameterized as , is defined by

for all .

Definition 9. A triple distribution is called a delta shock wave, if it is represented in the form

and satisfies Definition 7, where and are piecewise smooth bounded solutions of the Suliciu relaxation system (2).

We set since the concentration in needs to travel at the speed of discontinuity. Hence, we say that a delta shock wave (78) is a measure solution to the Suliciu relaxation system (2) if and only if the following relation holds:

In fact, for any test function , from (76), we obtain

Relation (79) 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 (2)–(44) with left and right constant states and , respectively, satisfying the conditions (H1) and (H2), the fact , and

Thereby, the Riemann problem is reduced to solving (79) with initial data

under entropy condition

From (79) and (83), it follows that

Multiplying the first equation in (85) by and then subtracting it from the second one, we obtain that

that is,

which is equivalent to

From (88), one can find that is a constant and . Then, (88) can be rewritten

When , the situation is very simple and one can easily calculate the solution

which obviously satisfies the entropy condition (84), since by condition (82),

Similarly we can deduce that

because

When , the discriminant of the quadratic equation (89) is

and then we can find

or

Next, with the help of the entropy condition (84), we will choose the admissible solution from (95) and (96). Observe that, by the entropy condition and since the system is strictly hyperbolic, we have that

Observe that

then for the solution given in (95), we have

which imply that the entropy condition (84) is valid. When , we have trivially that .

Now, for the solution (96), when we have

and when , that

showing that the solution (96) does not satisfy the entropy condition (84).

Thus we have proved the following result.

Theorem 10. Given left and right constant states and , respectively, such that satisfy the conditions (H1), (H2), and (82), that is,

Then, the Riemann problem (2)–(44) admits a unique entropy solution in the sense of measures. This solution is of the form

where , , and are shown in (90) for or (95) for .

The above result includes the cases

or

that according to our knowledge is the best case of the delta shock waves (see (68)).

We began by analyzing the case in (104). Assume left and right constant states and , respectively, such that satisfy the conditions (H1) and (H2), , and (104).

It is easy to see that if , then the inequality (82) is trivially satisfied. Suppose that . Then, , and so

meaning that (82) is satisfied. The analysis is similar for (105).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor Yunguang Lu for suggesting this problem, pointing out the simplification of the system proposed by Bouchut and Boyaval, and also communicating to us the paper [2]. Also, They are grateful to Professor Philippe G. LeFloch for bringing to our attention the references [4, 21, 22].

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