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Advances in Mathematical Physics
Volume 2014 (2014), Article ID 354349, 11 pages
Delta Shock Wave for the Suliciu Relaxation System
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.
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.
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  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  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 , 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  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 .
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 , 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  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 .
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
Now, we can do a transformation for . If solve
Also is a weak solution of
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).
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