Advances in Mathematical Physics
Volume 2018 (2018), Article ID 4925957, 11 pages
https://doi.org/10.1155/2018/4925957

Research Article

The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System

Xinli Han¹ and Lijun Pan²

¹School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
²Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Correspondence should be addressed to Lijun Pan; 98010149@163.com

Received 30 June 2018; Accepted 26 August 2018; Published 5 September 2018

Academic Editor: Carlo Bianca

Copyright © 2018 Xinli Han and Lijun Pan. 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.

In this paper, we study the perturbed Riemann problem with delta shock for a hyperbolic system. The problem is different from the previous perturbed Riemann problems which have no delta shock. The solutions to the problem are obtained constructively. From the solutions, we see that a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity under a perturbation of the Riemann initial data. This shows the instability and the internal mechanism of a delta shock. Furthermore, we find that the Riemann solution of the hyperbolic system is instable under this perturbation, which is also quite different from the previous perturbed Riemann problems.

1. Introduction

In this paper, we are concerned with the following nonstrictly hyperbolic system of conservation laws in the form

Equations (1) can be derived from a two-dimensional hyperbolic system of conservation laws

The above system is the mathematical simplification of Euler equations of gas dynamics. In 1991 and 1994, Yang and Zhang [1] and Tan and Zhang [2] obtained both numerical and analytical solutions to the Riemann problem of system (2). The form of Dirac delta functions supported on shocks was found necessary and was used as parts in the Riemann solutions. We call it a delta shock. A delta shock is the generalization of an ordinary shock. It is more compressive than an ordinary shock in the sense that more characteristics enter the discontinuity line. Mathematically, the delta shocks are new type singular solutions such that their components contain delta functions and their derivatives. Physically, they are interpreted as the process of formation of the galaxies in the universe, or the process of concentration of particles [3].

To investigate the validity of delta shock, in 1994, Tan et al. [4] considered the Riemann problem for the one-dimensional model (1). They found that there exist delta shock as the limit of vanishing viscosity for system (1). As for delta shock, there are numerous excellent papers. We refer readers to [3–14] and the references cited therein. There are still many open and complicated problems in the delta shock theories. Study of this area gives a new perspective in the theory of conservation law systems.

In this paper, we are interested in the internal mechanism and instability of a delta shock. For this purpose, we study system (1) with the following initial data:

where and are all bounded functions with the following property:

Here and are constants with . The initial value (3) is a perturbation of Riemann initial value (5) at the neighborhood of the origin in the plane. The perturbation on the Riemann initial data is reasonable. For example, error is unavoidable in computation and the error forms a perturbation of the initial data.

We divide our work into two parts according to the presence of delta shock or not. When the delta shock is not involved, the perturbed Riemann problem (1) and (3) is classical. More importantly, there is no delta shock in the corresponding Riemann solutions for the previous work on the perturbed Riemann problem. Therefore, we only pay attention to the perturbed Riemann problem (1) and (3) when the delta shock is involved. To overcome the difficulty caused by delta shock, we adopt the method of characteristic analysis and the local existence and uniqueness theorem proposed by Li Ta-tsien and Yu Wen-ci [15]. We construct the solution to the perturbed problem (1) and (3) locally in time.

Our result shows that a perturbation of initial data may bring essential change when a delta shock appears in the corresponding Riemann solution. A delta shock may turn into a shock and a contact discontinuity. This shows the instability of the delta shock, which allows us to better investigate the internal mechanism of a delta shock. Furthermore, the previous works [15, 16] about the perturbed Riemann problem pay more attention to the stability of the corresponding Riemann solution. In other words, the Riemann solution has a local structure stability with respect to the perturbation of Riemann initial data. A distinctive feature for this paper is that, for some initial data (3), the preceding local structure stability fails. We pay more attention to the differences between Riemann solution and perturbed Riemann solution.

The paper is organized as follows. In Section 2, we present some preliminary knowledge about the hyperbolic system (1). Then, the construction and proof of the solution to the perturbed Riemann problem (1) and (3) with delta shock are presented in Section 3.

2. Preliminaries

In this section, we recall the main properties of system (1) with Riemann initial data

where and are constants with (see [4, 17, 18] for a more detailed study of the model).

The eigenvalues of the hyperbolic system (1) are

with the corresponding left eigenvectors,

and the corresponding right eigenvectors,

By a direct calculation,

Therefore, is always linearly degenerate; is genuinely nonlinear if and linearly degenerate if .

The Riemann invariants of system (1) along the characteristic fields are

Definition 1 (see [3, 19]). A pair of is called a generalized delta shock solution to (1) with the initial data (3) on local time , if there exists a smooth curve and a weight such that and are represented in the following form:

in which is the delta function, , and satisfy

for all the test functions . Here is the tangential derivative of the curve , and stands for the tangential derivative of the function on the curve .

Definition 2. For an matrix , define

and

3. The Perturbed Riemann Problem with Delta Shock

In this section, we construct the perturbed Riemann solutions of hyperbolic system (1) with initial data (3) for local time and investigate the internal mechanism and instability of delta shock. About the perturbed Riemann problem, we have six cases according to the different constructions of the solutions to the corresponding Riemann problem (1) and (5) as follows:

(1)When , the Riemann solution is (2)When , the Riemann solution is delta shock (3)When , the Riemann solution is (4)When , the Riemann solution is (5)When , the Riemann solution is (6)When , the Riemann solution is

Here “+” means “followed by”; the capitals , , and denote shock, contact discontinuity, and rarefaction wave, respectively.

Since the perturbed Riemann problem (1) and (3) with no delta shock is classical and well known, it will not be pursued here. In this section, we mainly study the differences between the perturbed Riemann solution and the corresponding Riemann solution. Thus we only consider the perturbed Riemann problem (1) and (3) with delta shock, a.e., .

It is known from classical theory that the classical solution and can be defined in a strip domains and for local time, respectively (see Figure 1). Here and are local smooth solutions to the initial problem (1) with corresponding initial data and on both sides of , respectively. The right boundary of domain is a characteristic ; namely,

The left boundary of domain is characteristic ; namely,

Figure 1: The local time.

Now we turn our attention to the solution of (1) and (3) between the right boundary of domain and the left boundary of domain . We note that the corresponding Riemann solution is a delta shock with speed separating two states and (see Figure 2(a)). If the perturbed Riemann problem (1) and (3) has a solution by using a delta shock connecting two states and on local time , we must choose and to be

Where, here and below, we use the usual notation with and the values of the function on the left-hand and right-hand sides of the discontinuity , etc.; is the Heaviside function, that is, 0 when and 1 when ; and are the weight and the tangential derivative of curve , which can be defined by

From (19), one can get that the propagating speed of the delta shock

(a) The Riemann solution

(b) The perturbed Riemann solution

Figure 2: Case 1, Case 2 with , and Case 3 with .

We will prove that the delta shock solution constructed in (18) is a solution of the initial value problem (1) and (3) in the sense of distributions on .

Proposition 3. The delta shock solution constructed in (18) satisfies (1) and (3) in the sense of distributions on a domain

where is a finite time.

Proof. Let

Then the delta shock solution (18) can be reduced to

We need to check that satisfies (1), which is (12) and (13).
For any test function , we plug (23) into the left-hand side of equation (12) and get

Now using the fact that and are solutions to problem (1) with initial data and in the domains and , respectively, the divergence theorem gives

Similarly, we plug (23) into the left-hand side of (13) and get

which gets equality (13). Then we complete the proof of the proposition.

Furthermore, to guarantee uniqueness of the solution, the delta shock solution constructed in (18) should satisfy the entropy condition (27) on the discontinuity .

Definition 4. The delta shock solution constructed in (18) is an admissible solution of the initial value problem (1) and (3) in the sense of distributions on , if satisfies Definition 1 and the entropy condition

on the discontinuity .

We now turn to check the entropy condition (27) on . The discussion is divided into three cases: , , and . We start with the following case.

Case 1 (). Due to , there exist constants and so small that, for any and , the functions and satisfy the following condition:

Let (resp., ) be the downwards left (resp., right) characteristic from any point on the delta shock curve (see Figure 2(b)). When the time is small enough, the characteristic (resp., ) starting at will intersect at a point (resp., ) on the initial axis with (resp., ). Since the Riemann invariant must be a constant along characteristic and , we have

and

Using Rankine-Hugoniot condition and the above two expressions, we obtain the propagation speed of the delta shock at the point

Together with (28) and (31), we get the entropy condition

which is valid at any point on the curve locally in time. Then we have the following.

Theorem 5. In case of , the perturbed Riemann problem (1) and (3) has a delta shock solution constructed in (18) for local time. The admissible solution has a structure similar to that of the corresponding Riemann problem (1) and (5) (see Figure 2). Furthermore, the delta shock curve possesses the following property:

(a)If , then the curve is convex(b)On the other hand, if , then the curve is concave

Proof. By Proposition 3 and inequality (32), we know that the delta shock solution constructed in (18) satisfies (1) and (3) in the sense of distributions and the entropy condition (27) for local time, respectively. Obviously, the delta shock curve retains its form in a neighborhood of the origin . Namely, the solution of the perturbed Riemann problem (1) and (3) has a structure similar to the corresponding Riemann solution of (1) and (5) for local time.
In order to show the behavior of the delta shock curve near the origin, we need a certain a priori estimate on the value of . From (20), it can be easily checked that

Differentiating the above equality with respect to and letting , one obtains

Next, we compute the values of and , respectively. From (1), it can be checked that

Along the delta shock curve , by using the fact that and condition (35), we have

and

Substituting (36) and (37) into (34), we transform (34) into the form

With in mind, (38) shows the second derivative of the delta shock curve at origin when ; otherwise, when . Thus the proof of Theorem 5 is completed.

Case 2 (). When , due to , there exist constants and so small that, for any and , the functions and also satisfy the inequality (28). Then a discussion similar to that for Case 1 shows that the delta shock solution defined by (18) also satisfies Definition 1 and the entropy condition (27). The delta shock can retain its form in a neighborhood of the origin. That is, the perturbed Riemann solution of (1) and (3) has a structure similar to the Riemann solution of (1) and (5) for this case (see Figure 2).

When , we can prove (29)~(31) correspondingly for this subcase. In the same way as Case 1, by virtue of , there exist constants and so small that, for any and , we have

Then, from (39), we get

on the delta shock curve . Inequality (40) shows that the entropy condition (27) does not hold on the curve . Therefore, the perturbed Riemann solution to (1) and (3) should not be a delta shock solution for this subcase.

Now we prove that the perturbed Riemann solution should be a backward shock followed by a contact discontinuity for this subcase. The structure of the perturbed solution can be indicated in Figure 3(b), in which and are free boundaries. Furthermore, on , we have

On , we have

On the domain so small), the perturbed Riemann solution to (1) and (3) is . On the domain , the perturbed Riemann solution is . On the domain , the perturbed Riemann solution is denoted by , which is an unknown regular solution to problem (1) and (3); moreover,

Noticing the corresponding results in [15, 16], the above perturbed Riemann problem is equivalent to the free boundary problem (1) with boundary conditions (41)~(44) on the fan-shaped domain so small).

(a) The Riemann solution

(b) The perturbed Riemann solution

Figure 3: Case 2 with .

Set

Boundary condition (42) on then reduces to

Boundary condition (44) on can be written as

Thus, the characterizing matrix of this problem is of the form [15]

According to the local existence and uniqueness theorem (c.f. Chapter 6 in Li Ta-tsien and Yu Wen-ci [15]), if the minimal characterizing number

then there exists a unique solution on the fan-shaped domain , where is suitably small.

By Remark 4.4 in the introduction of [15], it is not hard to prove that the minimal characterizing number of this problem for this case is

Then the free boundary problem under consideration admits a unique piecewise smooth solution on the fan-shaped domain . Hence we have the following.

Theorem 6. In case of and , the solution to the perturbed Riemann problem (1) and (3) is composed of a backward shock followed by a contact discontinuity for local time. The perturbed solution is dramatically different from the corresponding Riemann solution of (1) and (5), which is a delta shock (see Figure 3).

Next, we proceed to prove that the perturbed Riemann problem (1) and (3) for this subcase admits a solution that contains a backward shock and a contact discontinuity near the origin in another way.

Let be the upwards right characteristic from any point on the shock curve (see Figure 3(b)). The point is the intersection point of the characteristic curve and the contact discontinuity . Since the Riemann invariant must be a constant along characteristic, we have

From (52), it is easy to see that the propagating speed of characteristic

is a constant. That is, the characteristic is a straight line. Using the above equality, we arrive at

Noting (54), we get

Differentiate the above equation with respect to and let ; then one obtains

Substituting

into (56), with , we get

Moreover, differentiating the last equality in (52) with respect to and letting , we obtain

In view of (58) and (59), it is easy to see that

In the following, firstly, we will accumulate the second derivative of the shock at the origin . Along , differentiating the above equality (41) with respect to and letting , by (44) and (52), we have

On the one hand, from (35), we have

Noting (57), (62), and , one can get that

On the other hand, by (60) and (44), we have

Along the contact discontinuity wave curve , from (35), (57), and (64), it follows that

Then, substituting (63) and (65) into (61), we get

Secondly, we now have an estimate of the second derivative of the contact discontinuity wave curve at the origin, a.e. . Along , with (43), it holds that

Differentiating (67) with respect to and letting give

Thirdly, combining (66) and (68), it follows that

Finally, due to , (57), and (69), the perturbed Riemann solution to (1) and (3) in this subcase clearly consists of a backward shock connecting to , followed by a contact discontinuity connecting to near the origin (see Figure 3(b)). Thus we have completed the construction and proof of the perturbed Riemann solution for this subcase.

Case 3 (). When , there exist constants and so small that, for any and , the functions and also satisfy inequality (28). By a similar argument used in Case 1, we get that the delta shock solution defined by (18) satisfies the entropy condition (27). That is, in a neighborhood of the origin, the perturbed Riemann solution of (1) and (3) has a structure similar to the Riemann solution to (1) and (5) for this subcase (see Figure 2).

When , by virtue of , there exist constants and so small that, for any and , we have

Then in the same way as Case 1, we can prove (29)~(31) corresponding for this subcase. Finally, from (70), we get

on the delta shock curve . Equality (53) shows that the entropy condition (27) does not hold for . Therefore, we should not expect the perturbed Riemann solution to (1) and (3) to be a delta shock solution for this subcase.

We will prove that the perturbed Riemann solution is a contact discontinuity followed by a forward shock locally in time for this subcase. The structure of the perturbed solution to (1) and (5) can be indicated in Figure 4(b). As shown in Figure 4(b),

and

are free boundaries. Furthermore, is a contact discontinuity, on which we have

is a forward shock, on which we have

(a) The Riemann solution

(b) The perturbed Riemann solution

Figure 4: Case 3 with .

On the domain so small), the perturbed Riemann solution to (1) and (3) is . On the domain , the perturbed Riemann solution is . On the domain , the perturbed Riemann solution is denoted by , which is an unknown regular solution to problem (1) with (3) and satisfies condition (45).

Since and are known, in order to get the perturbed Riemann solution, we have to solve the free boundary problem (1) and (74)(77) on the fan-shaped domain so small).

We introduce the change of variables

Boundary condition (75) on then reduces to

Boundary condition (77) on can be written as

Hence, the characterizing matrix of this problem is of the form [15]

According to the local existence and uniqueness theorem, what remains is to prove that .

By Remark 4.4 in the introduction of [15], it is not hard to prove that for this subcase

Then the free boundary problem under consideration admits a unique piecewise solution on the fan-shaped domain ( so small). We can conclude the following.

Theorem 7. In case of and , the solution of the perturbed Riemann problem (1) and (3) is composed of a contact discontinuity followed by a forward shock for local time. The perturbed solution is different from the corresponding Riemann solution of (1) and (5), which is a delta shock (see Figure 4).

Next, we proceed to prove that the perturbed Riemann problem (1) and (3) for this subcase admits a solution which contains a contact discontinuity followed by a forward shock near the origin. Let be the upwards right characteristic from any point on the contact discontinuity curve (see Figure 4). The point is the intersection point of the characteristic curve and the shock curve . Since the Riemann invariant must be a constant along characteristic, we have

From (83), it is easy to see that the propagating speed of the characteristic

is a constant. That is, the characteristic is a straight line. Using the above equality, we arrive at

Noting (85), we get

Differentiate the above equation with respect to and let ; then one obtains

Substituting

into (87), with , we get

Moreover, differentiating the last equality in (83) with respect to and letting , we obtain

From (89) and (90), we derive

In the following, firstly, we will accumulate the second derivative of the contact discontinuity curve at the origin . Along , differentiating (74) with respect to and letting , it yields

From (35), we have

Noting (88) and (93), one can get that

Secondly, we now estimate the second derivative of the shock curve at the origin . Along , with (76), it holds that

Differentiating (95) with respect to and letting give

On the one hand, by (75) and (91), we get

Along the delta shock curve , it follows from (35), (88), and (97) that

On the other hand, based on (35) and (88), we obtain

We substitute (98) and (99) into (96) and get

Thirdly, combining (94) and (100), it follows that

Finally, by virtue of , (88), and (101), the perturbed Riemann solution to (1) and (3) in this subcase clearly consists of a contact discontinuity connecting to , followed by a forward shock connecting to near the origin. Thus we have completed the construction and proof of the perturbed Riemann solution for this subcase (see Figure 4).

4. Conclusion

So far, we have locally constructed the solution to the perturbed Riemann problem (1) and (3) with delta shock. From the above discussion, we discover a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity after the perturbation of Riemann initial data. This shows the internal mechanism and instability of a delta shock. Then we can obtain the main theorem of this paper.

Theorem 8. There is a unique local solution to the perturbed Riemann problem (1) and (3) with delta shock. The Riemann solution of (1) and (5) is instable under a perturbation of the Riemann initial data, which is quite different from the previous perturbed Riemann problems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported by NSF of China (11301264), China Scholarship Council, China Postdoctoral Science Foundation (2013M531343), the Fundamental Research Funds for the Central Universities (NZ2014107), Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents, and the NSF of Jiangsu Province of China (BK20130779).

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