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 formEquations (1) can be derived from a two-dimensional hyperbolic system of conservation lawsThe 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 datawhere and are constants with (see [4, 17, 18] for a more detailed study of the model).
The eigenvalues of the hyperbolic system (1) arewith 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 satisfyfor 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,
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 beWhere, 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 byFrom (19), one can get that the propagating speed of the delta shock
(a) The Riemann solution
(b) The perturbed Riemann solution
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 domainwhere is a finite time.
Proof. LetThen the delta shock solution (18) can be reduced toWe 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 conditionon 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 haveandUsing 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 conditionwhich 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 thatDifferentiating the above equality with respect to and letting , one obtainsNext, we compute the values of and , respectively. From (1), it can be checked thatAlong the delta shock curve , by using the fact that and condition (35), we haveandSubstituting (36) and (37) into (34), we transform (34) into the formWith 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 haveThen, from (39), we geton 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 haveOn , we haveOn 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
SetBoundary condition (42) on then reduces toBoundary condition (44) on can be written asThus, 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 numberthen 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 isThen 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 haveFrom (52), it is easy to see that the propagating speed of characteristicis a constant. That is, the characteristic is a straight line. Using the above equality, we arrive at
Noting (54), we getDifferentiate the above equation with respect to and let ; then one obtainsSubstitutinginto (56), with , we get
Moreover, differentiating the last equality in (52) with respect to and letting , we obtainIn 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 haveNoting (57), (62), and , one can get thatOn the other hand, by (60) and (44), we haveAlong the contact discontinuity wave curve , from (35), (57), and (64), it follows thatThen, 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 thatDifferentiating (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 haveThen in the same way as Case 1, we can prove (29)~(31) corresponding for this subcase. Finally, from (70), we geton 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
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 variablesBoundary condition (75) on then reduces toBoundary condition (77) on can be written asHence, 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 subcaseThen 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 haveFrom (83), it is easy to see that the propagating speed of the characteristicis a constant. That is, the characteristic is a straight line. Using the above equality, we arrive at
Noting (85), we getDifferentiate the above equation with respect to and let ; then one obtainsSubstitutinginto (87), with , we get
Moreover, differentiating the last equality in (83) with respect to and letting , we obtainFrom (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 yieldsFrom (35), we haveNoting (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 thatDifferentiating (95) with respect to and letting giveOn the one hand, by (75) and (91), we getAlong the delta shock curve , it follows from (35), (88), and (97) thatOn the other hand, based on (35) and (88), we obtainWe 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).