INTERACTIONS OF DELTA SHOCK WAVES FOR A CLASS OF NONSTRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS∗

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter ε tend to zero, we analyze the stability of Riemann solutions.


Introduction
In the past over two decades, the investigation for delta shock waves has been an increasingly active topic on the study of conservation laws. As a generalization of an ordinary shock wave, the delta shock wave is a new kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its support. Physically, people often use the delta shock wave to represent the process of concentration of the mass [2], or express the galaxies in the universe [22,30].
For delta shock waves, there are numerous excellent papers for various hyperbolic systems. In 1977, Korchinski first [13] used generalized delta-functions to construct Riemann solution and called it overcompressive singular shock. Afterwards, more and more scholars started to concern about the existence and uniqueness of nonclassical solutions involving delta functions developing in the state variables. For instance, Keyfitz and Kranzer [12,14] introduced the singular shock wave. Le Floch [8] introduced the measure solutions. Tan and Zhang [27] independently discovered the delta shock wave. Sheng and Zhang [24] discussed the Riemann problem for the zero-pressure flow. Huang and Wang [11] proved the uniqueness of the weak solution when the initial data are a Radon measure for the zero-pressure flow. Yang [31] solved the Riemann problem for the generalized zero-pressure flow. See also the results in [1,4,6,20,28] and the references cited therein.
However, it is observed that these investigations on delta shock waves mentioned above have mostly been focused on the case that only one state variable develops the Dirac delta function and the others have bounded variations. Different from these ones, the theory of delta shock waves with Dirac delta functions developing in both state variables has been established by Yang and Zhang [32] for the following class of nonstrictly hyperbolic system of conservation laws where φ(u, v) satisfies the following assumption: (H 1 ) φ = φ(r) is a given smooth function of r = au + bv satisfying a 2 + b 2 = 0, a and b are constants.
The Riemann problem for system (1.1) with initial data (u, v)(0, x) = (u ± , v ± ) (±x > 0) (1.2) was constructively solved. In solutions, a special kind of delta shock waves on which both state variables simultaneously contain the Dirac delta functions was found. Moreover, when φ(u, v) is a smooth function satisfying φ(u, v) = φ(αu, αv), α > 0 is a constant, this kind of delta shock waves also appears in [33]. More on the theory of delta shock waves with Dirac delta functions developing in multiple state variables, see also [17][18][19]35], etc.
When the system (1.1) satisfies assumption (H 1 ), many important systems can be obtained with assignment for φ(r), a and b. For example, if φ(au + bv) = 1 2 u, it is reduced to the system investigated in [13]; while taking φ(au + bv) = u, it becomes one dimensional transport equations studied in [23,28]. Moreover, if φ(au + bv) = 1 + 1 1−u+v or 1 + 1 1+v , it corresponds to the nonlinear chromatography equation investigated in [5,9,34]. Letting φ(au + bv) = 1 1−u+v or 1 1+v , it is just another form of nonlinear chromatography system considered in [25,26,29]. For these special models, the Riemann solutions are constructed completely. The delta shock waves occur in these systems, but they are different. Only one state variable contains the Dirac delta function in [13,23,25,26,28,34], while both state variables simultaneously develop Dirac delta functions in [5,9,29]. In addition, there are still many other examples, and we will not list them one by one.
For these specific systems, besides the investigation of delta-shock solution, the interaction of delta shock waves is also a very interesting topic. It has significance in the general mathematical theory, numerical calculation and practical applications of quasi-linear hyperbolic equations. For system (1.1), when φ(au + bv) = u, Shen and Sun [23] discussed the interactions of the delta shock waves with the shock waves and rarefaction waves when the initial data are three piece constant states. When φ(au + bv) = 1 + 1 1−u+v and 1 + 1 1+v , Guo et al. [9] and Zhang [34] consider the perturbed Riemann problem. When φ(au + bv) = 1 1+v , Sun [26] studied the Riemann problem with initial data of three piece constant states and constructed the global structures of solutions completely.
Motivated by [9,23,26,34], in which they studied the interaction of delta shock waves and elementary waves by splitting delta function, we in this paper aim to studying the interactions of delta shock waves for the generalized system (1.1). To this end, we consider (1.1) with the initial data of three piecewise constant states as follows where u i , v i (i = ±, m) are arbitrary constants and ε is arbitrarily small. Obviously, the initial data (1.3) is a local perturbation of the Riemann initial data (1.2).
As delta shock waves interact with the other elementary waves, it will give rise to the product of δ(x) and H(x). So we use the method of splitting delta function along a regular curve in R 2 + , which is proposed by Nedeljkov and Oberguggenberger [15,16], to study the Riemann problem (1.1) and (1.3). Benefited from this method, the product of the piecewise smooth function and discontinuity along such a curve makes sense. This method has been applied to the various systems to investigate the interactions of delta shock waves, such as [9,10,21,23,26,34,36]. Specially, when delta shock wave interact with the rarefaction wave, using the wavefront tracking algorithm [3], we approximate the rarefaction wave by a set of small non-admissible shocks.
After attentive analysis of interaction between delta shock wave and elementary wave, we find that the delta contact discontinuity appears in solutions, on which both state variables simultaneously contain the Dirac delta functions. Generally speaking, the delta contact discontinuity is a kind of nonclassical wave with at least one Dirac delta function supported on it. The delta function propagates along the line of the contact discontinuity and thereby the propagation speed and strength do not change as it travels in space. This kind of discontinuity also appears in [16,23,26,34], etc. In particular, the delta contact discontinuity with both state variables simultaneously containing the Dirac delta functions was also found in [9] for the nonlinear chromatography equations. However, the system in [9] is specific and can be included in (1.1).
As dealing with the Riemann problem (1.1) and (1.3), the expression of the Riemann solution can not be concretely and explicitly formulated, here we use some new ideas and skills to obtain the existence and uniqueness of Riemann solutions qualitatively and abstractly. Moreover, it can be established that the solutions of the perturbed Riemann problem converge to nothing but the corresponding Riemann solutions as ε → 0, from which the stability of the Riemann solutions with respect to this local small perturbations of the Riemann initial data is obtained.
It is not hard to find that the solutions we obtained presents generality, because they contain some exact analytic solutions that are given in papers [9,23,26,34]. That is, this paper to some extent extends their results and proofs. Therefore, we get more abroad and more common results, which is main objective and innovation of this paper. This paper is arranged as follows. In Section 2, the Riemann solutions of (1.1) with two piecewise constant states is reviewed. Section 3 discusses the interactions of the delta shock waves and classical waves. The Riemann solutions of (1.1), (1.3) are constructed globally. By taking ε → 0, the stability of Riemann solutions of (1.1), (1.3) is analyzed.
Besides the constant state solution, the self-similar waves (u, v)(ξ)(ξ = x/t) of the first family are contact discontinuities and those of the second family are rarefaction waves R : or shock waves S : For the case r − 0 r + , the delta shock wave appears. In order to define the measure solutions, a two-dimensional weighted delta function β(s)δ S supported on a smooth curve S parameterized as t = t(s), x = x(s)(c s d) can be introduced as With this definition, we introduce a delta-shock solution to construct the solution of (1.1), which can be expressed as in which [p] = p + − p − denotes the jump of function p across the discontinuity, σ the velocity of the delta shock wave, and H(x) the Heaviside function. As shown in [32], the solution (u, v) constructed above satisfy and v has the similar integral identities as above.
Then, a unique delta-shock solution of (1.1) can be constructed as in which x(t), β(t) and σ satisfy the following generalized Rankine-Hugoniot relation and Moreover, the delta shock wave should satisfy the entropy condition: which means that all characteristic curves run into the delta shock curve from both sides. Overcompressibility reflects the fact that delta shock wave should arise only from local concentrations of the quantities u and v due to the conservation law. Under the entropy condition (2.9), solving the generalized Rankine-Hugoniot relation (2.7) and (2.8) with the initial data x(0) = 0 and w(0) = 0 yields that For the convenience of the discussion in the next section, we briefly review the concept of left-and right-hand side delta functions, further details of which can be found in [15,16].
Let R + 2 be divided into two disjoint open sets Ω 1 and Ω 2 with piecewise smooth boundary curve Γ, that is, Ω 1 ∩ Ω 2 = 0 and Ω 1 ∪ Ω 2 = R + 2 . Let C(Ω i ) and M (Ω i ) be the space of bounded and continuous real-valued functions equipped with the L ∞ -norm and the space of measures on Ω i (i = 1, 2), respectively. Suppose can be defined as the usual product of a continuous function and a measure. It is obvious that the above-defined product makes sense.
Every measure on Ω i can be considered as a measure on R + 2 with support in Ω i (i = 1, 2). Based on this, we can obtain the mapping m : The solution concept used in this paper can be described as follows: perform all nonlinear operations on functions in the space C Γ , carry out the multiplication and composition in the space M Γ and then take the mapping m : M Γ → M (R + 2 ) before differentiation in the space of distributions, and require that the equation is satisfied in the weak sense of distribution. Now we rewrite the solution (2.6) by employing the above definition of the split delta function. At this moment, the delta-shock solution of (1.1) and (1.2) can be expressed as where H is the Heaviside function, β(t)D = β − (t)D − + β + (t)D + is a split delta function, and all of them are supported by the line x = σt. What is more, D − is the delta measure on the set R 2 + ∩{(x, t)|x σt}, and D + on the set R 2 + ∩{(x, t)|x σt}, β − (t) and β + (t) are to be determined.
From (2.11), we can compute that Substituting the above two equations into the first equation of (1.1) yields that By calculation, we get and Similarly, from (2.11) and the second equation of (1.1), the same results can be obtained. Using classical waves and delta shock waves, we now construct the solutions of the Riemann problem (1.1) and (1.2) as follows: (a) when r + < r − < 0, the solution is ← − S + J; (e) when r − > r + > 0, the solution is J + − → S .
(f) when r − 0 r + , the solution is a delta shock wave.

Interactions of delta shock waves
In this section, we investigate the Riemann problem (1.1) and (1.3). Considering that the problem is classical when the delta shock wave does not appear in the interaction process, so in the presented paper, we mainly focus on the interactions of waves involving delta shock waves. In order to cover all the cases completely, we discuss this problem case by case according to the relations among r − , r m and r + .
At the intersection (x 1 , t 1 ), a new Riemann problem for (1.1) with initial data will be formed, where bα(t 1 ) and −aα(t 1 ) denote the strengths of the incoming delta shocks δ 1 and δ 2 on u and v at the time t 1 , and α(t 1 ) can be expressed as (3.5) The interaction of δ 1 and δ 2 results in a new delta shock wave, denoted by δ 3 , which is determined by where H is the Heaviside function and β(t)D = β − (t)D − + β + (t)D + is a split delta function. All of them are the functions of x − x 1 − (t − t 1 )σ 3 , in which σ 3 is the propagating speed of δ 3 . What is more, D − is the delta measure on the set R 2 + ∩ {(x, t)|x x 1 + (t − t 1 )σ 3 }, and D + is the delta measure on the set , β + (t) and σ 3 are to be determined. From (3.6), by a simple calculation, we obtain that and Substituting (3.7)-(3.10) into the system (1.1) and comparing the coefficients of δ and δ , it yields that and Considering the initial conditions (3.3) and (3.4), from (3.11)-(3.14), it leads to Obviously, the entropy condition (2.9) is satisfied, this is, the new formed delta shock wave δ 3 is overcompresive. Furthermore, the strengths of δ 3 about u and v are bβ(t) and −aβ(t), respectively. Therefore, the conclusion is that two delta shock waves conclude with each other to form a single delta shock wave. Moreover, it is easy to see that (x 1 , t 1 ) → (0, 0) and α(t 1 ) → 0 as ε → 0. In other words, the limit of Riemann solution of (1.1) and (1.3) is just the corresponding one of (1.1) and (1.2) as ε → 0.
In addition, from (3.21), we can get (3.22) and (3.26). So the strengths of δ 2 about u and v are bβ(t) and −aβ(t). Entropy condition is obviously satisfied.
The delta shock wave δ 2 will penetrate over the whole rarefaction wave ← − R in a finite time because r − 0 > r + . The interaction point of δ 2 and the head of the rarefaction wave ← − R is denoted by (x 2 , t 2 ), which is determined by (3.27) The strengths of δ 2 about u and v at (x 2 , t 2 ) can be calculated by (3.26). After the time t 2 , δ 2 will propagate with an invariant speed σ 3 = r * φ(r * )−r−φ(r−) r * −r− , and be denoted by δ 3 , where r * = r + . The strengths of it about u and v are bγ(t) and −aγ(t), where Because r * = r + and um vm = u * v * , the above formula can be also written as Since the speed of contact discontinuity J is φ(r + ), δ 3 must meet J at some point (x 3 , t 3 ). At this moment, a new initial value problem is formed. The intersection point (x 3 , t 3 ) is determined by the equations (3.28) After the interaction of δ 3 and J, a new delta shock wave is formed and denoted by δ 4 , whose velocity is . The strengths of δ 4 about u and v can be obtained from the following formula Obviously, the entropy condition is hold. From (3.17), (3.27) and (3.28), we can get that (x 1 , t 1 ), (x 2 , t 2 ) and (x 3 , t 3 ) all tend to (0, 0) as ε → 0. Furthermore, we also have lim ε→0 α(t 1 ) = lim ε→0 β(t 2 ) = lim ε→0 γ(t 3 ) = 0. It is easy to know that the limit of the solution of (1.1) and (1.3) is just the delta shock wave, which is the corresponding Riemann solution of (1.1) and (1.2).
Case 3. r − 0 > r m > r + . (If r − > r m > 0 r + , the structure of the solution is similar.) In this case, a delta shock wave δ 1 emits from (−ε, 0) and a shock wave ← − S followed by a contact discontinuity J emits from (ε, 0) (see Fig. 3). The propagating speed of δ 1 is . So δ 1 and ← − S will overtake in a finite time. The intersection (x 1 , t 1 ) is determined by which yields that A new Riemann problem is formed at (x 1 , t 1 ) with the following initial data where (u * , v * ) = ( au++bv+ aum+bvm u m , au++bv+ aum+bvm v m ) is the intermediate state between ← − S and J, and α(t 1 ) has the same expression as (3.18).
We solve the new initial data problem as we have done before. Then, a new delta shock wave δ 2 appears at (x 1 , t 1 ) expressed by where the propagating speed of δ 2 is σ 2 = r * φ(r * )−r−φ(r−) r * −r− with r + = r * , and . After the time t 1 , the situation is completely similar to Case 2. That is to say, the delta shock wave δ 2 will pass through J with the same speed. The only difference lies in that the strengths about u and v varies due to the difference choice of (u * , v * ) and (u + , v + ). So, the delta shock wave is wholly denoted by δ 2 when t > t 1 , and the strengths about u and v can be calculated by Similarly to the analysis in Case 2, the limits of the solution to (1.1) and (1.3) as ε → 0 is just the corresponding ones of (1.1) and (1.2). In this case, there are a delta shock wave δ 1 and two rarefaction waves ← − R and − → R near t = 0 on (x, t)-plane (see Fig. 4). The propagating speed of δ 1 is σ 1 = rmφ(rm)−r−φ(r−) rm−r− and that of the wave back of the rarefaction wave ← − R is λ 2m = φ + rφ r | r=rm . Moreover, the strengths of δ 1 about u and v are bα(t) and −aα(t) with α(t) = φ(rm)−φ(r−) Similar to Case 2, a new delta shock wave will be generated after δ 1 meets the rarefaction waves ← − R at (x 1 , t 1 ), which is determined by (3.17). The delta shock wave will cross the whole rarefaction wave ← − R and be denoted by δ 2 , whose curve is expressed by Γ : {(x(t), t) : t t 1 }, with (u − , v − ) on the left-hand side and (u(ξ), v(ξ)) on the right-hand side. Here λ 2m x−ε t = ξ = φ + rφ r 0, r m r(ξ) = au(ξ) + bv(ξ) 0, u(ξ) v(ξ) = um vm . The speed and strengths about u and v of δ 2 are determined by (3.25) and (3.26).
The interaction of δ 2 and the head of the rarefaction wave − → R begins at (x 2 , t 2 ), which can be calculated by    x 2 = x(t 2 ), At the time t = t 2 , we again obtain a new local Riemann problem with the initial data where 0 x−ε t = ξ = φ + rφ r λ 2+ , 0 r(ξ) = au(ξ) + bv(ξ) r + , u(ξ) v(ξ) = u+ v+ , and β(t 2 ) can be calculated by (3.26). In order to deal with the above initial value problem, we assume that the rarefaction wave − → R is approximated by a set of non-physical shock waves(a method proposed in [3]). Now the value (u(ξ), v(ξ)) in − → R is determined by the right state (u + , v + ) with u(ξ) v(ξ) = u+ v+ . In fact, we can construct the solution of the initial value problem (1.1) with (3.36) and (3.37) in the form: , and x 2 (t) is the shock wave curve satisfying (3.40) Now we prove that (3.38) and (3.39) are indeed the weak solution of the initial value problem (1.1) with (3.36) and (3.37). For every ϕ ∈ C ∞ 0 (R × R + ), the following equations Otherwise, we prove that (3.38) and (3.39) is still the weak solution of (1.1) with (3.36) and (3.37) near the support of the delta function. Substituting (3.38) and (3.39) into the first equation of (1.1), we have Similarly, Substituting (3.38) and (3.39) into the second equation of (1.1), we obtain which means that (1.1) is also satisfied near the line x = x 1 (t) in the weak (or distributional) sense. For the solution (3.38) and (3.39), the Dirac delta function is now supported on the contact discontinuity line and overcompressibility is lost. As in [16], we introduce the delta contact discontinuity as follows.
Definition 3.1. Consider a region Ω and a curve Γ 1 of slope λ 1 = φ(r) = φ(au + bv) in Ω. A pair of distributions (u, v) ∈ C(Ω) × D (Ω) is called a delta contact discontinuity, if u and v are a sum of a locally integrable functions on Ω and a delta functions on Γ 1 which solves (1.1) in the sense of distribution.
Solving the Riemann problem at (x 2 , t 2 ), we obtain the solution involving a delta contact discontinuity δJ and a shock wave S. In other words, when t > t 2 , δ 2 decomposes and the state (u * , v * ) = ( u+(au−+bv−) au++bv+ , v+(au−+bv−) au++bv+ ) lies between δJ and S. For the delta contact discontinuity, both state variables simultaneously contain the Dirac delta functions.
Then, the delta contact discontinuity δJ will continue to move forwards with a constant speed φ(r − ), while the strengths about u and v are invariant. However, the over-compressibility is lost.
At the same time, the shock wave S will continue to penetrate the rarefaction wave − → R . Its curve should satisfy the differential equations (3.40). Based on the comparison between the values r − and r + , we should divide our discussion into the following three subcases. In this subcase, the curve of S will penetrate over the whole rarefaction wave fan − → R at (x 3 , t 3 ), the intersection can be calculated by (3.44) After the time t 3 , the shock wave S propagates with an invariant speed r + φ(r + )−r − φ(r − ) r + −r − . As ε → 0, (x 1 , t 1 ), (x 2 , t 2 ) and (x 3 , t 3 ) coincide with each other at the point (0, 0). In addition, we have α(t 1 ) → 0 and β(t 2 ) → 0. Thus, the limit of the solution of (1.1) and (1.3) is the contact discontinuity J : x = φ(r − )t and the shock wave S : x = r+φ(r+)−r−φ(r−) r+−r− t, which is exactly the corresponding Riemann solution of (1.1) and (1.2) in this subcase. Subcase 4.2. r + > r − At this moment, the shock wave curve S can not penetrate the rarefaction wave fan − → R completely and ultimately has x = λ 2− t + ε as its asymptote (see Fig. 4(b)). Moreover, when ε → 0, the limit of the solution of (1.1) and (1.3) is the contact discontinuity and the rarefaction wave plus the intermediate state (u * , v * ) = ( u+(au−+bv−) au++bv+ , v+(au−+bv−) au++bv+ ) between them. Thus, the limit situation is also true for our assertion in this subcase. For this special subcase, the curve of S has x = λ 2+ t + ε as its asymptote, which is exactly the wave front of the rarefaction wave − → R . When ε → 0, the limit is a contact discontinuity J = φ(r − )t connecting (u − , v − ) and (u + , v + ) directly, and the conclusion is obviously identical with our assertion.
Through the discussions in case 4, we know that delta contact discontinuity is introduced beyond a point at which a delta shock wave loses overcompressibility. Its existence can be justified by two facts. First, a contact discontinuity emerges in the case when one of the characteristic fields is linearly degenerate. Second, if the conservation law has a delta function as initial data, it propagates along the characteristic lines. Now we have finished the discussion for all kinds of interactions when the delta shock wave is included. The global solutions for the perturbed initial value problem (1.1) and (1.3) have been constructed. From the results above, it is easy to see that the limits of the perturbed Riemann solutions of (1.1) and (1.3) are exactly the corresponding Riemann solutions of (1.1) and (1.2) as ε → 0, and the asymptotic behavior of the perturbed Riemann solutions is governed completely by the states (u ± , v ± ), that is, the Riemann solutions of (1.1) and (1.2) are stable with respect to such a local small perturbation.
Remark It is noticed that, as pointed in [32], the theory does not work in some typical systems. For instance, the following equations of geometrical optics which were proposed by Engquist and Runborg [7] in 1996. To our knowledge, the interactions of delta shock waves for the above geometrical optics (3.45) have not been discussed so far. Therefore, it is nature to consider the interaction of delta shock waves for the system (1.1) with the assumption that φ(u, v) is a given smooth function satisfying φ(u, v) = φ(αu, αv), α > 0 is a constant.
Obviously, under this assumption, one can find that the system (3.45) is the very prototype of (1.1) by taking φ(u, v) = u √ u 2 +v 2 . We leave them for our future study.