Nonlinear Analysis 73 (2010) 3284–3294

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Stability of the Riemann solutions for a nonstrictly hyperbolic system ofconservation lawsI

Chun Shen, Meina Sun ∗School of Mathematics and Information, Ludong University, Yantai 264025, PR ChinaLaboratory of Mathematics Physics, Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, PR China

a r t i c l e i n f o

Article history:Received 13 May 2009Accepted 7 July 2010

MSC:35L6535L6735B30

Keywords:Delta shock waveDelta contact discontinuityWave interactionRiemann problemSplit delta functionNonstrictly hyperbolicity

a b s t r a c t

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system ofconservation laws under local small perturbations of the Riemann initial data. The proofis based on the detailed analysis of the interactions of delta shock waves with shock wavesand rarefaction waves. During the interaction process of the delta shock wave with therarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, isdiscovered here, which is a Dirac delta function supported on a contact discontinuity andhas already appeared in the interaction process for the magnetohydrodynamics equations[M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictlyhyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143–1157].Moreover, the global structures and large time asymptotic behaviors of the solutions areconstructed and analyzed case by case.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we are concerned with the following nonstrictly hyperbolic system of conservation laws in the formut + (u2)x = 0,vt + (uv)x = 0.

(1.1)

It is well known that the Riemann problem for system (1.1) is a special Cauchy problem with initial data

(u, v)(x, 0) = (u±, v±), ±x > 0, (1.2)

where u± and v± are all given constants.The first equation in (1.1) is just the inviscid Burgers equation and the solutions to the Riemann problem are the classical

entropy solutions. The Dirac function is introduced as a part for v when the characteristic velocity u is discontinuous [1]. In1994, Tan et al. [2] considered the Riemann problem of (1.1) and they discovered that the form of the standard Dirac deltafunction supported on a shock wave was used as a part in their Riemann solutions for certain initial data. The concept of

I This work is partially supported by the National Natural Science Foundation of China (10901077, 10871199) and the China Postdoctoral ScienceFoundation (20090451089).∗ Corresponding author at: School of Mathematics and Information, Ludong University, Yantai 264025, PR China. Tel.: +86 535 6697510; fax: +86 5356681264.E-mail addresses: shenchun3641@sina.com (C. Shen), smnwhy0350@163.com (M. Sun).

0362-546X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2010.07.008

C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294 3285

the Dirac function was introduced into the classical weak solution by Korchinski [3] in 1977 when he studied the Riemannproblem for

ut +(12u2)x= 0,

vt +

(12uv)x= 0,

(1.3)

which has the trivial difference u→ 2u from (1.1).In this paper, our main purpose is to investigate the stability of the Riemann solutions of (1.1) and (1.2) under local small

perturbations of the Riemann initial data. Thus, we consider the initial value problem of (1.1) with the following three piecesconstant initial data:

(u, v)(x, 0) =

(u−, v−), −∞ < x < −ε,(um, vm), −ε < x < ε,(u+, v+), ε < x < +∞,

(1.4)

where ε > 0 is arbitrarily small. The initial data (1.4) is a local perturbation of the corresponding Riemann initial data(1.2). We will face the interesting question of determining whether the Riemann solutions of (1.1) and (1.2) are the limitsof (uε, vε)(x, t) as ε → 0, where (uε, vε)(x, t) are the solutions of (1.1) and (1.4) for given ε > 0. We will deal with thisproblem case by case along with constructing the perturbed solutions. This methodwas also adopted in [4,5] and referencestherein.The problem of weak uniqueness to (1.1) can also be proved by employing the weak asymptotics method [6,7], where

it can be treated as a problem of limit passage in the corresponding integral identity by adding a small perturbation to theright hand sides in (1.1). In [8], it was assumed that the continuity equation is understood in the sense of D′. From thisassumption, an integral identity defining a generalized solution with a possible Dirac delta function was derived. In fact,from [9], we know that the weak uniqueness to (1.1) can also be proved if (um, vm) depends on ε but satisfies the additionalassumptions: um → 0, ε−1vm → 0 as ε → 0. Here our main interest is to study the interaction of the delta shock wavewith the other elementary waves and discover the new nonlinear phenomenon, thus for the convenience of computationwe assume that (um, vm) does not depend on ε in the present paper.Eqs. (1.1) are the simplest Temple type equations, i.e., the shock curves coincide with the rarefaction curves in the phase

plane due to the special form Yt + (uY )x = 0 [10]. The Riemann problem for these equations can be explicitly solvedin the large data and the wave interactions have a simplified structure. Yao and Sheng [11] considered the interaction ofelementary waves including the delta shock wave on a boundary for (1.1). In this paper, through investigating how the deltashockwave penetrates the rarefactionwave for (1.1), a new kind of nonlinear hyperbolic wave, a delta contact discontinuitycan be captured for (1.1), i.e., the delta function propagates along the line of the contact discontinuity and thereby thepropagation speed and strength do not change as it travels in space. The appearance of the delta contact discontinuitycan be understood more clearly and the interactions can be dealt with more easily than those in [12] for the reason that(1.1) belongs to Temple class. Thus, we take three pieces of constant initial data instead of the Riemann data, and then thesolutions beyond the interactions are constructed globally. The interaction results can also be used in a procedure similar tothe wavefront tracking approximation for general initial data. The strengths of delta shock waves are computed completelyand in detail during the process of the interaction. Furthermore, we prove that the solutions of the perturbed initial valueproblem (1.1) and (1.4) converge to the corresponding Riemann solutions of (1.1) and (1.2) as ε → 0, thus showing thestability of the Riemann solutions with respect to this local small perturbation of the Riemann initial data.Solving the initial value problem (1.1) and (1.4) gives rise to the interaction problem concerning delta shock waves with

other elementary waves, where the product of δ(x) andH(x) occurs. It is difficult to deal with the delta shockwave problem,especially an interaction including a delta shock wave; for the related references we can see [13–17]. Here we mainlyadopt the method of splitting the delta function along a regular curve in R2+ proposed by Nedeljkov [18]. By employingthis method, the interaction problems of the delta shock wave and singular shock wave with the classical waves werewidely investigated in [19,12,20] for the Keyfitz–Kranzer equations [21], themagnetohydrodynamics equations [22] and thesimplified transport equations recently. With the method of splitting of delta function, the product of the piecewise smoothfunction and discontinuity along such a curve makes sense and the differentiation is defined by mapping into the usualRadon measure space. To deal with the interaction of the delta shock wave with the rarefaction wave, we approximate therarefaction wave by a set of small non-admissible shocks like in the wavefront tracking algorithm proposed by Bressan [23].The paper is organized in the following way. In Section 2, some preliminaries are expressed, which include the Riemann

problem to (1.1) and the solution concept based on splitting of delta measures along a regular curve in R2+, in order to makethe treatment self-contained. In Section 3,mainly discuss the interactions of the delta shockwaveswith the shockwaves andrarefaction waves for four typical cases when the initial data are three piece constant states; the solutions are constructedglobally and the stability of the Riemann solutions is analyzed by letting ε tend to zero. Further discussions are made andconclusions are drawn in Section 4.

3286 C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294

2. Preliminaries

In this section, we firstly describe some results on the Riemann solutions to (1.1) which were obtained by Tan et al. [2].The eigenvalues of the Eqs. (1.1) are λ1 = u and λ2 = 2u and the corresponding right eigenvectors are

−→r1 = (0, 1)T and−→r2 = (1, v/u)T , respectively. It is noted that here λ1 < λ2 for u > 0 and λ1 > λ2 for u < 0. Thus, (1.1) is nonstrictlyhyperbolic, λ1 is linearly degenerate and λ2 is genuinely nonlinear.Besides the constant state, it can be found that the self-similar waves (u, v)(ξ) (ξ = x/t) of the first family are contact

discontinuities

J : ξ = ul = ur ,

and those of the second family are rarefaction waves

R : ξ = 2u,uv=ulvl, ul < ur ,

or shock waves

S : ξ = ul + ur ,urvr=ulvl, ul > ur > 0 or 0 > ul > ur ,

where the indices l and r denote left and right states respectively. All the waves J , R and S are called classical waves here.For the case u− ≥ 0 ≥ u+, a solution containing a weighted δ-measure supported on a line will be constructed. Like

in [24,17,25], we can consider a piecewise smooth solution of (1.1) in the form

(u, v)(x, t) =

(u−, v−), x < σ t,(uδ, β(t)δ(x− σ t)), x = σ t,(u+, v+), x > σ t,

(2.1)

where

σ = uδ = u− + u+, β(t) = (u−v+ − u+v−)t. (2.2)

In order to define themeasure solutions, the two-dimensional weighted δ-measure β(s)δΓ supported on a smooth curveΓ = (x(s), t(s)) : a < s < b can be defined by

〈β(s)δΓ , ψ(x, t)〉 =∫ b

aβ(s)ψ(x(s), t(s))ds, (2.3)

for any test function ψ(x, t) ∈ C∞0 (R× R+).With the above definition, a family of δ-measure solutions (u, v) of (1.1) and (1.2) with parameter σ in the case

u− ≥ 0 ≥ u+ can be expressed as

u(x, t) = u− + [u]H(x− σ t), v(x, t) = v− + [v]H(x− σ t)+ β(t)δΓ , (2.4)

in which Γ = (σ t, t) : 0 ≤ t < +∞, β(t) = (σ [v] − [uv])t and H(x) is the Heaviside function.The measure solution (2.1) with (2.2) satisfies the generalized Rankine–Hugoniot condition

dxdt= uδ(t),

dβ(t)dt= [v]uδ(t)− [uv],

[u]uδ(t) = [u2],

(2.5)

where [u] = u(x(t)+ 0)− u(x(t)− 0) denotes the jump of u across the discontinuity x = x(t), etc.In order to ensure uniqueness, the δ-entropy condition λ2r ≤ λ1r ≤ σ ≤ λ1l ≤ λ2l should be imposed, which means

that all characteristics on both sides of the δ-shock wave curve are incoming.The above constructed δ-measure solution (2.1) with (2.2) should satisfy

〈v, ψt〉 + 〈uv, ψx〉 = 0, (2.6)

for any ψ(x, t) ∈ C∞0 (R× R+), in which

〈v, ψ〉 =

∫∞

0

∫∞

−∞

v0ψdxdt + 〈β(t)δS, ψ〉, (2.7)

〈uv, ψ〉 =∫∞

0

∫∞

−∞

u0v0ψdxdt + 〈σβ(t)δS, ψ〉, (2.8)

here v0 = v− + [v]H(x− σ t) and u0v0 = u−v− + [uv]H(x− σ t).

C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294 3287

There are exactly six configurations of the Riemann solutions of (1.1) and (1.2) under the entropy conditions of S and δSas follows:

δS(u− ≥ 0 ≥ u+), J +−→S (u− > u+ > 0),

←−S + J(0 > u− > u+),

←−R +−→R (u− < 0 < u+),

←−R + J(u− < u+ ≤ 0), J +

−→R (0 ≤ u− < u+).

Now, we briefly recall the concept of left- and right-hand side delta functions which will be extensively used later and adetailed study of which can be found in [18].Let R2+ be divided into two disjoint open sets Ω1 and Ω2 with a piecewise smooth boundary curve Γ , which satis-

fies Ω1 ∩ Ω2 = ∅ and Ω1 ∪ Ω1 = R2+. Let C(Ωi) and M(Ωi) be the space of bounded and continuous real-valuedfunctions equipped with the L∞-norm and the space of measures on Ωi (i = 1, 2), respectively. Let us assume thatCΓ = (C(Ω1),C(Ω2)) andMΓ = (M(Ω1),M(Ω2)), then the product of G = (G1,G2) ∈ CΓ and D = (D1,D2) ∈ MΓ isdefined as an element GD = (G1D1,G2D2) ∈MΓ , where GiDi (i = 1, 2) can be defined as the usual product of a continuousfunction and a measure. Thus, it is obvious to see that the above-defined product makes sense.Everymeasure onΩi can be viewed as ameasure on R2+ with support inΩi. From this viewpoint, themappingm : MΓ →

M(R2+) can be obtained by takingm(D) = D1 + D2. Similarly, we havem(GD) = G1D1 + G2D2.For example, the delta function δ(x − γ (t)) ∈ M(R2+) along the piecewise smooth curve x = γ (t) can be split in a

non-unique way into a left-hand component D− ∈M(Ω1) and a right-hand one D+ ∈M(Ω2) such thatδ(x− γ (t)) = β−(t)D− + β+(t)D+,β−(t)+ β+(t) = 1.

(2.9)

The solution concept used in this paper can be described as follows: perform all nonlinear operations on functions in thespaceCΓ , carry out themultiplication and composition in the spaceMΓ and then take themappingm : MΓ → M(R2+) beforedifferentiation in the space of distributions, and require that the equation is satisfied in the weak sense of distribution.Now we employ the definition of split delta function as above to rewrite the solutions (2.1) and (2.2) for the case

u− ≥ 0 ≥ u+. The solution can be rewritten asu(x, t) = u− + (u+ − u−)H,v(x, t) = v− + (v+ − v−)H + β−(t)D− + β+(t)D+,

(2.10)

where H is the Heaviside function, all the H , D− and D+ are supported by the line x = σ t = (u− + u+)t .According to the above definition, we have

vt(x, t) = (−σ(v+ − v−)+ β ′−(t)+ β′

+(t))δ − σ(β−(t)+ β+(t))δ′, (2.11)

(uv)x(x, t) = (u+v+ − u−v−)δ + (u−β−(t)+ u+β+(t))δ′. (2.12)

Substituting (2.11) and (2.12) into the second equation of (1.1) and comparing the coefficients of δ and δ′, we can obtain

β−(t) =u−β(t)u− − u+

, β+(t) =−u+β(t)u− − u+

. (2.13)

3. Interactions of delta shock waves with classical waves

It is well known that the delta shock wave occurs in the Riemann solution of (1.1) for certain Riemann initial data. Inorder to cover all the cases completely, our discussion should be divided into two parts according to the appearance of thedelta shock wave or not. When the delta shock wave does not appear in the interaction process, the problem is classical andwell known, hence it will not be pursued here. Otherwise, we have seven possibilities for an interactionwith the delta shockwave to occur, according to the different combinations from (−ε, 0) and (ε, 0) as follows:

(1) δS and←−S + J; (2) δS and R; (3) δS and

←−R + J; (4) J +

−→S and δS; (5) R and δS; (6) J +

−→R and δS; (7) δS and δS.

Case 1. δS and←−S + J .

First, we consider the interaction of the delta shock wave δS emanating from (−ε, 0)with the backward shock wave←−S

plus the contact discontinuity J emanating from (ε, 0). The occurrence of this case depends on the condition u− ≥ 0 >um > u+ (see Fig. 1).The propagating speed of the delta shock wave δS1 is σ1 = u− + um and that of the shock wave is τ = um + u+. Thus, it

is easy to see that δS1 will overtake S at a finite time. The intersection (x1, t1) is determined byx1 + ε = (u− + um)t1,x1 − ε = (um + u+)t1,

(3.1)

3288 C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294

Fig. 1. u− ≥ 0 > um > u+ .

which implies that

(x1, t1) =(ε(u− + 2um + u+)

u− − u+,

2εu− − u+

). (3.2)

The new initial data will be formulated at the intersection (x1, t1) as follows:

u|t=t1 =u−, x < x1,u∗, x > x1,

v|t=t1 =

v−, x < x1v∗, x > x1

+ β(t1)δ(x1,t1). (3.3)

Here the intermediate state between S and J can be easily obtained by (u∗, v∗) = (u+,u+vmum) and β(t1) denotes the strength

of δS1 at the time t1 and can be calculated by

β(t1) = (u−vm − umv−)t1. (3.4)

A new delta shock wave will be generated after the interaction of δS1 and S; let us denote it by δS2 here. Before δS2meetsthe contact discontinuity J , the solution can be expressed as

u(x, t) = u− + (u∗ − u−)H,v(x, t) = v− + (v∗ − v−)H + β−(t)D− + β+(t)D+,

(3.5)

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 supported bythe line x = x1 + (t − t1)σ2, i.e., they are the functions of x− x1 − (t − t1)σ2; here σ2 = u− + u+ is the propagation speedof δS2. Although they are supported by the same line, D− is the delta measure on the set R2+ ∩ (x, t)|x ≤ x1 + (t − t1)σ2and D+ is the delta measure on the set R2+ ∩ (x, t)|x ≥ x1 + (t − t1)σ2 respectively.From (3.5), we can compute

vt(x, t) = (−σ2(v∗ − v−)+ β ′−(t)+ β′

+(t))δ − σ2(β−(t)+ β+(t))δ′, (3.6)

(uv)x(x, t) = (u∗v∗ − u−v−)δ + (u−β−(t)+ u∗β+(t))δ′. (3.7)

Substituting (3.6) and (3.7) into the second equation of (1.1) and comparing the coefficients of δ and δ′, it leads to

−σ2(v∗ − v−)+ β′

−(t)+ β ′

+(t)+ u∗v∗ − u−v− = 0, (3.8)

−σ2(β−(t)+ β+(t))+ u−β−(t)+ u∗β+(t) = 0. (3.9)

Noting the initial condition (3.4), it can be easily derived from (3.8) that

β(t) = β−(t)+ β+(t) = β(t1)+u+um(u−vm − umv−)(t − t1), (3.10)

which is the strength of δS2 after the interaction of δS1 and S and before the occurrence of the interaction of δS2 and J .Obviously, β−(t) =

u−u−−u+

β(t) and β+(t) =−u+u−−u+

β(t) can be derived from (3.9) and (3.10).Then, δS2 and J will intersect at the point (x2, t2)which can be calculated by

x2 − x1 = (u− + u+)(t2 − t1),x2 − ε = u+t2.

(3.11)

An easy calculation leads to

(x2, t2) =(ε +

2εu+(u− − um)u−(u− − u+)

,2ε(u− − um)u−(u− − u+)

). (3.12)

C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294 3289

After the time t2, the delta shock wave will pass through J with the same speed as before, only the added rate of itsstrength changes due to the difference between v∗ and v+. Thus we still denote it with δS2 after time t2 and the strengthcan be calculated by

β(t) = β(t2)+ (u−v+ − u+v−)(t − t2), (3.13)

in which β(t2) can be calculated by (3.10).It is easy to see that both (x1, t1) and (x2, t2) tend to (0, 0) as ε→ 0 from (3.2) and (3.12), moreover we have β(t2)→ 0

as ε→ 0 from (3.10). Thus, the limit of the solution of (1.1) and (1.4) is still a single delta shock wave, which is exactly thecorresponding Riemann solution of (1.1) and (1.2) in this case.

Remark 1. The situation is similar and the final result after interaction is also a single delta shock wave when we considerthe interaction of the contact discontinuity J followed by the forward shock wave

−→S issuing from (−ε, 0) and the delta

shock wave δS emitting from (ε, 0). This case occurs if the condition u− > um > 0 ≥ u+ is satisfied.

Case 2. δS and R.In this case, we consider the interaction of the delta shock wave δS1 starting from (−ε, 0) and the rarefaction wave R

starting from (ε, 0). This case happens if and only if u− ≥ 0 ≥ um and u+ ≥ 0 ≥ um. The propagating speed of the deltashock wave δS1 is σ1 = u− + um and the wave back in the rarefaction wave propagates with speed 2um. Thus, it is easy tosee that δS1 and Rwill intersect at a finite time. The intersection (x1, t1) is determined by

x1 + ε = (u− + um)t1,x1 − ε = 2umt1,

(3.14)

which means that

(x1, t1) =(ε(u− + 3um)u− − um

,2ε

u− − um

). (3.15)

The strength of δS1 at (x1, t1) can be calculated by

β(t1) = (u−vm − umv−)t1 =2ε(u−vm − umv−)

u− − um. (3.16)

The delta shock wave will enter the rarefaction wave fan when the interaction of δS1 and R begins to occur. At thesame time, a new delta shock wave δS2 is generated and propagates forwards until it meets the line x = ε. Here we useΓ : x = x(t) to express the curve of δS2 with values (u−, v−) on the left-hand side and ( x−ε2t ,

vm(x−ε)2umt

) on the right-handside. The propagation speed of δS2 can be determined by the Rankine–Hugoniot condition of the first equation in (1.1) andthe initial condition (3.15) as follows

σ2(t) =dxdt= u− +

x− ε2t

, x(t1) = x1. (3.17)

The unique solution can be written as

x(t) = 2u−t − 2√2ε(u− − um)t + ε, t ≥ t1. (3.18)

From (3.17) and (3.18), it is easy to see that

d2xdt2=

√ε(u− − um)2t3

> 0, (3.19)

which means that δS2 begins to accelerate and is no long a straight line after the interaction of δS1 and R occurs and beforeit meets the line x = ε.Now, we construct a delta shock wave supported on the curve Γ : x(t) = 2u−t − 2

√2ε(u− − um)t + ε as follows:

u(x, t) =

u−, x < x(t),x− ε2t

, x > x(t), (3.20)

v(x, t) =

v−, x < x(t)vm(x− ε)2umt

, x > x(t)

+ β−(t)D−Γ + β+(t)D+Γ , (3.21)

where β(t)DΓ = β−(t)D−Γ + β+(t)D+

Γ is a split delta function supported on Γ and β(t) = β−(t)+ β+(t) is the strength ofδS2 at the time t .

3290 C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294

Substituting (3.20) and (3.21) into (1.1) yields

−σ2(t)(vm(x− ε)2umt

− v−

)+ β ′

−(t)+ β ′

+(t)+

(vm

um

(x− ε2t

)2− u−v−

)= 0, (3.22)

−σ2(t)(β−(t)+ β+(t))+ u−β−(t)+x− ε2t· β+(t) = 0. (3.23)

Noting the initial condition (3.16), taking account into the relations (3.17) and (3.18), from (3.22), one can easily see that

β(t) = β−(t)+ β+(t) = β(t1)+u−vm − umv−

um

(u−(t − t1)− 2

√2ε(u− − um)(t − t1)

), (3.24)

for t ≥ t1 in the rarefaction wave fan. Moreover we have β−(t) = x−εx−ε+2u−t

β(t) and β+(t) =2u−t

x−ε+2u−tβ(t) from (3.23)

and (3.24).In the following we will see that the delta shock wave supported by Γ is an overcompressive wave only up to the point

where the curve Γ meets the line x = ε, for the propagation wave speed of R is greater than zero and the value of (u, v) in Ris calculated differently from before on the right-hand side of the line x = ε. The interaction point (x2, t2) = (ε,

2ε(u−−um)u2−

)

can be easily obtained from (3.18) and the strength of δS2 now becomes β(t2)which can be calculated by (3.24).At the time t = t2, we again have a new local Riemann problem with the initial data

u|t=t2 =u−, x < ε,η, x > ε,

v|t=t2 =

v−, x < εv+η

u+, x > ε

+ β(t2)δ(x2,t2), (3.25)

where we suppose that η = x−ε2t > 0 is sufficiently small, i.e., we still assume that the rarefaction wave R is approximated

by a set of non-physical shock waves. Let us notice that the rarefaction wave R is forwards on the right-hand side of the linex = ε, namely the value (u, v) in the rarefaction wave R now is determined by the right state (u+, v+)with uv =

u+v+.

In fact, we can construct the solution of the initial value problem (1.1) and (3.25) in the form:

u(x, t) =u−, x < x(t),η, x > x(t), (3.26)

v(x, t) =

v−, x < ε + u−(t − t2)u−v+u+

, ε + u−(t − t2) < x < x(t)

v+η

u+, x > x(t)

+ β(t2)δ(x− ε − u−(t − t2)), (3.27)

where x = x(t) is the shock wave curve and can be expressed as x = ε + (u− + η)(t − t2) in the local neighborhood of(x2, t2) for a given sufficiently small η > 0.Now we prove that (3.26) and (3.27) is indeed the weak solution of the initial value problem (1.1) and (3.25). For every

ϕ ∈ C∞0 (R× R+), it is obviously true that〈ut + (u2)x, ϕ〉 = 0,〈vt + (uv)x, ϕ〉 = 0,

(3.28)

if suppϕ ∩ (x, t)|x = ε + u−(t − t2), t > t2 = ∅. Otherwise we prove that (3.26) and (3.27) are still the weak solutionof (1.1) and (3.25) near the support of the delta function. The first equation of (1.1) does not contain v, so it is still satisfied.Substituting (3.26) and (3.27) into the second equation of (1.1), we have

vt + (uv)x = −u−

(u−v+u+− v−

)δ − u−β(t2)δ′ +

(u−u−v+u+− u−v−

)δ + u−β(t2)δ′ = 0, (3.29)

which shows that (1.1) is also satisfied near the line x = ε + u−(t − t2) in the weak (or distributional) sense.The type of admissible solutions known so far are not enough to explain the solution (3.26) and (3.27). In order to deal

with it, the delta contact discontinuity should be introduced, as in [12], as follows.

Definition 3.1. Let Ω be a region where u is a continuous function and the curve Γ in Ω of slope λ1 = u. A pair ofdistributions (u, v) ∈ C(Ω) × D′(Ω) is called a delta contact discontinuity, if u is a weak solution of the first equationin (1.1) and v is a sum of a locally integrable function onΩ and a delta function on Γ which solves the second equation in(1.1) in the sense of distribution.

C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294 3291

Fig. 2a. u+ > u− ≥ 0 ≥ um .

Solving the Riemann problem at (x2, t2), we can see the appearance of a delta contact discontinuity δJ and a shock waveS, i.e., the Dirac delta function is now supported on the contact discontinuity. Namely, when t > t2, δS2 decomposes and thestate (u∗, v∗) = (u−,

u−v+u+

) lies between δJ and S.The delta contact discontinuity δJ will continue to move forwards with a constant speed u− and the invariant strength

β(t2). This is due to the fact that both sides of the line x = ε+u−(t−t2) have the same speed u− and the over-compressibilitycondition is now lost.The shock wave curve S will continue to penetrate the rarefaction wave through the continuation of the curve Γ , for the

speed of S is only determined by the value u. We notice that the left-hand side value of u is still u− and the right-hand sidevalue of u is also x−ε2t . This implies that the shock wave curve should also satisfy the differential equation (3.17), moreoverwe notice that its beginning point (x2, t2) lies in the curve Γ . Thus its solution has the same representation as (3.18) whent ≥ t2, i.e., the shock wave curve is the continuation of the curve Γ . Based on the comparison between the values u− andu+, we should divide our discussion into the following three subcases.Subcase 2.1. u− < u+.Then the shockwave curve S can not penetrate the rarefactionwave fan R completely and below the line x(t) = 2u−t+ε

when u− < u+ (see Fig. 2a). As ε → 0, one can see that the four points (−ε, 0), (ε, 0), (x1, t1) and (x2, t2) coincide witheach other at the point (0, 0), moreover we have β(t2)→ 0. Thus, we can see that the limit of the solution of (1.1) and (1.4)is the contact discontinuity J and the rarefaction wave R plus the intermediate state (u∗, v∗) = (u−,

u−v+u+

) between them,which is exactly the corresponding Riemann solution of (1.1) and (1.2) in this subcase.Subcase 2.2. u− > u+.In this subcase, the curve of S will penetrate over the whole rarefaction wave fan at (x3, t3) (see Fig. 2b), the intersection

can be calculated byx3 = 2u−t3 − 2

√2ε(u− − um)t3 + ε,

x3 − ε = 2u+t3,(3.30)

i.e.,

(x3, t3) =(4εu+(u− − um)(u− − u+)2

+ ε,2ε(u− − um)(u− − u+)2

). (3.31)

After the time t3, the shock wave S propagates with an invariant speed u− + u+. Letting ε → 0, one can easily see thatthe limit of the solution of (1.1) and (1.4) is the contact discontinuity J : x = u−t and the shock wave S : x = (u− + u+)t ,which is exactly the corresponding Riemann solution of (1.1) and (1.2). Thus, the limit situation is also true for our assertionin this subcase.Subcase 2.3. u− = u+.For this special subcase, the curve of S is below the line x = ε + 2u−t , which is exactly the wave front of the rarefaction

wave R. As ε → 0, the limit is a contact discontinuity J : x = u−t connecting (u±, v±) directly, and the conclusion isobviously identical with our assertion.

Remark 2. The analysis and computation is similar when we consider the interaction of the rarefaction wave R emanatingfrom (−ε, 0) and the delta shock wave δS emanating from (ε, 0). This case occurs when um ≥ 0 ≥ u− and um ≥ 0 ≥ u+hold.

3292 C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294

Fig. 2b. u− > u+ ≥ 0 ≥ um .

Fig. 3. u− ≥ 0 ≥ u+ > um .

Case 3. δS and←−R + J .

In this case, we investigate the interaction of the delta shockwave δS1 starting from (−ε, 0) and the backward rarefactionwave

←−R followed by the contact discontinuity J starting from (ε, 0) (see Fig. 3). The intermediate state (u∗, v∗) = (u+,

u+vmum)

is between R and J . This case arises when u− ≥ 0 ≥ u+ > um.Like for Case 2, the interaction of δS1 and R results in a new delta shock wave δS2 when they meet at (x1, t1) which has

the same representation as (3.15). The strength β(t1) and the curve of δS2 are also the same as Case 2. But we notice thatu∗ = u+ < 0 ≤ u− here, thus δS2 will penetrate over R in a finite time. The terminal point (x2, t2) is also identical with(3.31) in subcase 2.2 due to the same expression of the curve Γ . The strength β(t2) can be calculated by the formula (3.24).When δS2 passes through (x2, t2), it will propagate with an invariant speed u− + u+, and we denote it by δS3. After thetime t2, the situation is now like Case 1, i.e., δS3 will pass through J with the same speed, just the strength varies due to thedifference between v∗ and v+.It is obvious to see that the limit of the solution of (1.1) and (1.4) is just the delta shock wave. Similarly to the analysis

in Case 1, the Riemann solution of (1.1) and (1.2) is also stable with respect to the local perturbation of the Riemann initialdata in this case.

Remark 3. The interaction situation is similarwhen the contact discontinuity J followed by the forward rarefactionwave−→R

emit from (−ε, 0) and the delta shock wave δS emanates from (ε, 0). The occurrence of this case depends on the conditionum > u− ≥ 0 ≥ u+.

Case 4. δS and δS.In the end, let us focus attention on the interaction of two delta shockwaves starting from (−ε, 0) and (ε, 0) respectively.

The occurrence of this case needs the special condition u− > um = 0 > u+ (see Fig. 4), namely, the intermediate statebetween (−ε, 0) and (ε, 0) is at rest now.The propagating speeds of δS1 and δS2 are σ1 = u− and σ2 = u+, respectively. Thus, it is easy to see that δS1 will

overtake δS2 at a finite time. The intersection of δS1 and δS2 can be calculated by (x1, t1) = (ε(u−+u+)u−−u+

, 2εu−−u+

) and thestrength at (x1, t1) is β(t1) = 2εvm, which is exactly the strength of δS1 together with that of δS2 at the time t = t1. Now,the new initial value problem is formulated at (x1, t1), which can be dealt with similarly to Case 1. A new delta shock waveδS3 will be generated after the coalescence of δS1 and δS2 at (x1, t1), whose speed and strength are σ3 = u− + u+ andβ(t) = β(t1)+ (u−v+ − u+v−)(t − t1).

C. Shen, M. Sun / Nonlinear Analysis 73 (2010) 3284–3294 3293

Fig. 4. u− > um = 0 > u+ .

Hence the result of interaction of two delta shock waves is still a single delta shock wave. It is clear that β(t1)→ 0 andt1 → 0 as ε → 0, i.e., the limit of the solution of (1.1) and (1.4) is the corresponding Riemann solution of (1.1) and (1.2) inthis special case.

4. Discussions and conclusions

In order to prolong the admissible solution after an interaction point at which a delta shockwave loses its overcompress-ibility, a new type of nonlinear wave, namely the delta contact discontinuity will be introduced as an reasonable generaliza-tion of the classical waves and delta shock wave known before. If we consider the Riemann initial data containing the Diracdelta function in a particular form as u(x, 0) = c (i.e. u− = u+) and v(x, 0) = v− + (v+ − v−)H(x)+mδ(0, 0), it is easy tosee that the Dirac function will propagate along the contact discontinuity line x = ct .By letting ε→ 0, it is easy to see that the Riemann solutions are stable under a local small perturbation of the Riemann

initial data (1.2) in the above typical four cases when the delta shock wave is included. Otherwise, when the delta shockwave is not involved, the conclusion is obviously true and can be dealt with by a similar procedure. So we have finished thediscussion for all kinds of interactions, and the global solutions for the perturbed initial value problem (1.1) and (1.4) havebeen constructed globally. In a short, we can summarize our results in the following theorem.

Theorem 4.1. The limits of the perturbed Riemann solutions of (1.1) and (1.4) are exactly the corresponding Riemann solutionsof (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±). Thus, we can draw the conclusion that the Riemann solutions of (1.1) and (1.2) are stable with respect to such a localsmall perturbation (1.4) of the Riemann initial data (1.2).

Acknowledgements

The authors are grateful to the two anonymous referees for pointing out unclear statements and improving thepresentation of this paper.

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