Global Stability of Multi-wave Configurations for the Compressible Non-isentropic Euler System?

Min DING

Abstract This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈(1,3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, consisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.

Keywords Structural stability, Multi-wave configuration, Shock, Contact discontinuity, Compressible non-isentropic Euler system, Wave front tracking method

1 Introduction

As is well known, Glimm has proved a global existence of weak solutions for strict hyperbolic conservation laws when the total variation of the initial data is sufficiently small via Glimm scheme in [7]. Bressan has established global existence, uniqueness of solutions to one-dimensional Cauchy problem for the general hyperbolic conservation laws when the total variation of the initial data is sufficiently small by wave front tracking method, and also proved the continuous dependence on the initial data in book [1]. In early works, Chern [4] initially studied Cauchy problem for general hyperbolic conservation laws, and proved the stability of a single large shock wave by Glimm scheme and wave front tracking method. Schochet [16]has shown the BV stability of the multi-wave configurations (a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave) for the non-isentropic gas dynamics for≈1.015, while forγ∈(1,γ0),there exist Riemann problems for which BV stability condition fails. Lewicka [11] solved the well-posedness of the solutions for Cauchy problem to the general hyperbolic conservation laws when the initial data is a small perturbation of wave patterns of large non-interacting waves. The result in [11] includes the following facets:

1. If the background wave patternU(x,t) satisfies the finite conditions, then Riemann problem with initial data close toU(x,0) admits a self-similar solution in the vicinity of the stateU(x,t).

2. If BV stability condition is satisfied, and the initial data is under a small perturbation ofU(x,0) with possibly large data, then Cauchy problem of general hyperbolic conservation laws admits a global entropy admissible solution. Section 8 has applied the general framework to non-isentropic Euler equations, and achieved the BV stability of wave patterns consisting of a strong 1-shock,strong 2-contact discontinuity,and strong 3-shock wave forγ∈[1.05576,8.7577].Meanwhile,L1stability condition holds forγ∈[1.05576,8.7577],then there exists a Lipschitz continuous semigroup of global entropy admissible solutions.

In short, comparing with the previous references, Schochet [16] was the first to introduce the finite condition, and formulate the stability ofMstrong shocks, 2 ≤M≤n, by means of matrix analysis and Glimm scheme. Bressan and Colombo [2] considered the general Riemann problem for systems of two equations and derived the correspondingL1stability condition of the large solutions. Lewicka [12] proved the BV andL1stability conditions for non-interacting two large shocks of general conservation laws. Later, Lewicka [10] has shown that BV stability conditions in [9] is equivalent to Schochet BV finite condition, as well as the equivalent ofL1stability condition from [9] with the one introduced in [2] for 2×2 system. For a single strong rarefaction wave, the stability of a strong rarefaction wave to Cauchy problem for the general hyperbolic conservation laws has been proved by Lewicka in [14], also see [13]. The structural stability of steady four-wave configurations for two-dimensional steady supersonic Euler flow has been established in Chen-Rigby [3] by Glimm scheme.

Remark 1.1Here strong or large wave means the strength of the wave is not sufficiently small, or else, it is a weak wave.

In this paper, we are concerned with the BV stability of the multi-wave configurations of Cauchy problem to the compressible non-isentropic Euler equations for 1<γ≤3, and to some extent, it fixes up the result of BV stability of such system forγ∈(1,γ0). To overcome the difficulties,some new nonlinear weights have been introduced and assigned to each perturbation wave, the total amount of weighted perturbation waves decreases at each interaction with any of the strong waves.

The system composes of the conservation laws of mass, momentum and energy which can be read as (see [5, 15])

whereρ,pandustand for the density, the pressure and the speed of the fluid, respectively,andeis the internal energy. The constitutive relations for the polytropic gas are given by

whereSrepresents the entropy, andκ,cvare positive constants, the adiabatic exponentγ∈(1,3].

For simplicity, system (1.1) can be written in the general conservation law form:

whereU=(ρ,u,p)?, and

By solving the polynomial det(λ?UW(U)??UH(U))=0, the eigenvalues of system (1.2)are respectively

wherec=is the local sonic speed and the corresponding right eigenvectors are

whererjis normalized by

The entropy-entropy flux pair (η,q)(U) of system (1.1) is a pair ofC1functions satisfying

In particular, if

thenη(U) is called a convex entropy.

We consider an unperturbed multi-wave configuration, called the background solution denoted byU, consisting of the four constant states (see Figure 1)

Figure 1 Background solution.

where the constant stateU1connects to the stateU2by a strong 1-shock with the speeds10,and the stateU2is separated from the stateU3by a strong 2-contact discontinuity with the strength |σ20|, and the stateU3joins to the stateU4by a strong 3-shock wave with the speeds30, satisfying (2.9)—(2.11).

Suppose that the initial data are given by

which is a small BV perturbation of the stateU(x,0).

Next, we define the entropy solutions to problem (1.2) and (1.4) as follows.

Definition 1.1A measurable function U(x,t)∈L∞(R2+)is an entropy solution to problem(1.2)and(1.4), provided that

(i)U(x,t)is a weak solution to problem(1.2)and(1.4), if for any φ(x,t) ∈C∞c(R2+), it holds that

(ii)The Clausius inequality holds in the sense of distributions:

for any a(S)∈C1and a′(S)≥0.

The main result of this paper is given by the following theorem.

Theorem 1.1There are some positive constants ε>0and C >0such that if

then there exists a global existence of entropy solution U(x,t)to problem(1.1)and(1.4), including a strong1-shock wave, a strong2-contact discontinuity, and a strong3-shock wave,which is a small perturbation of solution(1.3). In addition, for all t>0, there exists a positive constant M0such that

Moreover, denote the curves of the strong1-shock wave, the strong2-contact discontinuity, and the strong3-shock wave by x=χi(t), i=1,2,3, respectively. Then, it satisfies

Our motivation is to study the structural stability of multi-wave configurations to Cauchy problem of the compressible non-isentropic Euler system(1.1)under BV perturbation of initial data. Different from the results involving a single strong wave (shock, rarefaction wave or contact discontinuity),we not only need trace the locations of the strong waves,but also control the change of the strengths for the strong waves after each wave interaction with weak waves from left and right. In account of all the possible local wave interactions, we introduce some weights for the approaching waves,and construct a new Glimm functional, which measures the difference between the total variations of the approximate solutions and background solutions.We observe that the weak 1-waves collide with the strong 1-shock waves from the right, that the weak 3-waves interact with the strong 2-contact discontinuities from left, that the weak 1-waves collide with the strong 2-contact discontinuities from right and that the weak 3-waves interact with the strong 3-shock waves from left, among which the reflection coefficients are less than 1, which is essential for proving the monotone decreasing of Glimm functional.

This paper is organized as follows. In Section 2,we recall some basic properties of elementary waves (shock, rarefaction waves and contact discontinuities), and give the solvability of the Riemann problem for system (1.1), which is discussed in four cases. In Section 3, we construct the approximate solutions to the Cauchy problem by wave front tracking method. In Section 4, we consider all the local wave interaction estimates between weak waves, their reflections on the strong 1-shock waves, strong 2-contact discontinuities and strong 3-shock waves, and so on. In Section 5, we introduce some weighted strengths for the approaching waves, construct a new Glimm functional, and then prove the monotone decreasing of the functional. In Section 6, we derive some further estimates to show that the total strengths of strong wave fronts are bounded. The compactness and the convergence of the approximate solutions follow from the standard procedure.

2 Riemann Solutions

In this section, we study the Riemann problems and analyze the properties of the Riemann solutions to system(1.1), which are essential not only for the interaction estimates between the weak waves,but also for those involving the strong shock waves or strong contact discontinuities,etc..

Consider the Riemann problem of (1.1) with initial data

whereUL= (ρL,uL,pL)?andUR= (ρR,uR,pR)?represent the left and right states, respectively. The solvability of the Riemann problem can be found in [8, 17] when |UL?UR| is sufficiently small.

For any given left stateUl, the set of all possible statesUcan be connected toUlon the right by a 1 or 3-shock wave,the wave curves of which can be denoted byS1orS3, respectively.Similarly, we denote byR1orR3the 1 or 3-rarefaction wave curves. The rarefaction wave curveR1(Ul) throughUlsatisfies

Similarly, the rarefaction wave curveR3(Ul) throughUlis given by

The second characteristic field is linearly degenerate satisfying ?λ2·r2≡0. The 2-contact discontinuity throughUlsatisfies

The Rankine-Hugoniot conditions across the shock are given by

where [h] =h?hlstands for the jump of functionhacross the shock, andsis the speed of the shock. Therefore, eliminatingsfrom (2.4)—(2.6), the shock wave curves throughUlcan be respectively parameterized by

and

whereβ=

In addition, the Lax entropy conditions across the shock are

2.1 Background solution

In this subsection, we will present the unperturbed multi-wave configurations,consisting of a strong 1-shock,a strong 2-contact discontinuity and a strong 3-shock as the Riemann solutions for the compressible full Euler equations(1.1),called background solutionsUcomposing of four constant states shown in (1.3). From the parameterizations of the nonlinear elementary waves,it holds that

(i) Along the strong 1-shock wave curve, it satisfies that

(ii) The second characteristic family is linearly degenerate,and the strong 2-contact discontinuity throughU2is given by

(iii) Along the strong 3-shock wave curve, it satisfies that

Due to some BV perturbations of the initial state, we will take into account of four types of solvers to the Riemann problem (1.1) and (2.1).

2.2 Riemann problem involving only weak waves

In this subsection,we give the solvability of problem(1.1)and(2.1). As shown in[17],when|UL?UR| ?1, we can parameterize physically admissible wave curves in a neighborhood ofUk(k=1,2,3,4) byC2curves:αi→Φi(UL,αi) satisfying

which represents the left stateULand the right stateURcan be connected by a 1-waveα1, a 2-waveα2and a 3-waveα3. Moreover, it holds that

In addition,αi>0 along the rarefaction wave curveRi(UL), whileαi<0 along the shock wave curveSi(UL).

Hereinafter, we denote byαi,βi,γithe parameters of the correspondingi-waves,i=1,2,3,while by their absolute values the corresponding strengths of the waves. We also use the parameters to represent thei-waves provided no confusion occurs. In the sequel, we parameterize the strong shock by its velocitys, and ˙U(s) denotes the derivative ofUwith respect tosalong the strong shock wave curve. For convenience, letA(U,s)=?UH(U)?s?UW(U). Oεstands for a small neighbourhood, which will be used frequently later.

2.3 Riemann problem involving only a strong 1-shock wave

In this subsection,when|UL?UR|is not sufficiently small,we consider the Riemann problem(1.1)and(2.1),whereUL∈Oε(U1)andUR∈Oε(U2). The solvability of this Riemann problem can be given by the following lemma.

Lemma 2.1For any UL∈Oε(U1)and UR∈Oε(U2), there exists a strong1-shock wave,joining the left state ULto the right state URwith the speed s1. Moreover, s1∈Oε(s10).

ProofFrom Rankine-Hugoniot conditions (2.4)—(2.6), we have

Differentiating (2.13) with respect tos1, we can obtain that

By direct calculations, it holds that

From the Lax entropy conditions (2.7)—(2.8), we have detA|U=U2,s1=s10<0. We complete the proof of this lemma by the aid of the implicit function theorem.

The following lemma is important to estimate the strengths of the weak waves reflected on the strong 1-shock waves, and to estimate the changes to the strengths of the strong 1-shock waves (see the proofs of Lemmas 4.2—4.3).

Lemma 2.2It holds that

ProofFrom Lemma 2.1, we can obtain that detA<0. By direct calculations, one has

Then, it yields that

and

From the Rankine-Hugoniot conditions (2.4)—(2.5), one has

Therefore, we can derive that

Sinces

Hence, we complete the proof of this lemma.

2.4 Riemann problem involving only a strong 2-contact discontinuity

When |UL?UR| is not sufficiently small, whereUL∈Oε(U2) andUR∈Oε(U3), the solvability of (1.1) and (2.1) can be formulated in the following lemma.

Lemma 2.3For any given UL∈Oε(U2)and UR∈Oε(U3), there exists a strong2-contact discontinuity connecting the state ULto the state URwith strength|σ2|. Moreover, it satisfies that

This lemma can be easily proved by direct calculations. We omit the proof here.

2.5 Riemann problem involving only a strong 3-shock wave

In this subsection, we consider the Riemann problem (1.1) and (2.1), whereUL∈Oε(U3),andUR∈Oε(U4). The solvability of Riemann problem can be given by the following lemma.

Lemma 2.4For any UL∈Oε(U3)and UR∈Oε(U4), there exists a strong3-shock wave,separating the left state ULfrom the right state URwith speed s3. Moreover, s3∈Oε(s30).

Similar to Lemma 2.1,we can prove this lemma by the implicit function theorem. Thus, we omit the details here.

In the following,we present some properties of the strong 3-shock waves,which are essential to estimate the strengths of the weak waves reflected on the strong 3-shock waves, and to estimate the changes of the strengths of the strong 3-shock waves (see the proofs of Lemmas 4.6—4.7).

Lemma 2.5The following statements hold

ProofOne can calculate directly to obtain that

From the Lax entropy conditions (2.7) and (2.8), detA|U=U4,s=s30<0. Meanwhile,

Thus, the proof of this lemma is completed.

3 Approximate Solutions

In this section, we use the Riemann problem as building blocks to construct approximate solutions of Cauchy problem (1.1) and (1.4) by a modified wave front tracking scheme. First,we consider the solvability of Riemann problem (1.1) and (2.1).

As mentioned in Section 2,the solution to the Riemann problem(1.1)and(2.1)is composed of at most four constant states connected by shocks,rarefaction waves or contact discontinuities.By the wave front tracking method,there are two types of Riemann solvers to solve this Riemann problem.

Case 1. Accurate Riemann solver.

The accurate Riemann solver is given in Section 2, except that every rarefaction waveRi(i=1,3) is divided intoνequal parts.

Suppose that the left stateULand the middle stateUMare connected by a 1-rarefaction waveα1. Ifα1>0, then letU0,0=UL,U0,ν=UM.For any 1 ≤k≤ν,

Thus, the 1-rarefaction wave is replaced by

wherex1,k=x0+(t?t0)λ1(U0,k) andλ?1∈(max(x,t)λ1(U),min(x,t)λ2(U)).

Similarly, we can approximate 3-rarefaction wave byν3-rarefaction wave fronts in the domain {(x,t):x>x0+λ?2(t?t0)}, whereλ?2∈(max(x,t)λ2(U),min(x,t)λ3(U)).

Case 2. Simplified Riemann solver.

In order to keep the number of the wave fronts be finite for allt≥0, the simplified Riemann solver is introduced. Exactly speaking, an auxiliary wave, called a non-physical wave, is constructed with a constant speed, which is strictly larger than all the characteristic speeds of system (1.1). The strength of the non-physical wave measures the error of the simplified Riemann solver. It occurs in the following two cases:

Case a. Aj-waveβjand ani-waveαiinteract at (x0,t0), 1 ≤i≤j≤3. Suppose thatUL,UMandURare three constant states, satisfying

The auxiliary state is constructed by

then, the simplified Riemann solverUS(UL,UR)to Riemann problem(1.1)and(2.1)at(x0,t0)is given by

whereUνA(UL,R) is constructed by the accurate Riemann solver as shown in case 1. The non-physical wave can be defined by

whose strength is |UR?R|.

Case b. A non-physical waveεnpcollides with a weaki-wave frontαi(i=1,2,3) from left at the point (x0,t0). Suppose that the three statesUL,UMandURsatisfy

Then, the simplified Riemann solverUS(UL,UR) to problem (1.1) and (2.1) is defined by

3.1 Wave front tracking algorithm

The wave front tracking algorithm to construct the approximate solutions is given by:

Case 1. There are no more than two wave fronts interacting at one point by changing the speed of a single wave front with a quantityO(1)2?ν.

Case 2.If two wave frontsαiandβjinteract,then the generated Riemann problem is solved by the following rules.

Rule 1. If |αiβj|>μνand both are physical, whereμνis a fixed small parameter satisfyingμν→0 asν→+∞, then the accurate Riemann solver is adopted.

Rule 2. If |αiβj|<μνand either both are physical, or one of them is a non-physical wave,then the simplified Riemann solver is adopted.

Case 3. If a weak wave collides with the strong wave fronts, then the accurate Riemann solver is adopted.

Letτkbe the time when two wave fronts interact for thek-th time,k≥1. For any sufficiently largeν∈N, we can construct aν-approximate solutionUν(x,t), and assign each wave front with a generation order inductively as follows:

Step 1. For 0 ≤t < τ1, suppose thatUν(x,t) can be constructed by accurate Riemann solver to solve a series of Riemann problems, which can be carried out as shown in Section 2.All the wave fronts generated from Riemann problems att=0 have generation order 1.

Step 2. By induction, assume that the approximate solutionUν(x,t) has been constructed fort < τk, and thatUν|t<τkconsists of a finite number of wave fronts. As shown in Sections 2—3, when two wave fronts interact att=τk, a new Riemann problem is generated. More exactly speaking,let a weaki-wave frontαiof ordern1interact with aj-wave frontβjof ordern2. Suppose that each front has been assigned a generation order by the following rules.

Rule 1. Whenn1,n2<ν, the accurate Riemann solver is adopted to construct the outgoing wave front, and the generation order of the outgoingl-wave is assigned by

Rule 2. When max(n1,n2) =ν, the simplified Riemann solver is adopted to construct the outgoing wave fronts. The generation order of the outgoingl-wave front is assigned by (3.6),and that of the non-physical wave front isν+1.

Rule 3. Whenn1=ν+1 andn2≤ν,αiis a non-physical wave front. We adopt the simplified Riemann solver to construct the outgoing wave front. The generation order of the new non-physical wave front isν+1, while the generation order of the outgoing physical wave front is the same as that of the incoming waveβj.

Therefore,repeating the inductive process,we complete the construction of the approximate solutions in the whole domain.

4 Estimates on the Wave Interactions

In this section, we will make exact estimates of the wave interactions between the weak waves, the reflections on the strong shock waves and contact discontinuities, and so on.

Let (Ul,Ur) = (α1,α2,α3) represent that the Riemann problem with the left stateUland the right stateUris solved by a 1-waveα1, 2-waveα2and 3-waveα3.

4.1 Interaction between weak waves

In this subsection, without loss of generality, suppose that a weakj-physical waveβjinteracts with a weaki-physical waveαifrom left,and thatUB,UMandUA∈Oε(Uj)(j=1,2,3,4),satisfying

As shown in Section 3, if we adopt the accurate Riemann solver to solve the generated Riemann problem, then the outgoing wave fronts are physical waves,denoted byγ1,γ2andγ3,respectively. Otherwise, if the simplified solver is adopted, then the outgoing physical waves are denoted byγiandγj. Meanwhile, a non-physical wave is also introduced, denoted byεnp.By a standard process,see[1,p.133],we can obtain the wave interaction estimates for the weak waves by the following lemma.

Lemma 4.1If the accurate Riemann solver is adopted, then it holds

and

If the simplified Riemann solver is adopted, then it satisfies that

4.2 Interaction between the strong 1-shock and a weak 1-wave from right

In this case,we assume that a 1-weak waveα1interacts with the strong 1-shock waves1from the right. Let (Ul,Um) = (s1,0,0) and (Um,Ur) = (α1,0,0), whereUl∈Oε(U1),Um,Ur∈Oε(U2). From the construction of the wave front tracking algorithm, the accurate Riemann solver is adopted, and the generated wave fronts are denoted bys′1,γ2andγ3, respectively(see Figure 2). Then we can obtain the following estimates.

Figure 2 A weak 1-wave interacts with a strong 1-shock wave from right.

Lemma 4.2Assume that Ul,Umand Urare described as above. Then the generated Riemann problem with the left state Uland right state Uris solved by a strong1-shock s′1, a weak2-wave γ2and a weak3-wave γ3. Moreover, it holds

where Ks1and Ks2are bounded, depending only on the system and background solution, moreover, Ks3∈(?1,1).

ProofFrom the definition of Φ, it yields that

Based on Lemma 2.2, one can obtain that

By the theorem of the implicit function, close to the point(s1,α1)=(s10,0), there existC1functions of (s1,α1) such that

Using Taylor’s expansion formula, one can derive that

wheres′1(s1,0)=s1andγ2(s1,0)=γ3(s1,0)=0.

Next, we will show thatKsi|i=1,2,3is bounded. Differentiating (4.6) with respect toα1, it yields that

Multiplying (4.7) withA(Ur) from left, lettings1=s10,α1= 0 andUr=U2, one can obtain that

Therefore, from (4.8) and Lemma 2.2, we can formulate

where

Sinceλ3(U2)>s10>λ1(U2),s10

we haveSinceKsi,i= 1,2,3 are continuous with respect tos1,α1andUr, we can demonstrate thatKs3∈(?1,1) andKs1,Ks2are bounded.

Therefore, the proof of this lemma is completed.

4.3 Interaction between the strong 1-shock wave and a weak i-wave from left

Assume that the left stateUlis connected to the middle stateUmby a weaki-wave, and the stateUmis separated from the stateUrby a strong 1-shock with the speeds1, whereUl,Um∈Oε(U1),Ur∈Oε(U2). Then the generated wave fronts are a new strong 1-shock waves′1, a weak 2-waveγ2and a weak 3-waveγ3, respectively (see Figure 3). Therefore, we can obtain the following estimates.

Figure 3 A weak i-wave interacts with a strong 1-shock wave from left.

Lemma 4.3It holds that

whereare bounded, j=1,2,3, depending only on the system and background solution.

ProofAs we know,

Since

the condition of the implicit function theorem is satisfied close to the point (s1,αi) =(s10,0),and there existC1functions such that

From the Taylor’s expansion formula, one can derive that

wheres′1(s1,0) =s1andγ2(s1,0) =γ3(s1,0) = 0. In the rest, we will showare bounded. Without loss of generality,we suppose that a weak 1-waveα1interacts with a strong 1-shock wave from left. Differentiating (4.10) with respect toα1, one has

Multiplying (4.11) withA(Ur) from the left, and lettings1=s10,α1=0,Ur=U2, it yields

where

with

Therefore, we can obtain thatare bounded, depending only on the system and background solution.

4.4 Interaction between a strong 2-contact discontinuity and a weak 3-wave from left

Assume that the leftmost stateUlis joined to the middle stateUmby a weak 3-waveα3,and that the statesUmandUrare connected by a strong 2-contact discontinuityσ2, whereUl,Um∈Oε(U2) andUr∈Oε(U3). Let (Ul,Um) = (0,0,α3) andG(Um;σ2) =Ur. The outgoing wave fronts are respectively represented by a weak 1-waveγ1, a strong 2-contact discontinuityσ′2and a weak 3-waveγ3, see Figure 4. Meanwhile,we have the following interaction estimates.

Figure 4 A weak 3-wave interacts with a strong 2-contact discontinuity from left.

Lemma 4.4It holds that

where Ks4∈(?1,1)and Ksi|i=5,6are bounded, depending only on the system and background solution.

The proofs of Lemmas 4.4—4.5 are omitted in details, see Lemmas 4.2—4.3 in the reference[6].

4.5 Interaction between the strong 2-contact discontinuity and a weak 1-wave from right

Suppose that a strong 2-contact discontinuityσ2and a 1-weak waveα1interact at timet=τ. LetUm=G(Ul;σ2) and (Um,Ur)=(α1,0,0), whereUl∈Oε(U2) andUm,Ur∈Oε(U3).The outgoing wave fronts are denoted by a weak 1-waveγ1, a strong 2-contact discontinuityσ′2and a weak 3-waveγ3, respectively (see Figure 5). Then we have the following lemma.

Lemma 4.5It satisfies that

where Ks9∈(?1,1)and Ksi|i=7,8are bounded, depending on background solution.

4.6 Interaction between a strong 3-shock and a weak 3-wave from left

Suppose that the statesUlandUrare separated by a weak 3-waveα3and a strong 3-shock wave. Let (Ul,Um)=(0,0,α3) and (Um,Ur)=(0,0,s3) whereUl,Um∈Oε(U3),Ur∈Oε(U4).The generated wave fronts are respectively a weak 1-waveγ1, a weak 2-waveγ2and a strong 3-shock waves′3(see Figure 6). Then we have the following estimates.

Figure 5 A weak 1-wave interacts with a strong 2-contact discontinuity from right.

Figure 6 A 3-weak wave collides with 3-strong shock from left.

Lemma 4.6It holds that

where Ks10∈(?1,1),Ks11and Ks12are bounded, depending only on the system and background solution.

ProofAs we know, it satisfies that

By Lemma 2.5, one can derive that

From the implicit function theorem, close to the point (s3,α3) = (s30,0), there existC1functions of (s3,α3) such that

By Taylor’s expansion formula, it yields that

whereγ1(s3,0)=γ2(s3,0)=0 ands′3(s3,0)=s3.

In the following, we will showKs10,Ks11andKs12are bounded. We differentiate (4.14)with respect toα3to derive that

Multiply the above equality withA(Ur) from left and letα3=0, s3=s30, Ul=U3, then it yields that

Therefore, one can formulate that

where

Thus,Ks10,Ks11andKs12are bounded. Sinceλ3(U3)> s30> λ1(U3) ands30> λ2(U3),we have

which implies thatKs10∈(?1,1). This proof of this lemma is finished.

4.7 Interaction between the strong 3-shock and the i-weak waves from right

Suppose that a weaki-waveαicollides with the strong 3-shock waves3from right. Denote the left and right states of the strong 3-shock byUlandUm, respectively. The rightmost stateUris connected toUmby a weaki-physical waveαi, whereUl∈Oε(U3),Um,Ur∈Oε(U4).In other words, (Ul,Um) = (0,0,s3) andUr= Φi(Um,αi). The generated wave fronts are respectively a weak 1-waveγ1, a weak 2-waveγ2and a strong 3-shock waves′3(see Figure 7).Then we have the following estimates.

Figure 7 A weak i-wave interacts with strong 3-shock from right.

Lemma 4.7It holds that

where~Kisj, j=4,5,6, are bounded, depending only on the system and background solution.

ProofIt is easy to obtain that

Since

from the theorem of the implicit function, there exist someC1functions of (s3,αi) close to the point (s3,αi)=(s30,0), such that

We employ Taylor’s expansion formula to obtain that

whereγ1(s3,0)=γ2(s3,0)=0 ands′3(s3,0)=s3.Similarly to the proof of Lemma 4.3, we can also prove that ~Kisj,j=4,5,6, are bounded.

4.8 Interaction between the strong 1-shock and a non-physical wave from left

Suppose that a strong 1-shock waves1interacts with a non-physical waveεnpfrom right.Assume that the leftmost state isUl, and the strong 1-shock separates the stateUmfrom the stateUr. Thenεnp=|Um?Ul| andUr=Φ1(Um,s1). From the construction of the simplified Riemann solver, the outgoing physical wave is a strong 1-shocks′1, and a new auxiliary wave is denoted byε′np, as shown in Figure 8. Meanwhile, we have the following interaction estimates.

Figure 8 A non-physical wave collides with a strong 1-shock wave from left.

Figure 9 A non-physical wave collides with a strong 2-contact discontinuity from left.

Lemma 4.8It holds that

where K1npis bounded, depending only on the system and background solution.

ProofFrom the definition of simplified Riemann solver, one has

Therefore, the boundness ofK1npfollows easily, and we complete the proof of this lemma.

4.9 Interaction between a non-physical wave and the strong 2-contact discontinuity from left

Suppose that the strong 2-contact discontinuityσ2interacts with a non-physical waveεnp,and that the stateUmis joined to the stateUrby a strong 2-contact discontinuityσ2, i.e.,G(Um,σ2)=Ur. Let the leftmost state beUl, thenεnp=|Ul?Um|. In this case, the outgoing waves are respectively a strong 2-contact discontinuityσ′2and a non-physical waveε′np, see Figure 9, and the interaction estimates are given by the following lemma.

Lemma 4.9It holds that

where K2npis bounded, depending only on background solution.

The proof of this lemma can be found in [6, Lemma 4.4].

4.10 Interaction between the strong 3-shock and a non-physical wave from left

Suppose that a non-physical waveεnpinteracts with a 3-strong shock waves3from the left.From the construction of simplified Riemann solver, the outgoing physical wave is a strong 3-shock waves′3, and the auxiliary non-physical wave is denoted byε′np(see Figure 10). In a similar way to the argument of Lemma 4.8, we can draw the following interaction estimates.

Figure 10 Non-physical wave collides with a strong 3-shock wave from left.

Lemma 4.10It holds that

where K3npis bounded, depending only on the system.

Due to the construction of the approximate solutions, there is at most one of the following interactions occurring at timeτ.

Case 1.Two weak waves, denoted byαiandβj, 1 ≤i,j≤3,interact at timeτ.

Case 2.A 1-weak wave, denoted byα1,interacts with the strong 1-shock wave from right.

Case 3.A weaki-waveαicollides with a strong 1-shock wave from left,i=1,2,3.

Case 4.A weak 3-waveα3interacts with a strong 2-contact discontinuity from left.

Case 5.A weak 1-waveα1collides with a strong 2-contact discontinuity from right.

Case 6.A weak 3-waveα3interacts with a strong 3-shock wave from left.

Case 7.A weaki-waveαiinteracts with a strong 3-shock wave from right.

Case 8.A non-physical waveεnpcollides with a strong 1-shock wave from left.

Case 9.A non-physical waveεnpinteracts with a strong 2-contact discontinuity from left.

Case 10. A non-physical waveεnpcollides with a strong 3-shock wave from left.

Case 11. A non-physical waveεnpinteracts with a weaki-wave from left.

We denote

which measures the decreasing of the Glimm functional in Section 5.

5 Monotonicity of the Glimm Functional

In this section, we construct a new Glimm functional and prove its monotonicity based on the local wave interaction estimates in Section 4. Then the convergence of the approximate solutions is achieved by a standard procedure. Since the initial state is not a constant, we need to consider the interactions between the strong shock waves(2-contact discontinuity)and weak waves from left and right (see Lemmas 4.2—4.7) in the Glimm functional. We first define the approaching waves as follows.

Definition 5.1(Approaching Waves )

? (αi,βj) ∈A1:Two weak waves αiand βj(i,j∈{1,2,3})located at points xαiand xβjrespectively, with xαi

(i)i>j; (ii)i=j and one of them is a shock; (iii)αiis a non-physical wave.

?αi∈As1:A weak i-wave αiis approaching a strong1-shock wave if(xαi,tαi) ∈Ω1,i=1,2,3or i=1and(xα1,tα1)∈Ω2;

?αi∈Aσ2:A weak i-wave αiis approaching a strong2-contact discontinuity if(xαi,tαi)∈Ω2, i=3, or(xαi,tαi)∈Ω3,i=1;

?αi∈As3:A weak i-wave αiis approaching a strong3-shock wave, if(xαi,tαi) ∈Ω3,i=3, or(xαi,tαi)∈Ω4, i=1,2,3, with

where the curves of the strong1-shock wave, strong2-contact discontinuity and the strong3-shock wave are respectively denoted by x=χ1(t), x=χ2(t)and x=χ3(t).

Remark 5.1The approaching waves in A1are in fact the original approaching waves between weak waves (see Lemma 4.1).

Now we define a weighted Glimm functionalF(t) as follows:

and

whereKandK0are sufficiently large,andKj|1≤j≤6are determined by the following inequalities

Remark 5.2M?is a positive constant, depending on the background solution. Since the reflection coefficients are continuous with respect toUunder some small perturbations of the background solution, (5.1)—(5.6) are still valid.

In the following, we prove that the Glimm functional is decreasing based on the local interaction estimates. Before the interaction timeτ, we give the inductive hypotheses:

A1(τ?): Beforeτ, there exist a strong 1-shock wavea strong 2-contact discontinuityand a strong 3-shock wavesatisfying∈Oε(σ20)andwhose curves are respectively denoted bywhich divide the whole domain into four regions:where

A2(τ?): Beforeτ,Oε(U3) and

Remark 5.3Suppose that two waves interact for thek-th time at timeτ, and letτ?be the last interaction time toτ.

Theorem 5.1Suppose thatA1(τ?)andA2(τ?)hold for any interaction time τ. Then,there exists a positive constants δ1such that if

then,

and

ProofBased on all the possible local interaction estimates, the proof can be divided into the following several cases.

Case 1. Interaction between the weak waves.

Suppose that the two weak wavesαiandβjinteract at timet=τ. If the accurate Riemann solver is adopted, then the generated fronts are physical waves, denoted byγl,l= 1,2,3,respectively. If the simplified Riemann solver is adopted, then the auxiliary non-physical wave is denoted byεnp. Based on Lemma 4.1, we have

WhenLw(τ?) is sufficiently small, one can chooseK0andKlarge enough such that

Case 2. Interaction between the strong 1-shock and the weak 1-wave from right.

Based on Lemma 4.2, one has

provided thatLw(τ?) is sufficiently small and (5.1) holds. It satisfies

WhenKis chosen suitably large, we have

Case 3. Interaction between the strong 1-shock wave and the weaki-waves from left.

Based on Lemma 4.3, one has

Therefore, whenLw(τ?) is sufficiently small andK1,K3are chosen by (5.2), it yields

So, whenKis suitably large, one can obtain that

Case 4. Interaction between the strong 2-contact discontinuity and the weak 3-wave from left.

Based on Lemma 4.4, it satisfies

WhenLw(τ?) is sufficiently small andK2,K3,K5satisfy (5.3), one has

Thus, whenKis suitably large, it yields

Case 5. Interaction between the strong 2-contact discontinuity and the weak 1-wave from right.

Based on Lemma 4.5, one can obtain

Therefore, whenLw(τ?) is sufficiently small and (5.4) holds, one can derive

provided thatKis sufficiently large.

Case 6. Interaction between the strong 3-shock and the weak 3-wave from left.

Based on Lemma 4.6, one can obtain

IfLw(τ?) is sufficiently small and (5.5) holds, then

WhenKis suitably large, it satisfies

Case 7. Interaction between the strong 3-shock and a weaki-wave from right.

Based on Lemma 4.7, one can obtain

WhenLw(τ?) is sufficiently small and (5.6) holds, it yields

IfKis suitably large, then it satisfies

Case 8. Interaction between the strong 1-shock wave and a non-physical wave from left.

Based on Lemma 4.8, one has

WhenKis suitably large, it holds

Case 9. Interaction between the strong 2-contact discontinuity and a non-physical wave from left.

Based on Lemma 4.9, one has

WhenKis suitably large, it holds

Case 10. Interaction between the strong 3-shock wave and a non-physical wave from left.

Based on Lemma 4.10, it holds

WhenKis suitably large, it yields

Case 11. Interaction between a weaki-wave and a non-physical wave from left.

As shown in Bressan [1], when a weaki-waveαiinteracts with a non-physical waveεnp,the generated physicali-wave and non-physical wave are respectively denoted byγiandε′np, it holds

Then we can derive

Thus, whenK0andKare suitably large, it holds

and

In conclusion, we complete the proof of this theorem.

From Theorem 5.1, we can conclude the following proposition.

Proposition 5.1Under the assumptionsA1(τ?)andA2(τ?)given by Theorem5.1, if F(τ?)<δ1, then there exists a positive constantsuch that

6 Estimates on the Approximate Strong Fronts

In this section, we make some further estimates to study the total strength of the strong wave fronts of the approximate solutionUν(x,t). First, we give the estimates forEν(τ).

Lemma 6.1There exists a positive constant M1, independent of ν and Uν(x,t), such that

where the summation is taken over all interaction times and Eν(τ)is defined by(4.15).

ProofFrom Theorem 5.1, we know that for anyτ∈(τk?1,τk+1), k≥1, it holds that

Then, summing (6.2) with respect tok, we have

Hence, the proof of this lemma is completed.

Next, we will present some estimates of the strong wave fronts as follows.

Lemma 6.2There exists a positive constant M2, independent of ν and Uν(x,t), such that

ProofFrom the local wave interaction estimates involving the strong shock waves as shown in 5, it holds that

whereMandM2are positive constants, depending on the system, independent ofνand

Uν(x,t).

Finally, by Theorem 5.1 and Lemma 6.1, we have the following lemma.

Lemma 6.3There exists a positive constant M3, independent of ν and Uν(x,t), such that

By Theorem 5.1, the proof of the convergence of the approximate solutionUν(x,t) is a standard procedure, also see [1, 3]. The proof of Theorem 1.1 is completed.

Finally, we can make the conclusion of this article as follows.

Theorem 6.1Under the assumptions of the main theorem,there exist a subsequence{Uν(x,t)},aBVfunction U(x,t)and χj(t)∈Lip(R+,R), satisfying χj(0)=0, j=1,2,3, such that

(i)Uν(x,t)converges to U(x,t)a.e. inR2+, and the limit U(x,t)is a global entropy solution to system(1.1)and(1.4);

(ii)converges to χj(t)uniformly in any bounded t interval, j=1,2,3;

(iii)converges to sj∈BV(R+)a.e., and χj(t)=