Existence in the Large for Pressure-Gradient System?

Shuxin ZHANG Zejun WANG

Abstract In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data.To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.

Keywords Pressure-gradient system, Riemann problem, Diperna’s conditions,Glimm scheme, BV space

1 Introduction

In this paper, we study the following pressure-gradient system

Here u=u(x,t), p=p(x,t) are velocity, pressure, respectively. For smooth solution, it can be simplified as

We will study the Cauchy problem of (1.1) and the initial condition is given by

System (1.1) can be obtained from the following 1-dimensional Euler equations

by deleting the nonlinear convective terms,in the case of only considering the effect of differential pressure (see [1, 9]). Here u = u(x,t), p = p(x,t), ρ = ρ(x,t) and e = e(x,t) represent speed,pressure, density and internal energy, respectively. Pressure-gradient system is an important model in the theoretical research of conservation law system. We will study the existence of global BV solution to problem (1.1), (1.3) for large initial data.

In 1965, Glimm [7] used the method of random choice to establish the global existence of weak solutions of the hyperbolic conservation law system for small initial data. For the general system of conservation law

satisfying Diperna’s conditions, Diperna [4] considered the existence of global solutions for a class of nonlinear hyperbolic system by studying the shock curve described by the Riemann invariants of (1.5) and proved the existence of weak solution to the Cauchy problem. In the same year, Diperna [5] proved the existence of solutions for a class of quasi-linear hyperbolic conservation laws system with large initial data. Ding et al. [3] proved the global existence of solutions of p-system with γ >1 by using Glimm scheme for a special class of large initial data.Li et. al. [8] gave the existence of global entropy solutions to the relativistic Euler equations for a class of large initial data.

As a special and important system of conservation law, the following p-system

has been studied by many authors. The initial data is given by

Here v is the specific volume, v =ρ is the density and u is the velocity of the gas. p is the pressure satisfying p(v)=where γ >1 is a constant. For the p-system with γ =1, Nishida[10] proved the global existence of weak solutions to Cauchy problem via Glimm scheme for large initial data. In 1973, Nishida and Smoller [11] used Glimm scheme to obtain the global existence of solutions to problem (1.6)–(1.7) when

is sufficiently small. Frid [6] presented a periodic version of Glimm scheme applicable to psystem(for γ =1)and proved that the global BV solution always exists in L∞∩BVloc(R×R+).This result was further improved in [12].

For pressure-gradient system(1.1),Zhang and Sheng[15]studied the one-dimensional piston problem of (1.1). Yang and Sheng [9] studied the interaction of a class of waves of the aerodynamic pressure-gradient system. Xu and Huang [13] studied global existence of shock front solution to piston problem of pressure-gradient system. Ding [2] studied stability of rarefaction wave to the 1-dimensional piston problem for the pressure-gradient system. Zhang et al. [16]studied interactions between two rarefaction waves for the pressure-gradient system.

In this paper, we use Glimm scheme and the methods proposed by Diperna in [4] to prove the existence of weak solutions of problem (1.1), (1.3). The main theorem of this paper is as follows.

Theorem 1.1Suppose that the initial data U0(x) =(u0(x),p0(x))Tof (1.1) and the total variation of U0(x) are bounded. In addition, U0(x) satisfies

Then problem (1.1), (1.3) admits a solution U(x,t)∈L∞∩BVloc(R×R+).

The rest of this paper is arranged as follows. In Section 2, we study the shock curves of(1.1) and prove that the shock curves satisfy Diperna’s conditions in [4]. In Section 3, we use Glimm scheme to construct an approximate solution sequence and prove that the sequence and its total variation are uniformly bounded. Then we define the Glimm functional and prove its monotonicity under Diperna’s conditions. In Section 4, we finish the proof of Theorem 1.1 by combining the previous properties.

2 Properties of Shock Curves

Denote U =(u,p)T, then problem (1.2)–(1.3) can be rewritten as

It is easy to verify that

Thus both characteristics of (2.1) are genuinely nonlinear. 1-Riemann invariant w and 2-Riemann invariant z of (2.1), which are defined as two functions satisfying Dw·r1= 0 and Dz·r2=0, respectively, can be given explicitly as

The function pair (z,w) is also called Riemann invariant coordinates system. For the general definition of Riemann invariant coordinates system, one can refer to [14].

For a given left state U0= (u0,p0)T, the i-rarefaction wave curve Ri(U0) (i = 1,2) of(2.1) is defined as all the right states U = (u,p)Tthat can connect U0by an i-rarefaction wave. These two curves in the (u,p) plane can be given explicitly by w(u,p) = w(u0,p0) and z(u,p)=z(u0,p0), respectively, that is

where the range of p and u in (2.7) and (2.8) can be obtained by (2.4) and (2.5).

For a given left state U0= (u0,p0)T, the i-shock curve Si(U0) (i = 1,2) of (2.1) is defined as all the right states that can connect U0by an i-shock wave. It can be given in the (u,p)plane by Rankine-Hugoniot conditions, that is

where s is the speed of the i-shock.

Eliminating s from (2.9), (2.10), we can obtain

Taking the Lax entropy conditions, that is, λ1(U0)>λ1(U) for S1and λ2(U0)>λ2(U) for S2,into account, we can give the equations of the two shock curves as follows

For a shock or rarefaction wave with right state U = (u,p)Tand left state U0= (u0,p0)T,we denote

Combining (2.12)–(2.15) and by simple calculation, we can get the following lemma.

Lemma 2.1On the shock curves S1(U0), S2(U0), the changes of z, w satisfy

Denote σ =w+z, η =w ?z, by using (2.6), we have

Thus S1(U0), S2(U0) can be rewritten as

That is, along S1, c(ε) is monotonically increasing and satisfies ?1=c(1)

Thus we can get (2.25), and (2.26)–(2.28) can be similarly proved.

Figure 1

Figure 2

ProofWe only prove (i) since the proofs of the others are similar. The property (i) is shown in Figure 3.

Figure 3

Figure 4

By (2.6), Lemmas 2.2–2.4, the pressure-gradient system satisfies the following Diperna’s conditions.

(A1) ?w·r1=?z·r2=0, ?z·r1>0, ?w·r2>0.

(A2) In the (z,w) plane, shock curves satisfy:

Thus we can get ?σ ≤?σ′.

3 Construction of Approximate Solutions

We use Glimm scheme to construct an approximate solution sequence, which is denoted as {Uh(x,t)} for t ≥0. Fix a spatial mesh-length l = ?x > 0 and a temporal mesh-length h=?t>0 satisfying the Courant-Friedrichs-Lewy condition

Let αn(n=1,2,···) be a random point in (?1,1) and denote ym,n=(xm+αn)l, where m is an integer, and m+n is an even number.

For t=0, {Uh(x,0)} is defined as

Obviously {Uh(x,0)} satisfies

Assuming that{Uh(x,t)}has been constructed insk,we continue to construct approximate solution {Uh(x,t)} in sn+1. Define

Then we solve Riemann problem(2.1),(3.8)in t>tn,and the solution in between tn

In order to prove the convergence of the sequence {Uh(x,t)}, we need to prove that there exists some positive constant C, such that

Thus due to the Helly’s theorem,as h →0,there exists a convergent subsequence of{Uh(x,t)}.Denote the limit function as U(x,t), then by standard process we can verify that U(x,t) is a weak solution to problem (2.1)–(2.2).

Figure 5

As shown in Figure 5, let Dm,nbe a “diamond” with (ym?1,n,tn), (ym,n?1,tn?1), (ym+1,n,tn), (ym,n+1,tn+1) as its vertices. All of these “diamonds” cover the upper half of (x,t) plane.The elementary waves issuing from(xm?1,tn?1)and entering Dm,nare denoted as α=(α1,α2).The left, middle and right states of waves(α1,α2)are denoted by U1, U12and U2, respectively.The waves issuing from (xm+1,tn?1) and entering Dm,nare denoted as β =(β1,β2). The left,middle and right states of waves (β1,β2) are denoted by U2, U23and U3, respectively. The elementary waves issuing from (xm,tn) are denoted as γ =(γ1,γ2). The left, middle and right states of waves (γ1,γ2) are denoted by U1, U13and U3, respectively. Every elementary wave may be a shock wave or a rarefaction wave. Denote |α| = |α1|+|α2|, |β| = |β1|+|β2| and|γ|=|γ1|+|γ2|. Due to (2.18), we define σh=w(Uh)+z(Uh), and define the strengths of α1,α2as

The strengths of β1, β2, γ1, γ2can be similarly defined. The symbol “+” means the positive part of a number, that is, a+=max{a,0}.

A mesh curve, associated with Uh, is a polygonal graph with vertices that from a finite sequence of sample points(ym1,n1,tn1),···,(yml,nl,tnl). A mesh curve J is called an immediate successor of the mesh curve I when JI is the upper boundary of some diamond,and IJ is the lower boundary of diamond. Thus J has the same vertices as I, save for one, (ym,n?1,tn?1),which is replaced by (ym,n+1,tn+1). This induces a natural partial ordering in the family of mesh curves: J is a successor of I, denoted J >I, whenever there is a finite sequence, namely,I = I0,I1,··· ,In= J of mesh curves such that Ilis an immediate successor of Il?1, for l=1,··· ,n.

On the mesh curve J, define the Glimm functional as follows

Since σh= w(Uh)+z(Uh), it is known from (2.25)–(2.26) that crossing a shock curves S1, S2there holds [σh]+= (σl?σr)+≥0. From (2.18) that crossing a rarefaction wave curves R1,R2, we have [σh]+= (σl?σr)+= 0. Thus, F(J) represents the total variation of the shocks crossing J.

The following proposition gives the monotonicity of the Glimm functional defined by (3.12).

Proposition 3.1If J >I, we have

ProofWhen a mesh curve J is an immediate successor of I, which is shown in Figure 5,we need only to prove[σh(γ)]+≤[σh(α)]++[σh(β)]+. As shown in Figure 5,α1, α2,β1,β2, γ1,γ2are shocks or rarefaction waves. When γ1and γ2are both shocks, whatever the incoming waves are, (3.13) always hold. When γ1and γ2are both rarefaction waves, we can easily get 0 = F(J) ≤F(I). We need only consider the case when γ1is a 1-rarefaction wave and γ2is a 2-shock wave. According to [4], the waves α1, α2, β1, β2can be divided into 16 cases. In the following S or R is used to represent that α1, α2, β1, β2is a shock or a rarefaction wave,respectively. For example, when α1is a shock,denote α1as S. In addition, denote σ1=σ(U1),σ12=σ(U12), and so on.

(1) Cases RRRR, RRRS, RSRR and RSRS. These four cases are obviously impossible.

(2) Case SRRR. On the curve S1, by Lemma 2.1, we have |w12?w13|<|z12?z13|. Since z12=z2

F(J)?F(I)=(σ1?σ13)?(σ1?σ12)=w12+z12?z13?w13<0,

which is exactly [σh(γ)]+≤[σh(α)]++[σh(β)]+. The proofs of cases SSRR, SRRS, SSRS,SRSR and SRSS are similar.

(3) Case RRSR. In this case z3= z13= z23< z1< z12= z2. We can divide it into two subcases: w13≤w23(see Figure 6) and w13> w23(see Figure 7). When w13≤w23,z1?z13< z2?z23, from Lemma 2.6, we can get σ1?σ13< σ2?σ23, which implies that F(J)?F(I)=(σ1?σ13)+(σ23?σ2)<0. Thus [σh(γ)]+≤[σh(α)]++[σh(β)]+holds.

Figure 6

Figure 7

Next we show that w13>w23is impossible. If w13>w23and z13=z23,we have σ23=w23+z23< w13+z13=σ13. Choose a point (z0,w0), satisfying z0= z1and z23=(w23,z2,w2).Since (z13,w13)∈Q(z23,w23) and z0?z23=z1?z13, we see from Lemma 2.1 that w13?w1w23implies w1?w0>w13?w23>0, which is a contradiction since we have w1

(4) Case RSSR. In this case, we have z3= z13= z23< z1< z12< z2, then z1?z13 0, that is, σ12> σ2, thus σ1?σ13<σ2?σ23<σ12?σ23. Therefore, we can get

If w13>w23, it is obvious that [σh(γ)]+≤[σh(α)]++[σh(β)]+holds.

(5) Case SSSS. This case implies w23>w13, which is impossible for the similar reason to case (3).

Therefore, from Proposition 3.1, we have F(J) ≤F(0), where “0” is the unique I-curve between the two lines t=0 and t=s.

In the following we study the strength of the waves for Riemann problem(2.1)–(2.2), where Ul, Ur∈Q, Q = {U ∈R2;z(U) ≤z0,w(U) ≥w0}. For simplicity, we use (Ul,Ur) to denote the solution of Riemann problem(2.1)–(2.2). According to Lemma 2.5,there is an intermediate state Umwith Ulas the left state and Uras the right state.

Define the strength of elementary waves for the solution(Ul,Ur)to Riemann problem(2.1)–(2.2) in two different ways. The first one is

From the discussion in Section 3, we know that for rarefaction waves, (σk?σk+1)+= 0, and(3.14) only records the strengths of shocks. The second definition of the strength is

It also records only the strengths of the shocks.

Lemma 3.1Denote U =(u, p)Tas the solution to the Riemann problem (2.1) and (1.3),then there exists some positive constant c, such that

ProofFrom (2.18), (3.14) and (3.15), we have

Take c=2, then the lemma is proved.

Let DV(z(Uh(t))) and DV(w(Uh(t))) be the decreasing total variation of z(Uh(t)) and w(Uh(t)), respectively.

Lemma 3.2For any fixed t ∈[nh,(n+1)h), we have

ProofFrom the analysis of Section 3, we know that w(Uh(t)) and z(Uh(t)) can only increase or remain unchanged when they cross a rarefaction wave, and they can only decrease when they cross a shock. When two states Uland Umare connected by a 1-shock, there hold w(Ul)

ηm?ηl=wm?wl+zl?zm.

Therefore,

Similarly, it can be shown that when Umand Urare connected by a 2-shock, (3.17) still holds.

4 Proof of Main Theorem

To prove Theorem 1.1, we need only to prove that the approximate solution sequence{Uh(x,t)} constructed in Section 3 satisfies (3.9)–(3.11). We complete the proof by several lemmas and a main theorem.

Lemma 4.1Suppose that there exist some positive constants c, p0such that p < p0, then TV(Uh(x,t))|[0,∞)is uniformly bounded for t>0.

ProofCombining Lemmas 3.1–3.2, Proposition 3.1 and Lemma 2.2, we can get

where c is the same as in Lemma 3.1. The proof is complete.

Lemma 4.2For any t>0, we have

ProofFor any t>0, by Lemmas 3.1 and 4.1, we have

Thus (4.1) follows.

Theorem 4.1For any t1,t2>0, the approximate solution Uh(x,t) satisfies

where C depends only on c, p0.

ProofBy Lemmas 4.1–4.2, we can get

then in combination with (4.1), we can get (4.2), where N =max+1,i=1,2.

From Lemmas 4.1–4.2 and Theorem 4.1, according to Helly’s theorem, there exists a convergent subsequence {Uhm(x,t)} of {Uh(x,t)}, such that

By standard procedure, we can verify that U(x,t) is a global weak solution of (2.1)–(2.2).(3.9) can be obtained from (4.1). (3.10) can be obtained from Proposition 3.2 and Lemma 4.1.Finally, (3.11) can be obtained from Lemma 4.1 and Theorem 4.1. Thus, we have proved that U(x,t) is a weak solution to the problem (2.1)–(2.2).