Fei WU (吳菲) Zejun WANG (王澤軍)

Department of Mathematics，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，China

Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA)，MIIT，Nanjing 211106，China E-mail:wufei003@nuaa.edu.cn;wangzejun@gmail.com

Abstract In this paper，we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space.First，we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane.Based on these properties，we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme，and prove that there exists an entropy solution V (x，t) which belongs to L∞∩BVloc(R×R+).

Key words isothermal relativistic Euler system;Glimm scheme;Riemann problem;BV space;periodicity

1 Introduction

In this paper，we consider the following isothermal relativistic Euler system:

Here c is the speed of light，which is a constant，v is the velocity of particles，p is the pressure，and it satisfies p=k2ργwhere ρ is the density，γ is the adiabatic index and k is a constant representing the local sound speed.Under Newton’s limit，when c→+∞，(1.1) is formally transformed into a classical Euler system with one space variable:

The Euler system is an important mathematical model in ideal fluid dynamics.If the macroscopic velocity of the fluid is close to the speed of light，we must consider the relativistic effect.At the same time，even if the macro-velocity of the fluid does not reach the level at which the relativistic effect must be considered，the relativistic effect cannot be ignored if the micro-velocity of the fluid particles is very large.At such a time，the classical system of fluid dynamics is no longer valid，and it must be replaced by a relativistic form.

The first equation in (1.1) represents the energy conservation law in relativistic form，and the second one represents the momentum conservation law.In relativistic fluid mechanics，mass is no longer conserved but only represents a part of the total energy.Consequently，mass conservation and energy conservation are considered together in relativistic fluid mechanics.The two equations in (1.2) describe the conservation of mass and momentum of a compressible ideal gas flowing in a one-dimensional pipeline.In the Lagrange coordinate system，(1.2) can be transformed into the following form

Here v stands for specific volume，and v=1/ρ.(1.3) is the well-known p-system.

In 1965，Glimm proposed a difference scheme to establish the existence of BV solutions for the Cauchy problem of hyperbolic conservation laws for (1.3).In 1968，Nishida[1]defined the strength of elementary waves by Riemann invariants and gave the global existence of weak solutions for (1.3) when γ=1.This result was extended by Bakhvalov[2]，who identified a class of 2×2 systems，

to which Nishida’s reasoning could be applied.For (1.3)，Bakhvalov’s result can be regarded as the case when the state function p (v) satisfies

p′(v)＜0，p′′(v)＞0，3(p′′(v))2≤2p′(v) p′′′(v)，for v＞0.

For a polytropic gas，p (v)=γv-γ，this condition is equivalent to-1＜γ≤1，γ0.In 1973，by studying the global geometric properties of the shock curves and using the Glimm difference scheme，Diperna[3]constructed approximate solutions and proved the existence of global weak solutions for (1.4).For some other results of (1.3)，see[4-8].

For the relativistic form and γ=1，Smoller and Temple[9]proved the existence of BV solutions for (1.1) with corresponding Cauchy data

Chen[10]proved the existence of a BV solution when γ＞1 and (γ-1) TV{ρ0，v0}is small.Li，F(xiàn)eng and Wang[11]gave the existence of global entropy solutions for a class of initial data.Other relevant results on the existence of solutions are as follows:Makino and Ukai[12]gave the existence of locally smooth solutions.Ruan and Zhu[13]proved the existence of globally smooth solutions.Hsu，Lin and Makino[14]proved the existence of an L∞solution with a vacuum state for a class of special initial data，and the result is extended to the global existence of the spherical symmetric solution with some special initial data in[15].For isothermal gas，Geng[16]proved the existence of classical and weak solutions of steady state relativistic Euler equations with spherical symmetry structure.

In this paper，we study the periodic solution to an isothermal relativistic Euler system via a modified Glimm scheme in BV space.Frid[7]presented a periodic version of the Glimm scheme applicable to the p-system (1.3)(for γ=1) and proved that the global BV solution always exists in L∞∩BVloc(R×R+).However，this had a rather strong requirement for the total variation of the initial data.In[8]，Wang and Zhang got rid of that condition by making full use of the property of random points taking in the Glimm difference scheme.Frid and Perepelista[17]proved the existence of periodic BV solutions for system (1.1) when 1＜γ＜2 and the initial data consists of periodic BVlocfunctions with small oscillations，by applying and extending the theorems in[2，3].

For the non-periodic solution of this system，there are some other results concerning the uniqueness and stability of Riemann solutions with a large oscillation (Chen and Li[18])，limit problems (Min and Ukai[19]，Li and Geng[20]，Li and Ren[21])，non-homogeneous problems (Li and Wang[22]) and dynamic schemes[23，24].

In this paper，based on the relevant results in[7]and[8]，we study the existence of periodic solutions to an isothermal relativistic Euler system in BV space with γ=1.The paper is organized as follows:in Section 2，we discuss some basic properties of the relativistic Euler system and give the main theorem of this paper.In Section 3，inspired by[25]，we use an algebraic method to analyze the geometric properties of rarefaction wave and shock wave curves in the Riemann invariant coordinate plane，and the solvability of the Riemann problem is also obtained.In Section 4，we use the Glimm scheme to construct an approximate solution sequence and estimate the boundedness of the approximate solution sequence and its total variation.

2 Some Properties of Relativistic Euler System

In this section，we analyze some basic properties of the relativistic Euler system.First we rewrite (1.1) as

The Cauchy data of (2.1) is given by

Next，we introduce the notion of entropy solutions for Cauchy problem (2.1)(2.2)，which are in the physically relevant region

Π={(ρ，v):0≤ρ＜+∞，|v|＜c}.

Under the Lorentz transformation，

we have

where τ is the velocity with which barred coordinatesmove as measured in the unbarred coordinates (x，t)，v denotes the velocity of a particle as measured in the unbarred frame，anddenotes the velocity of a particle as measured in the barred frame.It is not difficult to check that (2.1) is invariant under the Lorentz transformation.It has been shown in[22]that TVis Lorentz invariant.

We rewrite Cauchy problem (2.1)(2.2) in the following general form of hyperbolic conservation laws:

Due to the nonlinearity of (2.3)，even if the initial data is smooth enough，the solution may produce strong discontinuity，and thus the definition of weak solutions makes sense.

Definition 2.1U (x，t) is a weak solution to Cauchy problem (2.3)(2.4) in R×R+if

Definition 2.2The weak solution of (2.1) defined above is an entropy solution to Cauchy problem (2.3)(2.4) if there is the appropriate convex entropy-entropy flux pair (η(U)，q (U))∈C2satisfying?η(U)?F (U)=?q (U)，and if

In this paper，we take the initial data as V0(x)=((ρ0(x)，v0(x))?，and assume that the initial data is periodic with period L，that is，

The initial data is assumed to satisfy

The main result of this paper is as follows:

Theorem 2.3For problem (2.1)(2.2)，we assume that initial data V0(x)∈Π∩{ρ＞0}satisfies conditions (2.9)-(2.12)，then there exists a global periodic entropy solution V (x，t) which belongs to L∞∩BVloc(R×R+) and is periodic with period L.Moreover，

To prove this theorem，one of our most important steps is to verify that the translation invariance of the rarefaction wave and the shock wave curves in the Riemann invariant coordinate plane;this is much more complicated than in the classical Euler system.Then we can obtain the solvability of the Riemann problem;see Lemmas 3.2-3.5.Next we use the Glimm scheme to construct an approximate solution sequence and estimate the boundedness of the approximate solution sequence and its total variation.We use the idea in[8]and let the initial data satisfy condition (2.12) in order to reduce the restriction in[17].Finally，the solution of (2.1)(3.1) can be obtained as the limit of a sequence of approximate solutions by using a standard diagonal process.

3 Geometric Properties of Shock Waves

In this section，we study the Riemann problem for (2.1).The initial data is given by

It is easy to verify that (2.1) is strictly hyperbolic and genuinely nonlinear.The corresponding right eigenvectors are

respectively.The Riemann invariants w1and w2of (2.1) satisfy

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From (3.2) we can get explicit expressions of w1and w2as

Next，we study the properties of the rarefaction wave curves and shock wave curves of (2.1).

Since the i-Riemann invariant is constant along the i-rarefaction wave，then for fixed left state (ρl，vl)?，the two rarefaction wave curves for (2.1) can be written as

The Rankine-Hugoniot conditions of (2.1) are

si(U-Ul)=F (U)-F (Ul)，i=1，2.

Here si(i=1，2) is the shock wave velocity.By eliminating si，we can get the following lemma，which can be also found in[11]:

Lemma 3.1The shock curves，which consist of all states V=(ρ，v)?which can connect Vlfrom the right by a shock wave，can be expressed as follows:

The four curves，R1，R2，S1，S2，are shown in Fig.1.

Figure 1

Lemma 3.2The shock curves S1，S2and the rarefaction wave curves R1，R2of (2.1) have the property of translation invariance in a Riemann invariant plane.That is，the four wave curves can all be written in the same form as follows:

According to (3.3)，we can deduce that

To avoid tedious calculation，we denote that

so (3.8) can be simplified as

To finish the proof，we need only to prove that the two shock wave curves have the form

To prove (3.9)，we write two shock curves S1and S2as

By using (3.8)，the left side of (3.10) can be written as

and the right side of (3.10) can be written as

Therefore (3.9) is proved，and the proof of the lemma is complete. □

In order to further study the properties of shock curves，we make the following transformation:

Lemma 3.3Under transformation (3.11)，the shock wave curves S1，S2take the form

ProofFrom (3.11)，we can deduce that

Thus，the left and right side terms in (3.5) and (3.6) are changed into

By substituting these two formulas into (3.5) and (3.6)，(3.12) is proved，and the proof is complete. □

Under transformation (3.11)，(3.3) can be rewritten as

we have the following lemma:

Lemma 3.4The function F defined in (3.17) is monotonically increasing and convex.

ProofWhen θ≤0，it is obvious.When θ＞0，we get

Because the range of the hyperbolic tangent function is (-1，1)，we know that the range of function g (θ) is[0，1).From (3.18)，we only need to prove that g′(θ)＞0，which implies that F′(θ)＞0.For the convenience of calculation，set

Then we have

Thus we get g′(θ)＞0，and the monotonicity of F (θ) follows.

To prove the convexity of F (θ)，direct calculation gives

We can also easily get that

Then the numerator in (3.20) can be expressed as

By direct calculation，we can get that

Let H denote the terms in the square bracket on the right side of (3.22).Denote A=k2-c2，B=k2+c2，C=k2c2，x=tanh2θ.Then x∈(0，1).H can be looked upon as a function of x and can be written as

Differentiating this with respect to x，we can get that

and thus that f (x) is monotonically decreasing.Because

f (1)=3(B2-A2)(4C-B2+A2)=0，

we can infer that f (x)＞f (1)=0，x∈(0，1);i.e.，H＞0.

Combining the above with (3.21)，(3.22) and (3.20)，we can obtain that F′′(θ)＞0，so F (θ) is a convex function.The proof is complete. □

Now consider Riemann problem (2.1)(3.1).For all Vlthat are fixed，we suppose that Vrlies on any of the four curves R1，R2，S1，S2as represented in (3.4) and (3.12).De fine

Ti=Si∪Ri，i=1，2.

For fixed Vl，consider the family of curves

F={T2(Vm):Vm∈T1(Vl)}.

Through each point Vrthere passes exactly one curve T2(Vm) of F.Then the solvability of Riemann problem (2.1)(3.1) can be obtained.

Lemma 3.5Riemann problem (2.1)(3.1) admits a solution that consists of two elementary waves，i.e.，there is a state Vmsuch that Vmand Vlcan be connected by a 1-wave (a shock or rarefaction wave)，and Vmand Vrcan be connected by a 2-wave (a shock or rarefaction wave).

ProofFrom the definition of Tiand by using (3.4)(3.12)，Tihas the following forms:

From (3.17)，the three states (ξl，ηl)，(ξm，ηm) and (ξr，ηr) satisfy

If (ξm，ηm)∈T1∩T2can be determined，then the Riemann problem (2.1)(3.1) is solvable.

Eliminating ξmfrom equations (3.23) and (3.24)，we have that

Since F is strictly increasing and convex，ηmcan be uniquely determined by (3.25).The proof is completed. □

Lemma 3.6([1，26，27]) The solution of Riemann problem (2.1)(3.1) satisfies

4 Estimation of the Solutions Using the Glimm Difference Scheme

In this section，we use the well known Glimm scheme to construct an approximate solution sequence of (2.1)(2.2).

Fix mesh widths l=Δx，h=Δt satisfying the Courant-Friedrichs-Lewy condition

Denote xr=rl，r=0，±1，±2，···，and ts=sh，s=0，1，2，···，and build the staggered grid of mesh-points (xr，ts)，with s=0，1，2，···，and r+s being even.

From (3.1)，V0(x)=(ρ0(x)，v0(x)) is the initial data of (2.1) on

Π={(ρ，v):0≤ρ＜+∞，|v|＜c}.

We start by taking a piecewise constant approximation of the initial data by setting

Obviously，we have that

We define the approximate solution of (2.1)(2.2) in the strip R×[0，h) by solving Riemann problems (2.1)(4.2).The solutions are denoted as sequences{Vh(x，t)}，0≤t＜h.Since{Vh(x，t)}are piecewise constant with discontinuities at points (rl，0)，where r is even，this Cauchy problem consists of many local Riemann problems.Condition (4.1) guarantees that the waves in the solutions of these Riemann problems will not interact in this strip.

In order to define the solution in the strip R×[h，2h)，we choose a random number α1∈(0，1) and set

Then we solve Riemann problem (2.1)(4.4)，and the approximate solution sequence{Vh(x，t)}is defined as the approximate solutions in the strip R×[h，2h).

Assuming that the solutions have been constructed in the strip R×[(n-1) h，nh)，we solve the Riemann problem of (2.1) in R×[nh，(n+1) h) with the initial data

The approximate solution sequence{Vh}constructed by the above process satisfies (2.7)，which is the weak solution of (2.1).The details of this can be found in[26，27].

Next，we define the simplified strength of elementary waves (see[25，28]for more details)

S (Vl，Vr)=|ηm-ηl|+|ηm-ηr|.

Since η is monotone across the simple waves，we have that

Here η(x，t)=lnρ(x，t).In what follows，we denote V (Vl，Vr，) as the solution of Riemann problem (2.1)(3.1)，where

The following lemmas give an important property of (4.6):

Lemma 4.1Letting V1，V2，V3be three arbitrary states in R+{0}×R，we have

S (V1，V3)≤S (V1，V2)+S (V2，V3).

In particular，if V2is an intermediate state in the solution of the Riemann problem for initial data (V1，V3)，i.e.，V2=V (V1，V3，ξ2) for some ξ2∈R，then

S (V1，V3)=S (V1，V2)+S (V2，V3).

ProofFirst we need to introduce the intermediate state，say Vij，which appears in the solution of the Riemann problem for initial data (Vi，Vj).For (V1，V3)，(V1，V2) and (V2，V3)，we use (3.25) to obtain that

Eliminating ξifrom the above equations，i=1，2，3，we have that

We complete the proof of Lemma 4.1 in the next two lemmas. □

Lemma 4.2If F is a convex function with F (0)=0，then F (a+b)≥F (a)+F (b)，(a，b≥0) holds.

ProofIf F is a convex function，for any three points x1＜x2＜x3we have that

We assume that 0≤a＜b and take x1=a，x2=b，x3=a+b，then substitute them into (4.7) to get that

Then we take x1=0，x2=a，x3=b and substitute them into (4.7) to get that

Notice that for F (0)=0，we can get that

F (a+b)≥F (a)+F (b).

The proof is complete. □

Lemma 4.3([25]) Let F be a strictly increasing function and that satisfies

(i) F (a+b)≥F (a)+F (b)，for a，b≥0，

(ii) F (a+b)≤F (a)+F (b)，for a，b≤0.

Letting a，b，c，d，x，y be real numbers such that

we have that

Lemma 4.4Let Σ=.If Vl，Vr∈Σ，then we have V (Vl，Vr，ξ)∈Σ for all ξ∈R，where V is the solution of Riemann problem (2.1)(3.1).

ProofDenoting Vmas the intermediate state between two elementary waves，we will show that Vm∈Σ.From (3.23) and (3.24) we have that

Direct calculation gives that

Remark 4.5Lemma 4.4 shows that the solution of Riemann problem (2.1)(3.1) belongs to the wedge region Σ，thus，if ξ is bounded then η is bounded，and vice versa.

Let V be a function satisfing V=(ρ，v)∈BVloc(R，R+×R) and (3.6).Define

where the supremun is taken over all partitions a=x0＜x1＜···＜xN=b of the interval[a，b].Lemma 4.1 implies that μ extends to a Radon measure over R.Let Vh=(ρh，vh) be an approximate solution.The periodic version of the Glimm functional is defined by

Fper(Vh(t))=μh，t([a，a+L))，

where μh，tis the measure obtained from Vh(·，t) by (4.11)，and a∈R is arbitrary.

Lemma 4.6Fper(Vh(t)) does not depend on a;i.e.，μ([a，a+L))=μ([0，L)) for any a∈R.

ProofThis follows immediately from the definition of Fper(Vh(t)) and the periodicity of Vh. □

Lemma 4.7Fper(Vh(t)) is nonincreasing with respect to t.In particular，we have that

The proof is complete. □

The following lemmas give the consistency and boundedness of approximate solutions with respect to time:

ProofBy (3.8) and Lemma 3.6 we have that

Then，using Lemmas 4.3-4.4，we can get that

Next，we will prove the boundedness of the approximate solution.The following lemma gives an important property of an approximate solution in a period a similar result can be found in[8]:

Lemma 4.9For (2.1)，where U is given in (2.5)，and for any t＞0，there exists a subsequence of{Uh(x，t)}(still labeled as{Uh(x，t)}) such that

Recall that on every strip (-∞，+∞)×[kh，(k+1) h)，approximate solution Uh(x，t) is the solution of the Cauchy problem，with the initial data consisting of many constant states.Integrating the two equations in (2.1)，we have that

Considering the expression of U in (2.5)，we have that

so (4.15) can be further written as

Remembering the definition of an approximate solution as in (4.5)，we have，for kl，that

Thus，with the help of (4.14)-(4.16)，we have that

Here N and C depend on the diameter of the support set of test function φ(x，t).The proof is complete. □

Lemma 4.10For the approximate solution Vh(x，t) constructed above，we have，for any t＞0，that

ProofFor any t＞0，applying Lemmas 4.8-4.9 and (4.3)，we can get that

From this lemma，we can choose h to be small enough such that A (h)＜1，so we can take=R+1.

Lemma 4.11([26，27]) The approximate solution Vh(x，t) satisfies

From Lemmas 4.8 and 4.10，the approximate solution sequence{Vh(x，t)}，regarded as a result of the functions of x，is uniformly bounded and has a uniformly bounded total variation on each bounded interval on any line t=const.＞0.By Helly’s theorem，we can find a subsequence (still labelled as{Vh(x，t)}) which converges pointwisely on any bounded interval of this line.Combining Lemma 4.11 and a standard diagonal process，we can obtain a subsequenceof{Vh(x，t)}satisfying

Finally，the solution V (x，t) of (2.1)(3.1) can be obtained as the limit of a sequence of approximate solutions{Vh(x，t)}.The proof of Theorem 2.3 is finished.