Jianjun HUANG (黃健駿)

Department of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082, China

E-mail : mcshjj@hotmail.com

Zhenglu JIANG (姜正祿)?

Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

E-mail : mcsjzl@mail.sysu.edu.cn

Here p0=(1+|p|2)1/2is the energy of a dimensionless relativistic gas particle with the momentum p, and Q(f,f) is the relativistic Enskog collision operator describing the binary collision with the difference between the gain and loss terms. There are several representations of the collision operator (for example, [31, 44]). Corresponding to the collision operator of the relativistic Boltzmann equation given by Glassey and Strauss [24], the gain and loss terms can be expressed as follows:

In (1.2) and (1.4), a is the diameter of the hard sphere, of course, a>0,is a subset of the unit sphere surface S2, which is defined by= {ω ∈ S2: ω ·(p/p0? p?/p?0) ≥ 0}, and the collision factor Y is a functional depending on the local mass densities ρ±= ρ(t,x ± aω) with

The derivation of(1.1)is analogous to that of the relativistic Boltzmann equation[15,24]. The other parts of (1.2) and (1.4) are explained in what follows.

As usual, p and p?are dimensionless momenta of two relativistic particles immediately before collision,while p′andare dimensionless momenta after collision;p?0=(1+|p?|2)1/2denotes the dimensionless energy of the colliding relativistic gas particle with the momentum p?before collision, and as used below in the same way,represent the dimensionless energies of the two relativistic particles after collision. We remark that p′andcan be explicitly calculated by using the same scheme as given by Glassey and Strauss[23]. In terms of the momenta p, p?and the unit vector ω,the momenta after collision can be represented by

Here and below, we abbreviate the distribution functions in (1.2) and (1.4) byf and f??, respectively, corresponding to the momentap and p?.

q(p,p?,ω) is the scattering kernel which is represented by

where s = |p?0+p0|2? |p?+p|2, and s1/2is the total energy in the center-of-mass frame[17, 18],and σ is the differential scattering cross section which depends on the momenta(p,p?)and the unit vector ω. Here we introduce a quantity defined by2g is in fact the value of the relative momentum in the center-of-mass frame [17, 18], and it is related to s. It can also be seen that s=4+4g2.

In 1922, Enskog [20]derived the Enskog equation as a modification of the Boltzmann equation to describe moderately or highly dense gases. For the background of the Boltzmann equation, one can refer to the work of previous researchers such as Andréasson [2], Bichteler[9],Cercignani and Kermer[13],DiPerna and Lions[16],Dudyński and Ekiel-Je˙zewska[17,18],Glassey [22], Glassey and Strauss [23–25], Jiang [28–30, 32], and Strain [40–42].

Of course,there are many results concerning the Enskog equation. We briefly review a part of the history of this model in the classical and relativistic cases. In the classical case,the local existence of the Enskog equation was first investigated by Lachowicz[36]. The global existence of the Enskog equation was given by Cercignani[12]and Polewczak[37], respectively,for when the initial data are near vacuum data. Arkeryd [4]gave another existence proof of the global solution to the Enskog equation with large L1initial data. Polewczak[39]showed the existence of normalized solutions to the Enskog equation by use of DiPerna and Lions’ techniques [16].Jiang [33]obtained the global existence of the solution to the Enskog equation with external force for when the initial data is small enough. Asymptotic equivalence between the Enskog and Boltzmann equations was studied by Arkeryd and Cercignani [5, 6].

For the relativistic case, Jiang [31]extended the result of Glassey [22]from the relativistic Boltzmann case to the relativistic Enskog case; that is, the relativistic Enskog equation with small initial data admits a unique global solution for a class of differential scattering cross sections. Later,Wu[44]gave the L∞-stability of the solutions to the relativistic Enskog equation under the same assumption as that in the work of Jiang [31], and proved the global existence and L1-stability of the solutions under the assumption given by Cercignani [12].

There are also many authors who are contributed to the study of the Enskog equation,for example, Bellomo [7], Bellomo and Lachowicz [8], Cercignani [11], Esteban and Perthame[21], Ha [27], Jiang, Ma and Yao [34], Polewczak [38], Toscani and Bellomo [43]. Many other relevant results can be found in the references mentioned above.

In this article, using the same techniques as Arkeryd and Cercignani [6], we show a global existence theorem in the L1space for the relativistic Enskog equation in a periodic box, and give the asymptotic equivalence between the relativistic Enskog and Boltzmann equations as the diameter a tends to zero. Our results are an extension of the work of Arkeryd and Cercignani.Their proof of the global existence of the solutions to the Enskog equation is on the basis of DiPerna and Lions’techniques[16]. DiPerna and Lions’idea was to define three different types of solutions(distribution solutions,mild solutions,and renormalized solutions)and to construct the subsolution and supersolution with the help of the momentum-averaged properties given by Golse et al. [26]. However,Arkeryd and Cercignani’s idea was to define the iterated integral form of the solution to the Enskog equation given in [3]. In a fashion similar to the classical case, the iterated integral form of the solution to the relativistic Enskog equation is defined as follows:

Definition 1.1Let ?be a trajectory operator[35]such that f?(t,x,p)=f(t,x+tp/p0,p).Assume that CL is the linear space of all functions ψ ∈ C1((0,∞);L∞(T3×R3))with compact support and ψ(·,x,p) ∈ C1(0,∞) for almost all (x,p) ∈ T3× R3. f is said to satisfy the relativistic Enskog equation in the iterated integral form iffor almost every (x,p)∈T3×R3and

It can be seen in [5]that the solution of the iterated integral form is equivalent to those of the other three different types given by DiPerna and Lions [16].

According to the book of Chapman and Cowling [14], the collision factors Y(ρ±) approximate unity for a rare gas,and increase as the gas density increases. Throughout this article,we assume that Y(ρ±) are two of the same constants with Y(ρ±)= κ, as in the work of Arkeryd and Cercignani [6]. The initial data f0= f0(x,p) is periodic in each xi(i = 1,2,3) of period 2π, and required to have finite mass, energy and entropy; that is,

Furthermore, as in the work of Wu [44], we assume that the scattering cross section σ is nonnegative, continuous, symmetric with respect to p, p?and ω, and satisfies

for some k >0, which implies that

Our first result is expressed as follows:

Theorem 1.1Assume that f0satisfies (1.10). Then the Cauchy problem on the relativistic Enskog equation (1.1) with the initial data f0has a mild solution f satisfying that f ∈ C([0,∞);L1(T3× R3)). Furthermore, for t > 0, the mass and energy in the system are bounded by its initial data, and the entropy satisfies

In the proof of Theorem 1.1, we need the iterated integral form of the solution, and use a momentum-averaged regularity which is similar to the velocity-averaged regularity (see [26])employed by DiPerna and Lions [16]in the classical case. The momentum-averaged regularity is as follows:

Lemma 1.1Let (E,μ) be an arbitrary measure space, and

Then one has the following two properties:

(i) If gnand Gnbelong to a weakly compact set in L1(K) for any compact set K of[0,T]×T3×R3, andin the distribution sense, thenbelongs to a compact set of L1([0,T]×T3×E) for any compact set K of [0,T]×T3×R3, provided that supp? ? K × E;

(ii) If, in addition, gnbelongs to a weakly compact set of L1([0,T]× T3× R3), thenbelongs to a compact set of L1([0,T]×T3×E).

Our second result is about the asymptotic equivalence between the relativistic Enskog equation and the relativistic Boltzmann equation. As we know, the Enskog equation acts as a modification of the Boltzmann equation in systems of moderately or highly dense gases.Arkeryd and Cercignani[6]showed that in the classical case,the solution of the Enskog equation converges weakly to the solution of the related Boltzmann equation as the diameter a tends to zero. Now we can get a similar result for the relativistic Enskog equation; that is,we can prove the following theorem:

Theorem 1.2Assume that a2κ is always constant as a tends to zero. For any sequenceand the sequence of solutionsof Theorem 1.1, there is a subsequencefor whichweakly converges to a solution f of the related relativistic Boltzmann equation in L1.

The rest of this article is organized as follows: in Section 2, we show the conservation laws and the entropy in the relativistic system. The solutions of the approximate equations of the relativistic Enskog equation are constructed in Section 3,and the proof of Theorem 1.1 is given in Section 4. Finally, we prove Theorem 1.2 in Section 5.

2 Conservation Laws and Entropy

In this section, as in the case of the relativistic Boltzmann equation (see [32]), we show that the relativistic Enskog equation given by (1.1) makes the conservation of mass, momenta and energy, and that its entropy does not increase.

Note that Y(ρ±)= κ. By letting φ = φ(x,p) be the measurable function on T3×R3, it is at least formally found that the collision operator satisfies the following equality:

Here we abbreviate the distribution functions in the collision operator byrespectively. Because Q=Q+?Q?, we can denote Igand Ilby

We first consider the loss term Il. By interchanging the variables p and p?, replacing ω with?ω, and then replacing x with x ? aω, (2.3) can be rewritten as

Then we calculate the gain term Ig. Note that the momenta and energies of the two colliding particles is conserved before and after collision; that is,

In addition, because the collision process defined by (1.6) is reversible, the roles of (p,p?,ω)andin (1.6) are interchangeable. This leads to

It follows from (1.6) and (2.7) that

Hence, in view of (1.7), we have that

We swap variables (p,p?,ω) andby the use of (2.9) and the propertyIt follows that

Using the same kind of changes as in (2.5), we have that

To sum up, we have that

In view of (2.12), it follows thatwhere c1∈ R,c2∈ R3and c3∈ R. Then it can be found thatis independent of t for any distributional solution f to equation(1.1). This yields the conservation of mass,momentum and energy of the relativistic system.

The other properties we want to mention pertain to the estimate of the entropy functional of the relativistic Enskog equation (1.1). This is denoted by

and where (in all that follows),Let φ =log f in (2.12). Then it follows from (2.14) that

By use of (1.12) and the inequality h(log h ?log k) ≥h ?k with h = ff??andin(2.16), we have that

Additionally, it can be proven that the functional HB(t) is bounded from above if (1.10)holds. With the help of (1.12) and Fubini’s theorem, combining (2.15) and (2.17), we can get the following inequality:

where (and in all that follows), f?+= f(t,x+aω,p?), ρ(t,x) =and ξ(t,x) =are the density and the flow of the system, respectively. For simplicity,we denote ρ(t,x ± aω) and ξ(t,x ± aω) by ρ±and ξ±, respectively. Let Ba(x) be the ball{y ∈ R3:|y ? x|≤ a,x ∈ T3}. By the divergence theorem, it follows that

Note that the measure of Ba(x) is not larger than that of T3. Then we have that

3 Construction of Approximating Solutions

In order to prove Theorem 1.1,both collision operator and initial data need to be truncated by using an approximation which is similar to that given by Arkeryd and Cercignani [6]for the classical Enskog equation. Due to the conservation of energy of the relativistic system in(2.12), we can make a modification of the truncated collision operator in [6].

Consider the smooth test functionsuch that 0 ≤ χ(r) ≤ 1, where χ(r)=1 for r ∈ [0,1]and χ(r)=0 for r >2. Let fn=fn(t,x,p) satisfy

Here the truncated collision operator Qnis given by

with ? > 0 and χn= χ((p0+p?0)/n), and the initial data f0is truncated assuch that=0 for |p|≥2n.

This yields the non-increasing property of mass and energy of the system.

It can be found from (3.4) that the second term in (3.6) is non-positive, thus

Now we need to estimate Iifrom i=1 to 3, respectively. For the first term I1, by use of (2.5),(3.5) and the inequality

it follows that

For I2, we can use (2.16)–(2.19) to obtain that

For I3, by use of (3.4), we have that

Thus it follows from (3.9)–(3.11) that

as υ tends to infinity. Denote bythe solution of equation (3.1) with initial data fυ0. By enploying a technique which is similar to that of Arkeryd and Cercignani [6]in the classical case, it follows thatconverges to fnin L1(T3×R3). Note thatsatisfies (3.12) with initial data fυ0. Then, by use of the weak lower semicontinuous property of the entropy (see[10]), we have that

which implies that fnsatifies (3.12).

Given any time interval[0,T]and initial datasatisfying(1.10)and=0 for |p|≥2n,it follows that there exists a unique nonnegative solution of equation (3.1) satisfying (3.4),(3.5) and (3.12) on [0,T′]. Then we can repeat the above scheme on [T′,2T′]with initial data f(T′,x,p), and successively on subintervals of length T′covering[0,T]. Thus there is a unique solution of equation (3.1) satisfying (3.4), (3.5) and (3.12) on [0,T]for any T <∞.

4 Proof of Global Existence

In this section, we prove Theorem 1.1; that is, we show the global existence of a mild solution to the relativistic Enskog equation. This can be carried out following the arguments developed by Arkeryd and Cercignani [6]for the classical Enskog equation. Because most of them can be adapted with little modification, we give only a sketch of the proof here.

Given the initial data f0satisfying (1.10), we truncate f0with= χ(|p|/n)f0to get a sequence of nonnegative functions. It can be found that=0 for |p|≥2n. As shown by our discussion in the previous section,there exists a unique nonnegative solution fn=fn(t,x,p)to the problem regarding the approximate equation (3.1) in any time interval [0,T]. In addition,with the help of the properties of weak compactness in[19],(3.4),(3.5)and(3.12),it is implied thatis weakly compact in L1([0,T]× T3× R3). Thus we may assume without loss of generality that there exists an elementsuch that fnconverges weakly to f in L1([0,T]× T3× R3) for all T ∈ [0,∞) as n tends to infinity.

Now consider the following renormalized approximate equation:

with any δ >0, hδ= δ?1log(1+ δh) and

Equation (4.1) is similar to that given in [16]. It can be seen that equation (4.1) is satisfied in the distribution sense when h=fn, and that=δ?1log(1+δfn). Also,is weakly compact in L1([0,T]× T3×R3) becauseHere we denote by fδthe weak limit ofin L1.

We next show the weak precompactness of the sequencewhere

Let BR= {p ∈ R3: |p| ≤ R}. Then, by use of (1.12), (3.4), (3.5), (3.12) and the fact that the range of ω is a bounded set, it can be proven thatis weakly compact infor any δ >0 and R>0. By using a scheme which is similar to that given by Jiang (see Property 1 in [30]), it follows thatis weakly compact in L1([0,T]× T3× BR). For the gain termwhich is similar to the classical case (see Theorem 6 in [6]), we have that

Thus, by use of Lemma 1.1, it follows that

for any ? ∈ L∞([0,T]× T3× R3). Moreover, it can be proven thatconverges to fnin L1(T3× R3) uniformly in [0,T]as δ tends to zero because of the weak L1-compactness of fn(see [16]); that is,

as δ tends to 0. By taking a weak limit, it follows that fδconverges to f in L1([0,T]×T3×R3)as δ tends to 0. Therefore we have that

Note that the mild solutionsatisfies the truncated relativistic Enskog equation in the iterated integral form

with ψ ∈ CL as given in Section 1. Let n tend to infinity and δ to zero in (4.10). Then, by the previous discussion about the momentum-averaged compactness, it follows that the first three terms in (4.10) converge to

For the last term in (4.10), we study the loss and gain parts of the collision operator.

For the loss term in (4.10), we have that

Because fnconverges to f weakly in L1([0,T]×T3×R3), by use of Lemma 1.1, and taking a change of variables x to x+aω, it follows from (4.7)–(4.9) that

for t ≤ T. Furthermore, by letting ?1tend zero, it can be found that (4.14) holds for any ψ ∈ CL with compact support in [0,T]× T3× R3. The last term in (4.12) can be treated similarly, which leads to

To sum up, we have that

For the gain term in (4.10), by replacing the characteristic function with χλin the proofs of Lemma 10 and Lemma 11 in [6], it follows that

for t ≤T.

Therefore f satisfies the following equation in the integral form:

Note that

Thus, for any 0 ≤ t

as h tends to zero. In view of (4.8), this implies that

By taking weak limits,it follows that f?∈ C([0,∞];L1(T3×R3)). Then by using an elementary argument from integration theory, it follows that f ∈ C([0,∞];L1(T3× R3)).

The boundedness of the mass and energy of the system follows from (3.4) and (3.5) with the help of the Fatou lemma. Thus

at almost every t ∈ [0,T]for all T < ∞. The boundedness of the entropy of f follows from the weak lower semicontinuous property of the entropy (see [10]):

Moreover, in view of (3.5), we have that

as n tends to infinity. Thus, by keeping the term I3in (3.7) throughout the limit, it follows that

5 Convergence to a Solution of the Relativistic Boltzmann Equation

In this section we prove Theorem 1.2; that is, we prove that the solutions of the relativistic Enskog equation in Theorem 1.1 converge to the solution of the related relativistic Boltzmann equation

as the diameter a tends to zero.

The notation of collision operator of the relativistic Boltzmann equation is here different from those given in [18, 29, 30, 32]. It is written as

We can use the basic property of the relativistic Boltzmann equation given by Strain [42]:

Thus there is a global mild solution of the relativistic Boltzmann equation under the assumptions of (1.10) and (1.11).

Because the solution faiof the relativistic Enskog equation with the diameter aihas finite mass, energy and entropy, it can be found that the sequenceis weakly compact in L1((0,T) × T3× R3) for all T < ∞. Without loss of generality, there exists an elementsuch that faiconverges weakly to f in L1([0,T]× T3× R3) for all T ∈ [0,∞) as i tends to infinity. Throughout this section, we assume thatis always constant as aitends to zero. In order to show that the weak limit f is the global solution of the relativistic Boltzmann equation,one only needs to process the proof of Theorem 1.2 step by step by following the proof of Theorem 1.1. By truncating the initial data f0withit follows from our discussion in Section 3 that there exists a unique nonnegative solutionto the problem of approximate equation(3.1)with the diameter ai,and thatweakly converges to faias n tends to infinity. Denote the subsequenceby fi′. It can be found that fi′weakly converges to f in L1. Thus Theorem 1.2 follows from the proof of Theorem 1.1 by replacing (4.13) with the assertion that

in L1. This is the direct application of the momentum-averaged technique given by Golse et al.[26]. In fact, if we approximate ? byas m tends to infinity,because fi′converges weakly to f in L1, by using a proof which is similar to that of (4.7), it follows that

Let m tend to infinity. This leads to (5.8), and completes the proof of Theorem 1.2.