Global Existence for the Relativistic Enskog Equations
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
GLOBAL EXISTENCE FOR THE RELATIVISTIC ENSKOG EQUATIONS∗
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Jianjun HUANG (
Department of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082, China E-mail : [email protected]
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Zhenglu JIANG (
†
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China E-mail : [email protected] Abstract This article extends the results of Arkeryd and Cercignani [6]. It is shown that the Cauchy problem for the relativistic Enskog equation in a periodic box has a global mild solution if the mass, energy and entropy of the initial data are finite. It is also found that the solutions of the relativistic Enskog equation weakly converge to the solutions of the relativistic Boltzmann equation in L1 if the diameter of the relativistic particle tends to zero. Key words
global existence; relativistic Enskog equation; relativistic Boltzmann equation
2010 MR Subject Classification
1
76P05; 35Q75
Introduction
We consider the Cauchy problem for the relativistic Enskog equation in a periodic box with the initial data satisfying finite mass, energy and “entropy”. The relativistic Enskog equation can be regarded as a modification of the relativistic Boltzmann equation when the density of the gas increases. In this model, the relativistic gas is described by the distribution function f = f (t, x, p) of particles with the momentum p ∈ R3 at the space position x ∈ T3 and the time t ∈ (0, ∞), where R3 is a three-dimensional Euclidean space and T3 is the region of the space position taken as [−π, π]3 under some periodic boundary conditions. This distribution function satisfies the relativistic Enskog equation in the following form: ∂f p ∂f + = Q(f, f ). ∂t p0 ∂x
(1.1)
Here p0 = (1 + |p|2 )1/2 is 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 ∗ Received
January 19, 2019; revised February 19, 2020. This work was supported by NSFC (11171356). author: Zhenglu JIANG.
† Corresponding
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
expressed as follows: +
Q (f, f ) = a
2
Z
2 R3 ×S+
q(p, p∗ , ω)Y (ρ+ )f (t, x, p′ )f (t, x + aω, p′∗ )dωdp∗ , Q− (f, f ) = f L(f ),
(1.2) (1.3)
where L(f ) = a2
Z
q(p, p∗ , ω)Y (ρ− )f (t, x − aω, p∗ )dωdp∗ .
(1.4)
2 R3 ×S+
2 In (1.2) and (1.4), a is the diameter of the hard sphere, of course, a > 0, S+ is a subset of the 2 2 2 unit sphere surface S , which is defined by S+ = {ω ∈ S : 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 Z ρ(t, x) = f (t, x, p)dp. (1.5) R3
The derivation of (1.1
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