The Existence of a Bounded Invariant Region for Compressible Euler Equations in Different Gas States
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES∗
ö¸)
Weifeng JIANG (
College of Science, China Jiliang University, Hangzhou 310018, China E-mail : [email protected]
)
Zhen WANG (
†
College of Science, Wuhan University of Technology, Wuhan 430071, China E-mail : [email protected] Abstract In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special state equation, a bounded invariant region for the non-isentropic process may exist. Key words
Euler equations; gas dynamic; non-isentropic; existence of invariant region
2010 MR Subject Classification
1
35L65
Introduction
In this article, we are concerned with the existence of the invariant region of one-dimensional non-isentropic gas dynamic equations, which is characterized by ρt + (ρu)x = 0, (1.1) (ρu)t + (ρu2 + p)x = 0, (ρE)t + (ρuE + pu)x = 0, with the gas state
p = p(ρ, s),
(1.2)
p = RρT, e = cv T, p(s, ρ) = es/cv ργ ,
(1.3)
and
∗ Received
August 24, 2019; revised May 13, 2020. The first author was supported by the Natural Science Foundation of Zhejiang (LQ18A010004), the second author was supported by the Fundamental Research Funds for the Central Universities (WUT: 2020IB011). † Corresponding author: Zhen WANG.
1230
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
where ρ > 0, s and u are density, entropy and velocity, respectively, and E = 12 u2 + e is the energy, with e being the internal energy. R, k, cv and γ are positive constants, and γ > 1. The initial data of (1.1) is (ρ , m , q ) x < 0, l l l (ρ, m, q)(x, t)|t=0 = (1.4) (ρr , mr , qr ) x > 0;
here m = ρu, q = ρE, and ρl,r , ml,r , ql,r are all given constants. The gas dynamic equation is one of the core subjects of conservation law, in which the existence of global weak solutions with large initial data is the most important problem. As far as we know, compensated compactness [1] is one of the most effective methods for solving the problem; by using it, the framework of the existence of the solution for isentropic gas dynamic equations with Cauchy data is almost completed. Diperna [2, 3] established the existence of the weak entropy solution for the isentropic Euler equations with general L∞ initial 2 data for γ = 1 + 2n+1 , where n ≥ 2 and n is an integer. Ding, Chen and Luo [4, 5] also got the existence of the isentropic solution by the vanishing numerical viscosity for γ ∈ [1, 53 ]. Lions, Perthame, Tadmor and Souganidis [6–8] got the existence results for γ > 3, while Huang and Wang [9] got the existence results for γ = 1. However, the compensated compactness theory enco
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