Global Weak Solutions for a Nonlinear Hyperbolic System
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM∗
k)
Qingyou SUN (
º 1)
Yunguang LU (
†
K.K.Chen Institute for Advanced Studies, Hangzhou Normal University, Hangzhou 311121, China E-mail : [email protected]; [email protected]
Christian KLINGENBERG Department of Mathematics, Wuerzburg University, Wuerzburg 97070, Germany E-mail : [email protected] Abstract In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable s, the system about the variables ρ and u is the classical 1 isentropic gas dynamics in Eulerian coordinates with the pressure function P (ρ, s) = es e− ρ , which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants (w1 (x, 0), w2 (x, 0)) of system (1.1) is small, (w1 (x, 0), w2 (x, 0)) are nondecreasing and the third Riemann invariant s(x, 0) is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1]. Key words
global weak solutions; viscosity method; compensated compactness
2010 MR Subject Classification
1
35L15; 35A01
Introduction
In this paper, we study the global solutions of the nonlinearly conservation system of three equations ρt + (ρu)x = 0, (ρu)t + (ρu2 + P (ρ, s))x = 0,
with bounded initial data
(ρs)t + (ρus)x = 0
(ρ, u, s)|t=0 = (ρ0 (x), u0 (x), s0 (x)), ∗ Received
(1.1)
ρ0 (x) ≥ 0, s0 (x) ≥ 0,
(1.2)
December 28, 2019; revised May 9, 2020. This work was supported by the the NSFC (LY20A010023) and a professorship called Qianjiang scholar of Zhejiang Province of China. † Corresponding author: Yunguang LU.
1186
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
1
where P (ρ, s) is fixed as es e− ρ . System (1.1) with this special pressure is interesting because it is the unique example, in the systems of nonisentropic gas flow, which can be diagonalized. Substituting the first equation in (1.1) into the second and the third, we have, for the smooth solution, the following equivalent system about the variables (ρ, u, s), ρt + uρx + ρux = 0, 1 1 1 1 (1.3) ut + 3 es− ρ ρx + uux + es− ρ sx = 0, ρ ρ st + usx = 0. Let the matrix dF (U ) be
u
1 s− 1 dF (U ) = ρ3 e ρ 0
ρ
0
u
1 s− ρ1 e ρ
0
u
.
(1.4)
Then three eigenvalues of (1.1) are 1 1 s λ1 = u − e 2 e− 2ρ , ρ
1 1 s λ2 = u + e 2 e− 2ρ ρ
λ3 = u
(1.5)
w3 = s.
(1.6)
with corresponding three Riemann invariants s
1
w1 = u − 2e 2 e− 2ρ ,
s
1
w2 = u + 2e 2 e− 2ρ ,
Based on the following condition (H
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