Acta Mathematicae Applicatae Sinica, English Series
Vol. 33, No. 3 (2017) 659–678
http://www.ApplMath.com.cn & www.SpringerLink.com
Acta MathemaƟcae Applicatae Sinica,
The Editorial Office of AMAS &
Springer-Verlag Berlin Heidelberg 2017
Non-selfsimilar Global Solutions and Their Structure
for the Multi-dimensional Combustion Models
, Xiao-zhou YANG
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
Abstract We investigate the Chapman-Jouguet model in multi-dimensional space, and construct explicitly
its non-selfsimilar Riemann solutions. By the method we apply in this paper, general initial discontinuities can
be dealt with, even for complex interaction of combustion waves. Furthermore, we analyze the way in which the
area of unburnt gas shrinks.
Keywords chapman-Jouguet model; multi-dimensional Riemann problem; non-selfsimilar solutions; charac-
2000 MR Subject Classiﬁcation 35Lxx
In this paper we investigate the Chapman-Jouguet model, which is an ideal model for describing
(u + q)
q(0,x), if sup
u(s, x) ≤ u
where t ∈ R
and x ∈ R
. The scalar value u represents a lumped quantity of gas ﬂow such as
density, velocity or temperature. u
is the ignition temperature of the reactant such that the
unburnt gas is burnt instantaneously once u exceeds u
. q denotes the binding energy of the
reactive gas. We require, for simplicity, that
, if u(0,x) ≤ u
0, if u(0,x) >u
is a positive constant which is determined by the chemical properties of reactant. f
(i =1, ···,n) are nonlinear functions satisfying f
(u) ≥ const > 0.
The initial data
, 0), if M(x) > 0,
), if M(x) < 0.
Manuscript received December 16, 2011.
Supported by the National Natural Science Foundation of China (No. 10871199 and 11071246).