Acta mathematica scientia, Series B >
EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS
Received date: 2013-05-10
Revised date: 2013-10-19
Online published: 2014-01-20
Supported by
The research of Gui-Qiang CHEN was supported in part by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal Society-Wolfson Research Merit Award (UK). Changguo XIAO was supported in part by the NSFC under a joint project Grant 10728101. Yongqian ZHANG was supported in part by NSFC Project 11031001, NSFC Project 11121101, and the 111 Project B08018 (China).
We are concerned with the global existence of entropy solutions of the twodimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function (T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional
detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and
the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.
CHEN Gui-Qang , XIAO Chang-Guo , ZHANG Yong-Qian . EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS[J]. Acta mathematica scientia, Series B, 2014 , 34(1) : 1 -38 . DOI: 10.1016/S0252-9602(13)60123-X
[1] Chen G -Q, Wagner D. Global entropy solutions to exothermically reacting, compressible Euler equations. J Differ Equ, 2003, 191: 277–322
[2] Chen G -Q, Zhang Y Q, Zhu D W. Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Rational Mech Anal, 2006, 181: 261–310
[3] Chen G -Q, Zhang Y Q, Zhu D W. Stabilily of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls. SIAM J Math Anal, 2007, 38: 1660–1693
[4] Chen S -X. Asymptotic behavior of supersonic flow past a convex combined wedge. Chin Ann Math, 1998, 19B(3): 255–264
[5] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Wiley-Interscience, 1948
[6] Dafermos C. Hyperbolic Conservation Laws in Continuum Physics. Berlin: Springer-Verlag, 2005
[7] Dafermos C, Hsiao L. Hyperbolic systems of balance laws with inhomegeneity and dissipation. Indiana Univ Math J, 1982, 31: 471–491
[8] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18: 697–715
[9] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537–566
[10] Liu T -P. Solutions in the large for the equations of nonisentropic gas dynamics. Indiana Univ Math J, 1977, 26: 147–177
[11] Liu T -P. Large-time behaviour of initial and initial-boundary value problems of a general systems of hyperbolic conservation laws. Comm Math Phys, 1977, 55: 163–177
[12] Luskin M, Temple J B. The existence of a global weak solution to the nonlinear waterhammer problem. Comm Pure Appl Math, 1982, 34: 697–735
[13] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1983
[14] Temple J B. Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics. J Differ Equ, 1981, 41: 96–161
[15] Volpert A I. The space BV and quasilinear equations. Mat Sb (NS), 1967, 73: 255–302 (in Russian); Math USSR Sb, 1967, 2: 225–267 (in English)
[16] Ying L -A, Wang C -H. Global solutions of the Cauchy problem for a nonhomogeneous quasilinear hyper-bolic system. Comm Pure Appl Math, 1980, 33: 579–597
[17] Ying L -A, Wang C -H. Solutions in the large for nonhomogeneous quasilinear hyperbolic systems of equations. J Math Anal Appl, 1980, 78: 440–454
[18] Zhang Y Q. Global existence of steady supersonic potential flow past a curved wedge with piecewise smooth
boundary. SIAM J Math Anal, 1999, 31: 166–183
[19] Zhang Y Q. Steady supersonic flow past an almost straight wedge with large vertex angle. J Differ Equ, 2003, 192: 1–46
/
| 〈 |
|
〉 |