EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS

• Published in: Acta mathematica scientia Vol. 34; no. 1; pp. 1 - 38
• Main Author: 陈贵强 肖长国 张永前
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 2014; Mathematical Institute, University of 0xford, 0xford, 0X2 6GG, UK%School of Mathematical Sciences, Fudan University, Shanghai 200433, China
• Subjects: 35A01; 35B40; 35L45; 35L50; 35L65; 35L67; 76N10; asymptotic behavior; Cauchy问题; combustion; Detonation; detonation wave; Detonation waves; Entropy; entropy solutions; Euler equations; Euler方程; Exothermic reactions; fractional-step; Glimm scheme; Initial value problems; Mathematical analysis; reacting Euler flow; Riemann problem; stability; steady flow; supersonic flow; two-dimensional; Wedges; 二维; 初始边界值问题; 恒定; 放热反应; 整体存在性; 熵解; fractional-step; 35L50; detonation wave; Riemann problem; entropy solutions; 35A01; Glimm scheme; 35L45; 35L67; steady flow; 35L65; 76N10; asymptotic behavior; combustion; reacting Euler flow; two-dimensional; supersonic flow; 35B40; stability; su-personic flow
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1016/S0252-9602(13)60123-X

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（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 ad-ditional 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.