Two-dimensional steady supersonic exothermically reacting Euler flows with strong contact discontinuity over a Lipschitz wall

In this paper, we established the global existence of supersonic entropy solutions with a strong contact discontinuity over Lipschitz wall governed by the two-dimensional steady exothermically reacting Euler equations, when the total variation of both initial data and the slope of Lipschitz wall is sufficiently small. Local and global estimates are developed and a modified Glimm-type functional is carefully designed. Next the validation of the quasi-one-dimensional approximation in the domain bounded by the wall and the strong contact discontinuity is rigorous justified by proving that the difference between the average of weak solution and the solution of quasi-one-dimensional system can be bounded by the square of the total variation of both initial data and the slope of Lipschitz wall. The methods and techniques developed here is also helpful for other related problems.

[1]  P. LeFloch,et al.  A version of the Glimm method based on generalized Riemann problems , 2007, math/0701063.

[2]  Dianwen Zhu,et al.  Stability of Compressible Vortex Sheets in Steady Supersonic Euler Flows over Lipschitz Walls , 2007, SIAM J. Math. Anal..

[3]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[4]  Gui-Qiang G. Chen,et al.  Existence and Stability of Supersonic Euler Flows Past Lipschitz Wedges , 2006 .

[5]  Chenshuxing.  ASYMPTOTIC BEHAVIOUR OF SUPERSONIC FLOW PAST A CONVEX COMBINED WEDGE , 1998 .

[6]  Andrew J. Majda,et al.  Theoretical and numerical structure for unstable one-dimensional detonations , 1991 .

[7]  P. Lax,et al.  Decay of solutions of systems of nonlinear hyperbolic conservation laws , 1970 .

[8]  Gui-Qiang G. Chen,et al.  Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows , 2012, 1209.3806.

[9]  Donald Scott Stewart,et al.  Calculation of linear detonation instability: one-dimensional instability of plane detonation , 1990, Journal of Fluid Mechanics.

[10]  Gui-Qiang G. Chen,et al.  EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS , 2013, 1310.7151.

[11]  A. Corli,et al.  Stability of contact discontinuities under perturbations of bounded variation , 1997 .

[12]  Hairong Yuan,et al.  Stability of transonic jet with strong shock in two-dimensional steady compressible Euler flows , 2015 .

[13]  Shuxing Chen,et al.  Isentropic Approximation of Quasi-One-Dimensional Unsteady Nozzle Flow , 2009, SIAM J. Math. Anal..

[14]  Yongqian Zhang,et al.  Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls , 2015, SIAM J. Math. Anal..

[15]  Yongqian Zhang.  Steady supersonic flow over a bending wall , 2011 .

[16]  C. Dafermos,et al.  Hyperbolic Systems of Balance Laws with Inhomogeneity and Dissipation , 1980 .

[17]  Yongqian Zhang,et al.  Global Existence of Steady Supersonic Potential Flow Past a Curved Wedge with a Piecewise Smooth Boundary , 1999, SIAM J. Math. Anal..

[18]  M. Sablé-Tougeron,et al.  Méthode de Glimm et problème mixte , 1993 .

[19]  Ying Lung-an,et al.  Global solutions of the cauchy problem for a nonhomogeneous quasilinear hyperbolic system , 1980 .

[20]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

[21]  Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws , 2012, 1205.5930.

[22]  Yongqian Zhang.  Steady supersonic flow past an almost straight wedge with large vertex angle , 2003 .

[23]  F. Spellman.  Combustion Theory , 2020 .

[24]  A. Bressan.  Hyperbolic systems of conservation laws : the one-dimensional Cauchy problem , 2000 .

[25]  C. Christoforou.  A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws , 2015, 1603.08126.

[26]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[27]  Andrew J. Majda,et al.  Qualitative model for dynamic combustion , 1981 .

[28]  Gui-Qiang G. Chen,et al.  Global entropy solutions to exothermically reacting, compressible Euler equations , 2003 .

[29]  C. M. Dafermos,et al.  Hyberbolic [i.e. Hyperbolic] conservation laws in continuum physics , 2005 .

[30]  W. Fickett,et al.  Flow Calculations for Pulsating One‐Dimensional Detonations , 1966 .

[31]  D. Schaeffer.  Supersonic flow past a nearly straight wedge , 1976 .

[32]  J. Erpenbeck,et al.  STABILITY OF IDEALIZED ONE-REACTION DETONATIONS , 1964 .

[33]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[34]  Yongqian Zhang,et al.  Irrotational approximation to the quasi-1-d gas flow , 2009 .

[35]  Tai-Ping Liu.  Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws , 1977 .