STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

摘要/Abstract

摘要： We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

关键词: stability, multi-wave configuration, vortex sheet, entropy wave, shock wave, BV perturbation, full Euler equations, steady, wave interactions, Glimm scheme

Abstract: We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

Key words: stability, multi-wave configuration, vortex sheet, entropy wave, shock wave, BV perturbation, full Euler equations, steady, wave interactions, Glimm scheme

陈贵强, Matthew RIGBY. STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW[J]. 数学物理学报(英文版), 2018, 38(5): 1485-1514.

Gui-Qiang G. CHEN, Matthew RIGBY. STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW[J]. Acta Mathematica Scientia, 2018, 38(5): 1485-1514.

参考文献

[1] Artola M, Majda A. Nonlinear development of instability in supersonic vortex sheets, I:The basic kink modes. Phys D, 1987, 28:253-281
[2] Artola M, Majda A. Nonlinear development of instability in supersonic vortex sheets, Ⅱ:Resonant interaction among kink modes. SIAM J Appl Math, 1989, 49:1310-1349
[3] Bressan A. Hyperbolic Systems of Conservation Laws:The One-dimensional Cauchy Problem. Oxford:Oxford University Press, 2000
[4] Chang T, Hsiao L. The Riemann Problem and Interaction of Waves in Gas Dynamics. England:Longman Scientific & Technical, Essex, 1989
[5] Chen G Q. Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems. Sci China Math, 2017, 60:1353-1370
[6] Chen G Q. Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics Ⅲ. Acta Math Sci (Chinese), 1988, 8:243-276; Acta Math Sci (English ed), 1986, 6:75-120
[7] Chen G Q, Feldman M. The Mathematics of Shock Reflection-Diffraction and Von Neumann's Conjecture. Annals of Mathematics Studies, 197. Princetion:Princeton University Press:2018
[8] Chen G Q, Kuang J, Zhang Y. Two-dimensional steady supersonic exothermically reacting Euler flow past Lipschitz bending walls. SIAM J Math Anal, 2017, 49:818-873
[9] Chen G Q, Li T H. Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge. J Diff Eqs, 2008, 244:1521-1550
[10] Chen G Q, Wang Y G. Characteristic discontinuities and free boundary problems for hyperbolic conservation laws//Holden H, Karlsen K H, eds. Nonlinear Partial Differential Equations, The Abel Symposium 2010. Springer-Verlag:Heidelberg, 2012:53-82
[11] Chen G Q, Zhang Y, Zhu D. Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Ration Mech Anal, 2006, 181(2):261-310
[12] Chen G Q, Zhang Y, Zhu D. Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls. SIAM J Math Anal, 2006/07, 38(5):1660-1693(electronic)
[13] Chen S X. Stability of a Mach configuration. Comm Pure Appl Math, 2006, 59(1):1-35
[14] Chen S X. E-H type Mach configuration and its stability. Comm Math Phys, 2012, 315(3):563-602
[15] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. 4th ed. Berlin:Springer-Verlag:2016
[16] Ding X. Theory of conservation laws in China//Hyperbolic Problems:Theory, Numerics, Applications (Stony Brook, NY, 1994). River Edge, NJ:World Sci Publ, 1996:110-119
[17] Ding X (Ding Shia Shi), Chang T (Chang Tung), Wang J H (Wang Ching-Hua), Xiao L (Hsiao Ling), Li C Z (Li Tsai-Chung). A study of the global solutions for quasi-linear hyperbolic systems of conservation laws. Sci Sinica, 1973, 16:317-335
[18] Ding X, Chen G Q, Luo P. Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics I. Acta Math Sci (Chinese), 1987, 7:467-480, Acta Math Sci (English ed), 1985, 5:415-432; Ⅱ. Acta Math Sci (Chinese), 1988, 8:61-94, Acta Math Sci (English ed), 1985, 5:433-472
[19] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl Math, 1965, 18:697-715
[20] Lax P D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Philadelphia:SIAM-CBMS, 1973
[21] Lewicka M, Trivisa K. On the L¹ well posedness of systems of conservation laws near solutions containing two large shocks. J Diff Eqs, 2002, 179(1):133-177
[22] Liu T P. Large-time behaviour of initial and initial-boundary value problems of a general system of hyperbolic conservation laws. Comm Math Phys, 1977, 55:163-177
[23] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York:Springer-Verlag, 1994