Stability of transonic shock fronts in two-dimensional Euler systems

We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.

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

[2]  Tai-Ping Liu,et al.  Nonlinear Stability of a Self-Similar 3-Dimensional¶Gas Flow , 1999 .

[3]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[4]  A. Majda.  The stability of multi-dimensional shock fronts , 1983 .

[5]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[6]  K. Zumbrun.  Multidimensional Stability of Planar Viscous Shock Waves , 2001 .

[7]  A. Majda,et al.  Multi‐dimensional shock fronts for second order wave equations , 1987 .

[8]  A. Majda,et al.  The stability of multidimensional shock fronts , 1983 .

[9]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[10]  Z. Xin.  On the linearized stability of viscous shock profiles for systems of conservation laws , 1992 .

[11]  S. Čanić,et al.  Quasi‐One‐Dimensional Riemann Problems and Their Role in Self‐Similar Two‐Dimensional Problems , 1998 .

[12]  S. Čanić,et al.  Free Boundary Problems for the Unsteady Transonic Small Disturbance Equation: Transonic Regular Reflection , 2000 .

[13]  C. Morawetz.  Potential theory for regular and mach reflection of a shock at a wedge , 1994 .

[14]  A. Majda,et al.  The existence of multidimensional shock fronts , 1983 .

[15]  Shuxing Chen.  Existence of Stationary Supersonic Flows Past a Pointed Body , 2001 .

[16]  Cathleen S. Morawetz,et al.  On the non‐existence of continuous transonic flows past profiles II , 1956 .

[17]  Yuxi Zheng,et al.  Systems of conservation laws , 2001 .

[18]  Guy Métivier,et al.  Stability of Multidimensional Shocks , 2001 .

[19]  Suncica Canic,et al.  Riemann Problems for the Two-Dimensional Unsteady Transonic Small Disturbance Equation , 1998, SIAM J. Appl. Math..

[20]  Eun Heui Kim,et al.  A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks , 2002 .

[21]  Barbara Lee Keyfitz,et al.  A proof of existence of perturbed steady transonic shocks via a free boundary problem , 2000 .

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

[23]  Mikhail Feldman,et al.  MULTIDIMENSIONAL TRANSONIC SHOCKS AND FREE BOUNDARY PROBLEMS FOR NONLINEAR EQUATIONS OF MIXED TYPE , 2003 .

[24]  Andrew J. Majda,et al.  One Perspective on Open Problems in Multi-Dimensional Conservation Laws , 1991 .