Figure 1.
A transonic characteristic discontinuity separating supersonic flow behind the rarefaction wave and the surrounding static gas.
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June
2018, 38(6): 2911-2943.
doi: 10.3934/dcds.2018125
Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system
| 1. | Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China |
| 2. | Department of Mathematics, Center for PDE and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China |
We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.
Keywords:
Compressible Euler equations,
characteristic discontinuity,
transonic,
rarefaction wave,
wave front tracking,
interaction of waves,
Glimm functional.
Mathematics Subject Classification:
Primary: 35L50, 35L65, 35Q31, 35R35; Secondary: 76N10.
Citation:
Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete and Continuous Dynamical Systems,
2018, 38
(6)
: 2911-2943.
doi: 10.3934/dcds.2018125
![]()
References:
| [1] |
D. Amadori,
Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.
doi: 10.1007/PL00001406. |
| [2] |
A. Bressan,
Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. |
| [3] |
G.-Q. G. Chen, J. Kuang and Y. Zhang,
Two-dimensional steady supersonic exothermically reacting Euler flow past Lipschitz bending walls, SIAM J. Math. Anal., 49 (2017), 818-873.
doi: 10.1137/16M1075089. |
| [4] |
G. -Q. G. Chen, V. Kukreja and H. Yuan, Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows J. Math. Phys., 54 (2013), 021506, 24 pp.
doi: 10.1063/1.4790887. |
| [5] |
G.-Q. G. Chen, V. Kukreja and H. Yuan,
Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows, Z. Angew. Math. Phys., 64 (2013), 1711-1727.
doi: 10.1007/s00033-013-0312-6. |
| [6] |
G.-Q. G. Chen, Y. Zhang and D. Zhu,
Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls, SIAM J. Math. Anal., 38 (2006/07), 1660-1693.
doi: 10.1137/050642976. |
| [7] |
G.-Q. G. Chen, Y. Zhang and D. Zhu,
Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Rational Mech. Anal., 181 (2006), 261-310.
doi: 10.1007/s00205-005-0412-3. |
| [8] |
R. Courant and K. O. Friedrichs,
Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 12, Wiley-Interscience, New York, 1948. |
| [9] |
C. M. Dafermos,
Hyperbolic Conservation Laws in Continuum Physics, 4th edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2016. |
| [10] |
M. Ding,
Existence and stability of rarefaction wave to 1-D piston problem for the relativistic full Euler equations, J. Differential Equations, 262 (2017), 6068-6108.
doi: 10.1016/j.jde.2017.02.028. |
| [11] |
M. Ding, Stability of rarefaction wave to the 1-D piston problem for exothermically reacting Euler equations Calc. Var. Partial Differential Equations, 56(2017), Art. 78, 49 pp.
doi: 10.1007/s00526-017-1162-4. |
| [12] |
M. Ding, J. Kuang and Y. Zhang,
Global stability of rarefaction wave to the 1-D piston problem for the compressible full Euler equations, J. Math. Anal. Appl., 448 (2017), 1228-1264.
doi: 10.1016/j.jmaa.2016.11.059. |
| [13] |
M. Ding and Y. Li, Stability and non-relativistic limits of rarefaction wave to the 1-D piston problem for the relativistic Euler equations Z. Angew. Math. Phys. 68 (2017), Art. 43, 32 pp.
doi: 10.1007/s00033-017-0787-7. |
| [14] |
J. Glimm,
Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure. Appl. Math., 18 (1965), 697-715.
doi: 10.1002/cpa.3160180408. |
| [15] |
H. Holden and N. H. Risebro,
Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, 152, Springer-Verlag, Berlin Heidelberg, 2015. |
| [16] |
V. Kukreja, H. Yuan and Q. Zhao,
Stability of transonic jet with strong shock in two-dimensional steady compressible Euler flows, J. Differential Equations, 258 (2015), 2572-2617.
doi: 10.1016/j.jde.2014.12.017. |
| [17] |
L. Liu, G. Xu and H. Yuan,
Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations, Adv. Math., 291 (2016), 696-757.
doi: 10.1016/j.aim.2016.01.002. |
| [18] |
A. Qu and W. Xiang,
Three-Dimensional Steady Supersonic Euler Flow Past a Concave Cornered Wedge with Lower Pressure at the Downstream, Arch Rational Mech. Anal., 228 (2018), 431-476.
doi: 10.1007/s00205-017-1197-x. |
| [19] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. |
| [20] |
Y.-G. Wang and H. Yuan,
Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), 341-388.
doi: 10.1007/s00033-014-0404-y. |
| [21] |
Z. Wang and Y. Zhang,
Steady supersonic flow past a curved cone, J. Differential Equations, 247 (2009), 1817-1850.
doi: 10.1016/j.jde.2009.05.010. |
| [22] |
Y. Zhang,
Steady supersonic flow over a bending wall, Nonlinear Anal. Real World Appl., 12 (2011), 167-189.
doi: 10.1016/j.nonrwa.2010.06.006. |
show all references
References:
| [1] |
D. Amadori,
Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.
doi: 10.1007/PL00001406. |
| [2] |
A. Bressan,
Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. |
| [3] |
G.-Q. G. Chen, J. Kuang and Y. Zhang,
Two-dimensional steady supersonic exothermically reacting Euler flow past Lipschitz bending walls, SIAM J. Math. Anal., 49 (2017), 818-873.
doi: 10.1137/16M1075089. |
| [4] |
G. -Q. G. Chen, V. Kukreja and H. Yuan, Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows J. Math. Phys., 54 (2013), 021506, 24 pp.
doi: 10.1063/1.4790887. |
| [5] |
G.-Q. G. Chen, V. Kukreja and H. Yuan,
Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows, Z. Angew. Math. Phys., 64 (2013), 1711-1727.
doi: 10.1007/s00033-013-0312-6. |
| [6] |
G.-Q. G. Chen, Y. Zhang and D. Zhu,
Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls, SIAM J. Math. Anal., 38 (2006/07), 1660-1693.
doi: 10.1137/050642976. |
| [7] |
G.-Q. G. Chen, Y. Zhang and D. Zhu,
Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Rational Mech. Anal., 181 (2006), 261-310.
doi: 10.1007/s00205-005-0412-3. |
| [8] |
R. Courant and K. O. Friedrichs,
Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 12, Wiley-Interscience, New York, 1948. |
| [9] |
C. M. Dafermos,
Hyperbolic Conservation Laws in Continuum Physics, 4th edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2016. |
| [10] |
M. Ding,
Existence and stability of rarefaction wave to 1-D piston problem for the relativistic full Euler equations, J. Differential Equations, 262 (2017), 6068-6108.
doi: 10.1016/j.jde.2017.02.028. |
| [11] |
M. Ding, Stability of rarefaction wave to the 1-D piston problem for exothermically reacting Euler equations Calc. Var. Partial Differential Equations, 56(2017), Art. 78, 49 pp.
doi: 10.1007/s00526-017-1162-4. |
| [12] |
M. Ding, J. Kuang and Y. Zhang,
Global stability of rarefaction wave to the 1-D piston problem for the compressible full Euler equations, J. Math. Anal. Appl., 448 (2017), 1228-1264.
doi: 10.1016/j.jmaa.2016.11.059. |
| [13] |
M. Ding and Y. Li, Stability and non-relativistic limits of rarefaction wave to the 1-D piston problem for the relativistic Euler equations Z. Angew. Math. Phys. 68 (2017), Art. 43, 32 pp.
doi: 10.1007/s00033-017-0787-7. |
| [14] |
J. Glimm,
Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure. Appl. Math., 18 (1965), 697-715.
doi: 10.1002/cpa.3160180408. |
| [15] |
H. Holden and N. H. Risebro,
Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, 152, Springer-Verlag, Berlin Heidelberg, 2015. |
| [16] |
V. Kukreja, H. Yuan and Q. Zhao,
Stability of transonic jet with strong shock in two-dimensional steady compressible Euler flows, J. Differential Equations, 258 (2015), 2572-2617.
doi: 10.1016/j.jde.2014.12.017. |
| [17] |
L. Liu, G. Xu and H. Yuan,
Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations, Adv. Math., 291 (2016), 696-757.
doi: 10.1016/j.aim.2016.01.002. |
| [18] |
A. Qu and W. Xiang,
Three-Dimensional Steady Supersonic Euler Flow Past a Concave Cornered Wedge with Lower Pressure at the Downstream, Arch Rational Mech. Anal., 228 (2018), 431-476.
doi: 10.1007/s00205-017-1197-x. |
| [19] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. |
| [20] |
Y.-G. Wang and H. Yuan,
Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), 341-388.
doi: 10.1007/s00033-014-0404-y. |
| [21] |
Z. Wang and Y. Zhang,
Steady supersonic flow past a curved cone, J. Differential Equations, 247 (2009), 1817-1850.
doi: 10.1016/j.jde.2009.05.010. |
| [22] |
Y. Zhang,
Steady supersonic flow over a bending wall, Nonlinear Anal. Real World Appl., 12 (2011), 167-189.
doi: 10.1016/j.nonrwa.2010.06.006. |
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