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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

* Corresponding author: Hairong Yuan

Received  July 2017 Revised  December 2017 Published  April 2018

Fund Project: The research of Min Ding is supported by the Fundamental Research Funds for the Central Universities (WUT: 2016IVA074) and by the National Natural Science Foundation of China under Grant Nos. 11626176 and 11701435. The research of Hairong Yuan is supported by the National Natural Science Foundation of China under Grant No. 11371141, and by Science and Technology Commission of Shanghai Municipality (STCSM) under grant No. 13dz2260400.

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.

Citation: Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 12, Wiley-Interscience, New York, 1948.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, 152, Springer-Verlag, Berlin Heidelberg, 2015.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 12, Wiley-Interscience, New York, 1948.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, 152, Springer-Verlag, Berlin Heidelberg, 2015.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
A transonic characteristic discontinuity separating supersonic flow behind the rarefaction wave and the surrounding static gas.
Reflection on the free-boundary.
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