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We study supersonic flow past a convex corner which is surrounded past quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a stiff rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to split the supersonic flow behind the rarefaction wave from the static gas. In this paper, we use a wave front tracking method to establish structural stability of such a flow design under non-smooth perturbations of the upcoming supersonic catamenia. It is an initial-value/free-boundary trouble for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful assay of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the potent rarefaction wave has a large full variation.
Mathematics Bailiwick 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 arrangement. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2911-2943. doi: ten.3934/dcds.2018125
References:
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| [2] | A. Bressan, Hyperbolic Systems of Conservation Laws: The I-Dimensional Cauchy Problem, Oxford Lecture Serial in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. |
| [3] | 1000.-Q. Yard. Chen, J. Kuang and Y. Zhang, Two-dimensional steady supersonic exothermically reacting Euler menstruation by Lipschitz angle walls, SIAM J. Math. Anal., 49 (2017), 818-873. doi: 10.1137/16M1075089. |
| [4] | Chiliad. -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/i.4790887. |
| [5] | G.-Q. G. Chen, Five. Kukreja and H. Yuan, Well-posedness of transonic feature discontinuities in two-dimensional steady compressible Euler flows, Z. Angew. Math. Phys., 64 (2013), 1711-1727. doi: 10.1007/s00033-013-0312-vi. |
| [half-dozen] | G.-Q. Chiliad. 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] | M.-Q. G. Chen, Y. Zhang and D. Zhu, Beingness and stability of supersonic Euler flows past Lipschitz wedges, Curvation. Rational Mech. Anal., 181 (2006), 261-310. doi: x.1007/s00205-005-0412-3. |
| [eight] | R. Courant and 1000. 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, 4 th edition, Grundlehren der Mathematischen Wissenschaften [Primal 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. |
| [eleven] | M. Ding, Stability of rarefaction wave to the ane-D piston problem for exothermically reacting Euler equations Calc. Var. Partial Differential Equations, 56(2017), Fine art. 78, 49 pp. doi: ten.1007/s00526-017-1162-4. |
| [12] | Thousand. 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. |
| [xiii] | G. Ding and Y. Li, Stability and not-relativistic limits of rarefaction moving ridge to the 1-D piston problem for the relativistic Euler equations Z. Angew. Math. Phys. 68 (2017), Fine art. 43, 32 pp. doi: 10.1007/s00033-017-0787-7. |
| [fourteen] | J. Glimm, Solutions in the big for nonlinear hyperbolic systems of equations, Comm. Pure. Appl. Math., 18 (1965), 697-715. doi: 10.1002/cpa.3160180408. |
| [15] | H. Holden and North. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2 nd edition, Practical Mathematical Sciences, 152, Springer-Verlag, Berlin Heidelberg, 2015. |
| [16] | Five. 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. |
| [eighteen] | A. Qu and W. Xiang, Three-Dimensional Steady Supersonic Euler Catamenia By a Concave Cornered Wedge with Lower Pressure at the Downstream, Arch Rational Mech. Anal., 228 (2018), 431-476. doi: 10.1007/s00205-017-1197-10. |
| [xix] | J. Smoller, Shock Waves and Reaction-Diffusion Equations, ii nd edition, Springer-Verlag, New York, 1994. |
| [xx] | Y.-K. Wang and H. Yuan, Weak stability of transonic contact discontinuities in iii-dimensional steady not-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), 341-388. doi: ten.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: x.1016/j.jde.2009.05.010. |
| [22] | Y. Zhang, Steady supersonic flow over a bending wall, Nonlinear Anal. Existent World Appl., 12 (2011), 167-189. doi: 10.1016/j.nonrwa.2010.06.006. |
prove all references
References:
| [1] | D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., four (1997), 1-42. doi: 10.1007/PL00001406. |
| [two] | A. Bressan, Hyperbolic Systems of Conservation Laws: The Ane-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. |
| [3] | K.-Q. G. Chen, J. Kuang and Y. Zhang, Two-dimensional steady supersonic exothermically reacting Euler flow past Lipschitz angle walls, SIAM J. Math. Anal., 49 (2017), 818-873. doi: 10.1137/16M1075089. |
| [four] | G. -Q. Thousand. Chen, V. Kukreja and H. Yuan, Stability of transonic feature discontinuities in two-dimensional steady compressible Euler flows J. Math. Phys., 54 (2013), 021506, 24 pp. doi: 10.1063/i.4790887. |
| [5] | G.-Q. G. Chen, V. Kukreja and H. Yuan, Well-posedness of transonic characteristic discontinuities in ii-dimensional steady compressible Euler flows, Z. Angew. Math. Phys., 64 (2013), 1711-1727. doi: 10.1007/s00033-013-0312-6. |
| [vi] | 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. One thousand. Chen, Y. Zhang and D. Zhu, Existence and stability of supersonic Euler flows past Lipschitz wedges, Curvation. Rational Mech. Anal., 181 (2006), 261-310. doi: 10.1007/s00205-005-0412-iii. |
| [8] | R. Courant and Grand. O. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 12, Wiley-Interscience, New York, 1948. |
| [ix] | C. 1000. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4 thursday edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2016. |
| [ten] | M. Ding, Being and stability of rarefaction moving ridge to 1-D piston problem for the relativistic total Euler equations, J. Differential Equations, 262 (2017), 6068-6108. doi: ten.1016/j.jde.2017.02.028. |
| [11] | Chiliad. Ding, Stability of rarefaction moving ridge to the 1-D piston trouble for exothermically reacting Euler equations Calc. Var. Partial Differential Equations, 56(2017), Fine art. 78, 49 pp. doi: x.1007/s00526-017-1162-iv. |
| [12] | M. Ding, J. Kuang and Y. Zhang, Global stability of rarefaction moving ridge to the ane-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. |
| [xiii] | Chiliad. Ding and Y. Li, Stability and non-relativistic limits of rarefaction moving ridge 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., eighteen (1965), 697-715. doi: ten.1002/cpa.3160180408. |
| [15] | H. Holden and N. H. Risebro, Front end Tracking for Hyperbolic Conservation Laws, 2 nd edition, Applied Mathematical Sciences, 152, Springer-Verlag, Berlin Heidelberg, 2015. |
| [sixteen] | 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: ten.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 West. Xiang, 3-Dimensional Steady Supersonic Euler Menstruum 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-ten. |
| [19] | J. Smoller, Stupor Waves and Reaction-Improvidence Equations, 2 nd edition, Springer-Verlag, New York, 1994. |
| [20] | Y.-G. Wang and H. Yuan, Weak stability of transonic contact discontinuities in 3-dimensional steady not-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), 341-388. doi: ten.1007/s00033-014-0404-y. |
| [21] | Z. Wang and Y. Zhang, Steady supersonic flow by 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: ten.1016/j.nonrwa.2010.06.006. |
Effigy ane. A transonic characteristic discontinuity separating supersonic menstruation behind the rarefaction wave and the surrounding static gas.
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