# American Institute of Mathematical Sciences

August  2016, 36(8): 4179-4211. doi: 10.3934/dcds.2016.36.4179

## Transonic flows with shocks past curved wedges for the full Euler equations

 1 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcli e Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom 2 Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China 3 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388, United States

Received  April 2015 Revised  January 2016 Published  March 2016

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the transformation from Eulerian to Lagrangian coordinates and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for the nonlinear equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or equivalently the entropy function along the shock-front downstream, all the physical variables can be expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at infinity can be uniquely determined with a uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the nonlinear equation, and then update the entropy function to prove that the updating map has a fixed point. Another advantage in our analysis is in the context of the full Euler equations so that the Bernoulli constant is allowed to change for different fluid trajectories.
Citation: Gui-Qiang Chen, Jun Chen, Mikhail Feldman. Transonic flows with shocks past curved wedges for the full Euler equations. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4179-4211. doi: 10.3934/dcds.2016.36.4179
##### References:
 [1] G.-Q. Chen, J. Chen and M. Feldman, Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles, J. Math. Pures Appl. (9), 88 (2007), 191-218. doi: 10.1016/j.matpur.2007.04.008. [2] G.-Q. Chen and B. Fang, Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone, Discrete Contin. Dyn. Syst., 23 (2009), 85-114. doi: 10.3934/dcds.2009.23.85. [3] G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494. doi: 10.1090/S0894-0347-03-00422-3. [4] G.-Q. Chen and T. Li, Well-posedness for two-dimnsional steady supersonic Euler flows past a Lipschitz wedge, J. Diff. Eqs., 244 (2008), 1521-1550. [5] G.-Q. Chen, Y.-Q. Zhang and D.-W. 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. [6] S.-X. Chen, Supersonic flow past a concave wedge, Science in China, 10 (1997), 903-910. [7] S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge, Chin. Ann. Math., 19 (1998), 255-264. [8] S.-X. Chen, A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Z. Angew. Math. Phys., 54 (2003), 387-409. doi: 10.1007/s00033-003-2111-y. [9] S.-X. Chen, Stability of transonic shock front in multi-dimensional Euler system, Trans. Amer. Math. Soc., 357 (2005), 287-308. doi: 10.1090/S0002-9947-04-03698-0. [10] S.-X. Chen, Stability of a Mach configuration, Comm. Pure Appl. Math., 59 (2006), 1-35. doi: 10.1002/cpa.20108. [11] S.-X. Chen and B.-X. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Diff. Eqs., 233 (2007), 105-135. doi: 10.1016/j.jde.2006.09.020. [12] S.-X. Chen, Z. Xin and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228 (2002), 47-84. doi: 10.1007/s002200200652. [13] S.-X. Chen and H. Yuan, Transonic shocks in compressible flow assing a duct for three-dimensional Euler system, Arch. Rational Mech. Anal., 187 (2008), 523-556. doi: 10.1007/s00205-007-0079-z. [14] A. Chorin and J. A. Marsden, A Mathematical Introduction to Fluid Mechanics, $3^{rd}$ edition, Springer-Verlag, New York, 1993. [15] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley Interscience, New York, 1948. [16] B.-X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Methods Appl. Sci., 29 (2006), 1-26. doi: 10.1002/mma.661. [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [18] 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. [19] C. Gu, A method for solving the supersonic flow past a curved wedge, Fudan J.(Nature Sci.), 7 (1962), 11-14. [20] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSAM, 11. SIAM, Philadelphia, Pa., 1973. [21] P. D. Lax, Hyperbolic systems of conservation laws in several space variables, In: Current Topics in Partial Differential Equations, 327-341, Kinokuniya, Tokyo, 1986. [22] P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340. doi: 10.1137/S1064827595291819. [23] T. Li, On a free boundary problem, Chinese Ann. Math., 1 (1980), 351-358. [24] W. Lien and T.-P. Li, Nonlinear stability of a self-similar 3-dimensional gas flow, Commun. Math. Phys., 204 (1999), 525-549. doi: 10.1007/s002200050656. [25] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [26] L. Prandtl, Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten, Z. Angew. Math. Mech., 16 (1936), 129-142. [27] D. G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J., 43 (1976), 637-670. doi: 10.1215/S0012-7094-76-04351-9. [28] J. von Neumann, Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations [Reprinted from MR0044302], Bull. Amer. Math. Soc. (N.S.), 47 (2010), 145-154. doi: 10.1090/S0273-0979-09-01281-6. [29] Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle, J. Diff. Eqs., 192 (2003), 1-46. doi: 10.1016/S0022-0396(03)00037-8.

show all references

##### References:
 [1] G.-Q. Chen, J. Chen and M. Feldman, Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles, J. Math. Pures Appl. (9), 88 (2007), 191-218. doi: 10.1016/j.matpur.2007.04.008. [2] G.-Q. Chen and B. Fang, Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone, Discrete Contin. Dyn. Syst., 23 (2009), 85-114. doi: 10.3934/dcds.2009.23.85. [3] G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494. doi: 10.1090/S0894-0347-03-00422-3. [4] G.-Q. Chen and T. Li, Well-posedness for two-dimnsional steady supersonic Euler flows past a Lipschitz wedge, J. Diff. Eqs., 244 (2008), 1521-1550. [5] G.-Q. Chen, Y.-Q. Zhang and D.-W. 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. [6] S.-X. Chen, Supersonic flow past a concave wedge, Science in China, 10 (1997), 903-910. [7] S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge, Chin. Ann. Math., 19 (1998), 255-264. [8] S.-X. Chen, A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Z. Angew. Math. Phys., 54 (2003), 387-409. doi: 10.1007/s00033-003-2111-y. [9] S.-X. Chen, Stability of transonic shock front in multi-dimensional Euler system, Trans. Amer. Math. Soc., 357 (2005), 287-308. doi: 10.1090/S0002-9947-04-03698-0. [10] S.-X. Chen, Stability of a Mach configuration, Comm. Pure Appl. Math., 59 (2006), 1-35. doi: 10.1002/cpa.20108. [11] S.-X. Chen and B.-X. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Diff. Eqs., 233 (2007), 105-135. doi: 10.1016/j.jde.2006.09.020. [12] S.-X. Chen, Z. Xin and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228 (2002), 47-84. doi: 10.1007/s002200200652. [13] S.-X. Chen and H. Yuan, Transonic shocks in compressible flow assing a duct for three-dimensional Euler system, Arch. Rational Mech. Anal., 187 (2008), 523-556. doi: 10.1007/s00205-007-0079-z. [14] A. Chorin and J. A. Marsden, A Mathematical Introduction to Fluid Mechanics, $3^{rd}$ edition, Springer-Verlag, New York, 1993. [15] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley Interscience, New York, 1948. [16] B.-X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Methods Appl. Sci., 29 (2006), 1-26. doi: 10.1002/mma.661. [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [18] 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. [19] C. Gu, A method for solving the supersonic flow past a curved wedge, Fudan J.(Nature Sci.), 7 (1962), 11-14. [20] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSAM, 11. SIAM, Philadelphia, Pa., 1973. [21] P. D. Lax, Hyperbolic systems of conservation laws in several space variables, In: Current Topics in Partial Differential Equations, 327-341, Kinokuniya, Tokyo, 1986. [22] P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340. doi: 10.1137/S1064827595291819. [23] T. Li, On a free boundary problem, Chinese Ann. Math., 1 (1980), 351-358. [24] W. Lien and T.-P. Li, Nonlinear stability of a self-similar 3-dimensional gas flow, Commun. Math. Phys., 204 (1999), 525-549. doi: 10.1007/s002200050656. [25] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [26] L. Prandtl, Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten, Z. Angew. Math. Mech., 16 (1936), 129-142. [27] D. G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J., 43 (1976), 637-670. doi: 10.1215/S0012-7094-76-04351-9. [28] J. von Neumann, Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations [Reprinted from MR0044302], Bull. Amer. Math. Soc. (N.S.), 47 (2010), 145-154. doi: 10.1090/S0273-0979-09-01281-6. [29] Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle, J. Diff. Eqs., 192 (2003), 1-46. doi: 10.1016/S0022-0396(03)00037-8.
 [1] Gui-Qiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2515-2542. doi: 10.3934/cpaa.2013.12.2515 [2] Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 [3] Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049 [4] Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85 [5] Akisato Kubo. Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 59-74. doi: 10.3934/cpaa.2004.3.59 [6] Yinzheng Sun, Qin Wang, Kyungwoo Song. Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4899-4920. doi: 10.3934/cpaa.2020217 [7] Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433 [8] Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355 [9] Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573 [10] Yanbo Hu, Jiequan Li. On a supersonic-sonic patch arising from the frankl problem in transonic flows. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2643-2663. doi: 10.3934/cpaa.2021015 [11] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic and Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [12] Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607 [13] Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021166 [14] Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030 [15] Giovanni Gravina, Giovanni Leoni. On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4853-4878. doi: 10.3934/cpaa.2020215 [16] Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2661-2681. doi: 10.3934/dcdsb.2021153 [17] Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control and Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761 [18] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122 [19] Steinar Evje, Kenneth H. Karlsen. Hyperbolic-elliptic models for well-reservoir flow. Networks and Heterogeneous Media, 2006, 1 (4) : 639-673. doi: 10.3934/nhm.2006.1.639 [20] 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

2020 Impact Factor: 1.392