# 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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.matpur.2007.04.008.  Google Scholar [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.  doi: 10.3934/dcds.2009.23.85.  Google Scholar [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.  doi: 10.1090/S0894-0347-03-00422-3.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00205-005-0412-3.  Google Scholar [6] S.-X. Chen, Supersonic flow past a concave wedge,, Science in China, 10 (1997), 903.   Google Scholar [7] S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge,, Chin. Ann. Math., 19 (1998), 255.   Google Scholar [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.  doi: 10.1007/s00033-003-2111-y.  Google Scholar [9] S.-X. Chen, Stability of transonic shock front in multi-dimensional Euler system,, Trans. Amer. Math. Soc., 357 (2005), 287.  doi: 10.1090/S0002-9947-04-03698-0.  Google Scholar [10] S.-X. Chen, Stability of a Mach configuration,, Comm. Pure Appl. Math., 59 (2006), 1.  doi: 10.1002/cpa.20108.  Google Scholar [11] S.-X. Chen and B.-X. Fang, Stability of transonic shocks in supersonic flow past a wedge,, J. Diff. Eqs., 233 (2007), 105.  doi: 10.1016/j.jde.2006.09.020.  Google Scholar [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.  doi: 10.1007/s002200200652.  Google Scholar [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.  doi: 10.1007/s00205-007-0079-z.  Google Scholar [14] A. Chorin and J. A. Marsden, A Mathematical Introduction to Fluid Mechanics,, $3^{rd}$ edition, (1993).   Google Scholar [15] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Wiley Interscience, (1948).   Google Scholar [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.  doi: 10.1002/mma.661.  Google Scholar [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, $2^{nd}$ edition, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [18] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697.  doi: 10.1002/cpa.3160180408.  Google Scholar [19] C. Gu, A method for solving the supersonic flow past a curved wedge,, Fudan J.(Nature Sci.), 7 (1962), 11.   Google Scholar [20] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, CBMS-RCSAM, 11 (1973).   Google Scholar [21] P. D. Lax, Hyperbolic systems of conservation laws in several space variables,, In: Current Topics in Partial Differential Equations, (1986), 327.   Google Scholar [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.  doi: 10.1137/S1064827595291819.  Google Scholar [23] T. Li, On a free boundary problem,, Chinese Ann. Math., 1 (1980), 351.   Google Scholar [24] W. Lien and T.-P. Li, Nonlinear stability of a self-similar 3-dimensional gas flow,, Commun. Math. Phys., 204 (1999), 525.  doi: 10.1007/s002200050656.  Google Scholar [25] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-1116-7.  Google Scholar [26] L. Prandtl, Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten,, Z. Angew. Math. Mech., 16 (1936), 129.   Google Scholar [27] D. G. Schaeffer, Supersonic flow past a nearly straight wedge,, Duke Math. J., 43 (1976), 637.  doi: 10.1215/S0012-7094-76-04351-9.  Google Scholar [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.  doi: 10.1090/S0273-0979-09-01281-6.  Google Scholar [29] Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle,, J. Diff. Eqs., 192 (2003), 1.  doi: 10.1016/S0022-0396(03)00037-8.  Google Scholar

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.  doi: 10.1016/j.matpur.2007.04.008.  Google Scholar [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.  doi: 10.3934/dcds.2009.23.85.  Google Scholar [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.  doi: 10.1090/S0894-0347-03-00422-3.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00205-005-0412-3.  Google Scholar [6] S.-X. Chen, Supersonic flow past a concave wedge,, Science in China, 10 (1997), 903.   Google Scholar [7] S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge,, Chin. Ann. Math., 19 (1998), 255.   Google Scholar [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.  doi: 10.1007/s00033-003-2111-y.  Google Scholar [9] S.-X. Chen, Stability of transonic shock front in multi-dimensional Euler system,, Trans. Amer. Math. Soc., 357 (2005), 287.  doi: 10.1090/S0002-9947-04-03698-0.  Google Scholar [10] S.-X. Chen, Stability of a Mach configuration,, Comm. Pure Appl. Math., 59 (2006), 1.  doi: 10.1002/cpa.20108.  Google Scholar [11] S.-X. Chen and B.-X. Fang, Stability of transonic shocks in supersonic flow past a wedge,, J. Diff. Eqs., 233 (2007), 105.  doi: 10.1016/j.jde.2006.09.020.  Google Scholar [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.  doi: 10.1007/s002200200652.  Google Scholar [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.  doi: 10.1007/s00205-007-0079-z.  Google Scholar [14] A. Chorin and J. A. Marsden, A Mathematical Introduction to Fluid Mechanics,, $3^{rd}$ edition, (1993).   Google Scholar [15] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Wiley Interscience, (1948).   Google Scholar [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.  doi: 10.1002/mma.661.  Google Scholar [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, $2^{nd}$ edition, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [18] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697.  doi: 10.1002/cpa.3160180408.  Google Scholar [19] C. Gu, A method for solving the supersonic flow past a curved wedge,, Fudan J.(Nature Sci.), 7 (1962), 11.   Google Scholar [20] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, CBMS-RCSAM, 11 (1973).   Google Scholar [21] P. D. Lax, Hyperbolic systems of conservation laws in several space variables,, In: Current Topics in Partial Differential Equations, (1986), 327.   Google Scholar [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.  doi: 10.1137/S1064827595291819.  Google Scholar [23] T. Li, On a free boundary problem,, Chinese Ann. Math., 1 (1980), 351.   Google Scholar [24] W. Lien and T.-P. Li, Nonlinear stability of a self-similar 3-dimensional gas flow,, Commun. Math. Phys., 204 (1999), 525.  doi: 10.1007/s002200050656.  Google Scholar [25] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-1116-7.  Google Scholar [26] L. Prandtl, Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten,, Z. Angew. Math. Mech., 16 (1936), 129.   Google Scholar [27] D. G. Schaeffer, Supersonic flow past a nearly straight wedge,, Duke Math. J., 43 (1976), 637.  doi: 10.1215/S0012-7094-76-04351-9.  Google Scholar [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.  doi: 10.1090/S0273-0979-09-01281-6.  Google Scholar [29] Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle,, J. Diff. Eqs., 192 (2003), 1.  doi: 10.1016/S0022-0396(03)00037-8.  Google Scholar
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