American Institute of Mathematical Sciences

Advanced
November  2013, 12(6): 2515-2542. doi: 10.3934/cpaa.2013.12.2515

Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations

Gui-Qiang G. Chen 1, and  Hairong Yuan 2,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2. 

Department of Mathematics, East China Normal University, Shanghai 200241, China

Received  June 2012 Revised  December 2012 Published  May 2013

We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonlocal elliptic operators, and an iteration mapping which possesses locally a unique fixed point. The approach reveals an intrinsic structure of the steady Euler system and subtle interactions of its elliptic and hyperbolic part.
Keywords: transonic shock-front, nonlocal elliptic operators, decomposition, Uniqueness, Euler system, free boundary problem, intrinsic structure, three-dimensional, iteration mapping, mixed-composite elliptic-hyperbolic type, Venttsel problem, subtle interactions..
Mathematics Subject Classification: 5M20, 35J65, 35R35, 35B45, 76H05, 76L0.
Citation: Gui-Qiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2515-2542. doi: 10.3934/cpaa.2013.12.2515
References:
[1]

P. Amorim, M. Ben-Artzi and P. G. LeFloch, Hyperbolic conservation laws on manifolds: Total variation estimates and the finite volume method, Methods Appl. Anal., 12 (2005), 291-324.  Google Scholar

[2]

P. Amorim, P. G. LeFloch and W. Neves, A geometric approach to error estimates for conservation laws posed on a spacetime, Nonlinear Anal., 74 (2011), 4898-4917.  Google Scholar

[3]

D. Bleecker and G. Csordas, "Basic Partial Differential Equations," International Press: Boston, 1996.  Google Scholar

[4]

G.-Q. Chen, C. M. Dafermos, M. Slemrod and D. Wang, On two-dimensional sonic-subsonic flow, Comm. Math. Phys., 271 (2007), 635-647.  Google Scholar

[5]

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.  Google Scholar

[6]

G.-Q. Chen and M. Feldman, Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 310-356.  Google Scholar

[7]

G.-Q. Chen and M. Feldman, Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Rational Mech. Anal., 184 (2007), 185-242.  Google Scholar

[8]

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., 88 (2007), 191-218.  Google Scholar

[9]

S. Chen and H. Yuan, Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems, Arch. Rational Mech. Anal., 187 (2008), 523-556.  Google Scholar

[10]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience Publishers Inc., New York, 1948.  Google Scholar

[11]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Springer-Verlag, New York, 2000.  Google Scholar

[12]

T. Frankel, "The Geometry of Physicist, An Introduction," 2nd Ed., Cambridge University Press, Cambridge, 2004.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd Edition, Springer-Verlag, Berlin-New York, 1983.  Google Scholar

[14]

T.-T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley & Sons, Masson, Paris, 1994.  Google Scholar

[15]

L. Liu and H. Yuan, Stability of cylindrical transonic shocks for two-dimensional steady compressible Euler system, J. Hyper. Diff. Equ., 5 (2008), 347-379.  Google Scholar

[16]

Y. Luo and N. S. Trudinger, Linear second order elliptic equations with venttsel boundary conditions, Proc. Royal Soc. Edinburgh, 118A (1991), 193-207.  Google Scholar

[17]

L. M. Sibner and R. J. Sibner, Transonic flows on axially symmetric torus, J. Math. Anal. Appl., 72 (1979), 362-382.  Google Scholar

[18]

M. Taylor, "Partial Differential Equations," Vol. 3, Springer-Verlag, New York, 1996.  Google Scholar

[19]

B. Whitham, "Linear and Nonlinear Waves," John Wiley, New York, 1974.  Google Scholar

[20]

H. Yuan, Examples of steady subsonic flows in a convergent-divergent approximate nozzle, J. Diff. Eqs., 244 (2008), 1675-1691.  Google Scholar

[21]

H. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal., 38 (2006), 1343-1370.  Google Scholar

[22]

H. Yuan, A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows, Nonlinear Analysis: Real World Appl., 9 (2008), 316-325.  Google Scholar

show all references

References:
[1]

P. Amorim, M. Ben-Artzi and P. G. LeFloch, Hyperbolic conservation laws on manifolds: Total variation estimates and the finite volume method, Methods Appl. Anal., 12 (2005), 291-324.  Google Scholar

[2]

P. Amorim, P. G. LeFloch and W. Neves, A geometric approach to error estimates for conservation laws posed on a spacetime, Nonlinear Anal., 74 (2011), 4898-4917.  Google Scholar

[3]

D. Bleecker and G. Csordas, "Basic Partial Differential Equations," International Press: Boston, 1996.  Google Scholar

[4]

G.-Q. Chen, C. M. Dafermos, M. Slemrod and D. Wang, On two-dimensional sonic-subsonic flow, Comm. Math. Phys., 271 (2007), 635-647.  Google Scholar

[5]

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.  Google Scholar

[6]

G.-Q. Chen and M. Feldman, Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 310-356.  Google Scholar

[7]

G.-Q. Chen and M. Feldman, Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Rational Mech. Anal., 184 (2007), 185-242.  Google Scholar

[8]

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., 88 (2007), 191-218.  Google Scholar

[9]

S. Chen and H. Yuan, Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems, Arch. Rational Mech. Anal., 187 (2008), 523-556.  Google Scholar

[10]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience Publishers Inc., New York, 1948.  Google Scholar

[11]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Springer-Verlag, New York, 2000.  Google Scholar

[12]

T. Frankel, "The Geometry of Physicist, An Introduction," 2nd Ed., Cambridge University Press, Cambridge, 2004.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd Edition, Springer-Verlag, Berlin-New York, 1983.  Google Scholar

[14]

T.-T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley & Sons, Masson, Paris, 1994.  Google Scholar

[15]

L. Liu and H. Yuan, Stability of cylindrical transonic shocks for two-dimensional steady compressible Euler system, J. Hyper. Diff. Equ., 5 (2008), 347-379.  Google Scholar

[16]

Y. Luo and N. S. Trudinger, Linear second order elliptic equations with venttsel boundary conditions, Proc. Royal Soc. Edinburgh, 118A (1991), 193-207.  Google Scholar

[17]

L. M. Sibner and R. J. Sibner, Transonic flows on axially symmetric torus, J. Math. Anal. Appl., 72 (1979), 362-382.  Google Scholar

[18]

M. Taylor, "Partial Differential Equations," Vol. 3, Springer-Verlag, New York, 1996.  Google Scholar

[19]

B. Whitham, "Linear and Nonlinear Waves," John Wiley, New York, 1974.  Google Scholar

[20]

H. Yuan, Examples of steady subsonic flows in a convergent-divergent approximate nozzle, J. Diff. Eqs., 244 (2008), 1675-1691.  Google Scholar

[21]

H. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal., 38 (2006), 1343-1370.  Google Scholar

[22]

H. Yuan, A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows, Nonlinear Analysis: Real World Appl., 9 (2008), 316-325.  Google Scholar

[1]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[2]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[3]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122
[4]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85
[5]

Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182
[6]

Taebeom Kim, Sunčica Čanić, Giovanna Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. Communications on Pure & Applied Analysis, 2010, 9 (4) : 839-865. doi: 10.3934/cpaa.2010.9.839
[7]

Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049
[8]

Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021153
[9]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[10]

Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184
[11]

Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361
[12]

Chuanxin Zhao, Lin Jiang, Kok Lay Teo. A hybrid chaos firefly algorithm for three-dimensional irregular packing problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 409-429. doi: 10.3934/jimo.2018160
[13]

Victor Isakov, Shingyu Leung, Jianliang Qian. A three-dimensional inverse gravimetry problem for ice with snow caps. Inverse Problems & Imaging, 2013, 7 (2) : 523-544. doi: 10.3934/ipi.2013.7.523
[14]

O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449
[15]

Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2027-2046. doi: 10.3934/dcds.2016.36.2027
[16]

Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048
[17]

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053
[18]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340
[19]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks & Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655
[20]

Myoungjean Bae, Hyangdong Park. Three-dimensional supersonic flows of Euler-Poisson system for potential flow. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2421-2440. doi: 10.3934/cpaa.2021079

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences