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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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

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