Local uniqueness of steady sphericaltransonic shock-fronts for the three--dimensional full Eulerequations

Gui-Qiang G. Chen; Hairong Yuan

doi:10.3934/cpaa.2013.12.2515

Article Contents

2013, Volume 12, Issue 6: 2515-2542. Doi: 10.3934/cpaa.2013.12.2515

This issue Previous Article Super polyharmonic property of solutions for PDE systems and its applications Next Article Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations

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
2.
Department of Mathematics, East China Normal University, Shanghai 200241

Received: May 31, 2012

Revised: November 30, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

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:

Uniqueness,
transonic shock-front,
Euler system,
three-dimensional,
free boundary problem,
mixed-composite elliptic-hyperbolic type,
iteration mapping,
decomposition,
Venttsel problem,
nonlocal elliptic operators,
intrinsic structure,
subtle interactions.

Mathematics Subject Classification: 5M20, 35J65, 35R35, 35B45, 76H05, 76L05.

Citation:

Related Papers

Cited by

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.