×

Stability of cylindrical transonic shocks for the two-dimensional steady compressible Euler system. (English) Zbl 1158.35014

The author studies the stability with respect to the perturbations of upcoming supersonic flows of a class of cylindrical symmetric transonic shocks for two dimensional complete compressible steady Euler system. The approach is to present the circular transonic shock front as a free boundary, so that the problem reduces to a nonlinear free boundary problem of an elliptic-hyperbolic composite system. The existence of the shock is proved by finding the locally unique fixed point of a profile updating mapping. The problem is considered in polar coordinates as a two-dimensional boundary-value problem with two additional algebraic equations in an annulus.

MSC:

35B35 Stability in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
35F15 Boundary value problems for linear first-order PDEs
35J25 Boundary value problems for second-order elliptic equations
76H05 Transonic flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.0.CO;2-K · Zbl 1017.76040 · doi:10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.0.CO;2-K
[2] DOI: 10.1090/S0894-0347-03-00422-3 · Zbl 1015.35075 · doi:10.1090/S0894-0347-03-00422-3
[3] DOI: 10.1002/cpa.3042 · Zbl 1075.76036 · doi:10.1002/cpa.3042
[4] DOI: 10.1002/cpa.20108 · Zbl 1083.35064 · doi:10.1002/cpa.20108
[5] DOI: 10.1007/s00205-007-0079-z · Zbl 1140.76015 · doi:10.1007/s00205-007-0079-z
[6] Courant R., Supersonic Flow and Shock Waves (1948) · Zbl 0041.11302
[7] DOI: 10.1007/978-3-642-61798-0 · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[8] Grisvard P., Elliptic Problems in Nonsmooth Domains (1985) · Zbl 0695.35060
[9] Li T.-T., Duke University Mathematics Series 5, in: Boundary Value Problems for Quasilinear Hyperbolic Systems (1985)
[10] Li T.-T., Pitman Research Notes in Mathematics 382, in: Boundary Value Problems with Equivalued Surface and Resistivity Well-Logging (1998) · Zbl 0919.35001
[11] DOI: 10.1007/BF01976043 · Zbl 0576.76053 · doi:10.1007/BF01976043
[12] Whitham G. B., Linear and Nonlinear Waves (1974) · Zbl 0373.76001
[13] DOI: 10.1002/cpa.20025 · Zbl 1076.76043 · doi:10.1002/cpa.20025
[14] DOI: 10.1137/050642447 · Zbl 1121.35081 · doi:10.1137/050642447
[15] DOI: 10.1016/j.nonrwa.2006.10.006 · Zbl 1137.76029 · doi:10.1016/j.nonrwa.2006.10.006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.