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The transonic shock in a nozzle, 2-D and 3-D complete Euler systems. (English) Zbl 1165.35031

The authors study the complete steady Euler system in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) in the case when the symmetric supersonic incoming flow goes through the suitably curved nozzle and the scope of the exit pressure is given. The authors focus on the existence and uniqueness of steady and unsteady transonic shocks, the global existence, stability, and the long time asymptotic behavior of an unsteady symmetric transonic shock in a symmetric converging part of the nozzle. It is also shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
76N15 Gas dynamics (general theory)
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