Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle |
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Authors: | Feimin Huang Jie Kuang Dehua Wang Wei Xiang |
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Affiliation: | 1. College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China;2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;3. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China;4. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA;5. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China |
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Abstract: | In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity. |
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Keywords: | 35B07 35B20 35D30 76J20 76L99 76N10 Contact discontinuity Supersonic flow Free boundary Compressible Euler equation Finitely long nozzle |
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