Abstract
As a ladder step to study transonic problems, we investigate two families of degenerate Goursat-type boundary value problems arising from the two-dimensional pseudo-steady isothermal Euler equations. The first family is about the genuinely two-dimensional full expansion of gas into a vacuum with a wedge; the other is a semi-hyperbolic patch that starts on sonic curves and ends at transonic shocks. Both the vacuum and the sonic sets cause parabolic degeneracy that results in substantial difficulties such as singularities of solutions and uniform a priori estimates. Main ingredients in this study are various characteristic decompositions for the pseudo-steady Euler equations in order to obtain necessary a priori estimates. Furthermore, we are able to verify the uniform Hölder continuity of solutions with exponent 1/2 for the gas expansion problem and up to 2/7 for the semi-hyperbolic problem.
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Yanbo Hu and Wancheng Sheng were supported by NSFC (10971130), Shanghai Leading Academic Discipline Project (J50101) and Shanghai Municipal Education Commission of Scientific Research Innovation Project: 11ZZ84. Jiequan Li was supported by the Key Program from Beijing Educational Commission (KZ200910028002), 973 project (2006CB805902), PHR(IHLB) and NSFC (10971142).
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Hu, Y., Li, J. & Sheng, W. Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations. Z. Angew. Math. Phys. 63, 1021–1046 (2012). https://doi.org/10.1007/s00033-012-0203-2
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DOI: https://doi.org/10.1007/s00033-012-0203-2