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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References
Canic S., Keyfitz B.L., Kim E.H.: A free boundary problem for a quasi-linear degenerate elliptic equation: regular reflection of weak shocks. Commun. Pure Appl. Math. 55(1), 71–92 (2002)
Chen G.Q., Feldman M.: Global solutions of shock reflection by large-angle wedges for potential flow. Ann. Math 171(2), 1067–1182 (2010)
Chen S.X.: Mach configuration in pseudo-stationary compressible flow. J. Am. Math. Soc. 21(1), 63–100 (2008)
Chen X., Zheng Y.X.: The direct approach to the interaction of rarefaction waves of the two-dimensional Euler equations. Indian J Math 59(1), 231–256 (2010)
Cole J.D., Cook L.P.: Transonic aerodynamics. Elsevier, Amsterdam (1986)
Courant R., Friedrichs K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)
Dai Z., Zhang T.: Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics. Arch. Ration. Mech. Anal. 155(4), 277–298 (2000)
Elling V., Liu T.P.: Supersonic flow onto a solid wedge. Commun. Pure Appl. Math. 61(10), 1347–1448 (2008)
Glimm G., Ji X., Li J., Li X., Zhang P., Zhang T., Zheng Y.: Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations. SIAM J. Appl. Math. 69(3), 720–742 (2008)
Tesdall A., Hunter J.K.: Self-similar solutions for weak shock reflection. SIAM J. Appl. Math. 63(1), 42–61 (2002)
Lei Z., Zheng Y.X.: A complete global solution to the pressure gradient equation. J. Differ. Equ. 236(1), 280–292 (2007)
Levine L.E.: The expansion of a wedge of gas into a vacuum. Proc. Camb. Philol. Soc. 64, 1151–1163 (1968)
Li J.Q.: On the two-dimensional gas expansion for compressible Euler equations. SIAM J. Appl. Math. 62(3), 831–852 (2001)
Li J.Q.: Global solution of an initial-value problem for two-dimensional compressible Euler equations. J. Differ. Equ. 179(1), 178–194 (2002)
Li J.Q., Sheng W.C., Zhang T., Zheng Y.X.: Two-dimensional Riemann problems: from scalar conservation laws to compressible Euler equations. Acta Math. Sci. Ser. B Eng. Ed. 29(4), 777–802 (2009)
Li J.Q., Yang Z.C., Zheng Y.X.: Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations. J. Differ. Equ. 250, 782–798 (2011)
Li, J.Q., Zhang, T., Yang, S.L.: The Two-dimensional Riemann Problem in Gas Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 98, Longman (1998)
Li J.Q., Zhang T., Zheng Y.X.: Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commun. Math. Phys. 267(1), 1–12 (2006)
Li J.Q., Zheng Y.X.: Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Rat. Mech. Anal. 193, 623–657 (2009)
Li J.Q., Zheng Y.: Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations. Commun. Math. Phys. 296, 303–321 (2010)
Li M.J., Zheng Y.X.: Semi-hyperbolic patches of solutions of the two-dimensional Euler equations. Arch. Ration. Mech. Anal. 201, 1069–1096 (2011)
Li T.T., Yu W.C.: Boundary Value Problems of Hyperbolic System. Duke University, Durham (1985)
Song K., Zheng Y.X.: Semi-hyperbolic patches of solutions of the pressure gradient system. Disc. Cont. Dyna. Syst. 24(4), 1365–1380 (2009)
Suchkov V.A.: Flow into a vacuum along an oblique wall. J. Appl. Math. Mech. 27, 1132–1134 (1963)
Zhang T., Zheng Y.X.: Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics systems. SIAM J. Math. Anal. 21(3), 593–630 (1990)
Zheng, Y.X.: Systems of Conservation Laws: Two-dimensional Riemann Problems. In the series of Progress in Nonlinear Differential Equations. Birkhauser (2001)
Zheng Y.X.: Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. Engl. Ser. 22(2), 177–210 (2006)
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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