Abstract
A simple wave is defined as a flow in a region whose image is a curve in the phase space. It is well known that “the theory of simple waves is fundamental in building up the solutions of flow problems out of elementary flow pattern” see Courant and Friedrichs’s chassical book “Supersonic Flow and Shock Waves”. This paper mainly concerned with the geometric construction of simple waves for the 2D pseudo-steady compressible Euler system. Based on the geometric interpretation, the expansion or compression simple wave flow construction around a pseudo-stream line with a bend part are constructed. It is a building block that appears in the global solution to four contact discontinuities Riemann problems.
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References
Li, J., Zhang, T., and Yang, S. The Two-Dimensional Riemann Problem in Gas Dynamics, Addison Wesley Longman limited, London (1998)
Zheng, T. and Zheng, Y. Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics system. SIAM J. Math. Anal. 21(3), 593–630 (1990)
Zheng, Y. Systems of Conservation Laws: Two-Dimensional Riemann Problems, Bikhäuser Boston, Boston (2001)
John, F. Partial Differential Equations, Springer-Verlag, New York (1982)
Courant, R. and Friedrichs, K. O. Supersonic Flow and Shock Waves, Interscience, New York (1948)
Li, J., Zhang, T., and Zheng, Y. Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commu. Math. Phys. 267(1), 1–12 (2006)
Bang, S. Rarefaction Wave Interaction of Pressure Gradient System, Ph. D. dissertation, Pennsylvania State University (2007)
Dai, Z. and 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)
Lei, Z. and Zheng, Y. A complete global solution to the pressure gradient equation. Journal of Differential Equations 236(1), 280–292 (2007)
Song, K. and Zheng, Y. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete and Continuous Dynamic System 24(4), 1365–1380 (2009)
Li, J. On the two-dimensional gas expansion for the compressible Euler equations. SIAM J. Appl. Math. 62(3), 831–852 (2002)
Li, J. Global solutions of an initial value problem for two-dimensional compressible Euler equations. Journal of Differential Equations 179(1), 178–194 (2002)
Li, J. and Zheng, Y. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Rat. Mech. Anal. 193(3), 623–657 (2009)
Chang, T., Chen, G., and Yang, S. On the 2-D Riemann problem for the compressible Euler equation, I. interaction of contact discontinuities. Discrete and Continuous Dynamical Systems 6(2), 419–430 (2000)
Glimm, J., Ji, X., Li, J., Li, X., Zhang, P., Zhang, T., and 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)
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Communicated by Xing-ming GUO
Project supported by the National Natural Science Foundation of China (No. 10971130) and the Shanghai Leading Academic Discipline Project (No. J50101)
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Lai, G., Sheng, Wc. Simple waves for two-dimensional compressible pseudo-steady Euler system. Appl. Math. Mech.-Engl. Ed. 31, 827–838 (2010). https://doi.org/10.1007/s10483-010-1317-7
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DOI: https://doi.org/10.1007/s10483-010-1317-7
Key words
- self-similar Euler system
- simple wave
- generalized characteristic analysis
- pseudo-stream line
- sonic circle
- sonic edge