American Institute of Mathematical Sciences

September  2013, 12(5): 2319-2330. doi: 10.3934/cpaa.2013.12.2319

The expansion of gas from a wedge with small angle into a vacuum

 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  June 2012 Revised  October 2012 Published  January 2013

In this paper, the problem of the expansion of a wedge of gas into vacuum is investigated. Let $\theta$ be the half angle of the wedge. For a given $\bar{\theta}$ determined by the adiabatic exponent $\gamma$, we prove the global existence of the solution through the direct approach in the case $\theta\leq\bar{\theta}$, extending the previous result obtained by Li, Yang and Zheng. Our analysis relies on the special symmetric structure of the characteristic form as well as characteristic decompositions.
Citation: Weixia Zhao. The expansion of gas from a wedge with small angle into a vacuum. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2319-2330. doi: 10.3934/cpaa.2013.12.2319
References:
 [1] X. Chen and Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J, 59 (2010), 231-256. doi: 10.1512/iumj.2010.59.3752.  Google Scholar [2] R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,'' Reprinting of the 1948 original. Applied Mathematical Sciences, 21. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar [3] X. M. Ji and Y. X. Zheng, Characteristic decouplings and interactions of rarefaction waves of 2-D Euler equations, J. Math. Anal. Appl., (2012). doi: 10.1016/j.jmaa.2012.05.035.  Google Scholar [4] L. E. Levine, The expansion of a wedge of gas into a vacuum, Proc. Camb. Phil. Soc., 64 (1968), 1151-1163. doi: 10.1017/S0305004100043899.  Google Scholar [5] J. Q. Li, On the two-dimensional gas expansion for compressible Euler equations, SIAM J. Math. Anal., 62 (2001), 831-852. doi: 10.1137/S0036139900361349.  Google Scholar [6] J. Q. Li, Global solution of an initial value problem for two-dimensional compressible Euler equations, J. Differ. Eqs., 179 (2002), 178-194. doi: 10.1006/jdeq.2001.4025.  Google Scholar [7] J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, J. Differ. Eqs., 250 (2011), 782-798. doi: 10.1016/j.jde.2010.07.009.  Google Scholar [8] J. Q. Li, T. Zhang and S. L. Yang, "The Two-dimensional Riemann Problem in Gas Dynamics," $\pi$ Pitman Monographs and Surveys in Pure and Applied Mathematics, 98. Longman, Harlow, 1998.  Google Scholar [9] J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193 (2009), 623-657. doi: 10.1007/s00205-008-0140-6.  Google Scholar [10] M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096. doi: 10.1007/s00205-011-0410-6.  Google Scholar [11] T. Li and T. H. Qin, Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms, Chinese Annals of Mathematics, 6 (1985), 199-210. Google Scholar [12] T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems,'' Duke University Mathematics Series V, 1985. Google Scholar [13] P. Qu, $C^0$ sstimate for a kind of partially dissipative quasilinear hyperbolic systems and its applications,, in manuscript., ().   Google Scholar [14] V. A. Suchkov, Flow into a vacuum along an oblique wall, J. Appl. Math. Mech., 27 (1963), 1132-1134. doi: 10.1016/0021-8928(63)90195-3.  Google Scholar [15] Y. X. Zheng, "Systems of Conservation Laws: Two-dimensional Riemann Problems,'' Progress in Nonlinear Differential Equations and Their Applications, 38, Birkhäuser, Boston, 2001.  Google Scholar

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References:
 [1] X. Chen and Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J, 59 (2010), 231-256. doi: 10.1512/iumj.2010.59.3752.  Google Scholar [2] R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,'' Reprinting of the 1948 original. Applied Mathematical Sciences, 21. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar [3] X. M. Ji and Y. X. Zheng, Characteristic decouplings and interactions of rarefaction waves of 2-D Euler equations, J. Math. Anal. Appl., (2012). doi: 10.1016/j.jmaa.2012.05.035.  Google Scholar [4] L. E. Levine, The expansion of a wedge of gas into a vacuum, Proc. Camb. Phil. Soc., 64 (1968), 1151-1163. doi: 10.1017/S0305004100043899.  Google Scholar [5] J. Q. Li, On the two-dimensional gas expansion for compressible Euler equations, SIAM J. Math. Anal., 62 (2001), 831-852. doi: 10.1137/S0036139900361349.  Google Scholar [6] J. Q. Li, Global solution of an initial value problem for two-dimensional compressible Euler equations, J. Differ. Eqs., 179 (2002), 178-194. doi: 10.1006/jdeq.2001.4025.  Google Scholar [7] J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, J. Differ. Eqs., 250 (2011), 782-798. doi: 10.1016/j.jde.2010.07.009.  Google Scholar [8] J. Q. Li, T. Zhang and S. L. Yang, "The Two-dimensional Riemann Problem in Gas Dynamics," $\pi$ Pitman Monographs and Surveys in Pure and Applied Mathematics, 98. Longman, Harlow, 1998.  Google Scholar [9] J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193 (2009), 623-657. doi: 10.1007/s00205-008-0140-6.  Google Scholar [10] M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096. doi: 10.1007/s00205-011-0410-6.  Google Scholar [11] T. Li and T. H. Qin, Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms, Chinese Annals of Mathematics, 6 (1985), 199-210. Google Scholar [12] T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems,'' Duke University Mathematics Series V, 1985. Google Scholar [13] P. Qu, $C^0$ sstimate for a kind of partially dissipative quasilinear hyperbolic systems and its applications,, in manuscript., ().   Google Scholar [14] V. A. Suchkov, Flow into a vacuum along an oblique wall, J. Appl. Math. Mech., 27 (1963), 1132-1134. doi: 10.1016/0021-8928(63)90195-3.  Google Scholar [15] Y. X. Zheng, "Systems of Conservation Laws: Two-dimensional Riemann Problems,'' Progress in Nonlinear Differential Equations and Their Applications, 38, Birkhäuser, Boston, 2001.  Google Scholar
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