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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

Weixia Zhao 1,

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.
Keywords: direct approach, Goursat problem., characteristic decompositions, expansion of gas, Two-dimensional Riemann problem.
Mathematics Subject Classification: Primary: 35L65; Secondary: 35J70, 35R3.
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

show all references

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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