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

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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

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

Received June 2012 Revised October 2012 Published January 2013

Abstract
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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. 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, (1948). 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. 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. 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. 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. 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, (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. 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. 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. 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. 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, (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. 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, (1948). 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. 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. 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. 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. 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, (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. 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. 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. 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. 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, (2001). Google Scholar

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