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. 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

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. 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

[1]

Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431

[2]

Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733

[3]

Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609

[4]

Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661

[5]

Al-hassem Nayam. Constant in two-dimensional $p$-compliance-network problem. Networks & Heterogeneous Media, 2014, 9 (1) : 161-168. doi: 10.3934/nhm.2014.9.161

[6]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713

[7]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[8]

Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems & Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709

[9]

Faustino Maestre, Pablo Pedregal. Dynamic materials for an optimal design problem under the two-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 973-990. doi: 10.3934/dcds.2009.23.973

[10]

Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065

[11]

Mao Chen, Xiangyang Tang, Zhizhong Zeng, Sanya Liu. An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018164

[12]

Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057

[13]

Jiequan Li, Mária Lukáčová - MedviĎová, Gerald Warnecke. Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 559-576. doi: 10.3934/dcds.2003.9.559

[14]

Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008

[15]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292

[16]

Stéphane Brull. Problem of evaporation-condensation for a two component gas in the slab. Kinetic & Related Models, 2008, 1 (2) : 185-221. doi: 10.3934/krm.2008.1.185

[17]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

[18]

Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149

[19]

Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25

[20]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]