American Institute of Mathematical Sciences

Advanced
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.  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]

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
[12]

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, 2020, 16 (1) : 495-510. doi: 10.3934/jimo.2018164
[13]

Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020024
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences