Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-30T23:13:18.906Z Has data issue: false hasContentIssue false
  • English
  • Français

On the inhomogeneous system of isentropic gas dynamics by the viscosity method

Published online by Cambridge University Press:  14 November 2011

Weifu Fang  and
Kazufumi Ito
Weifu Fang
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV 26506, U.S.A.
Kazufumi Ito
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The existence of global weak solutions is shown for the equations of isentropic gas dynamics with inhomogeneous terms by the viscosity method. A generalised version of the method of invariant regions is developed to obtain the uniform L∞ bounds of the viscosity solutions, and the method of compensated compactness is applied to show the existence of weak solutions as limits of the viscosity solutions. The lower positive bound for the density function is also obtained. As an example, a hydrodynamic model for semiconductors is analysed

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 261 - 280
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chen, G.-Q.. Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics (III). Ada Math. Sci. 6(1986), 75–120.Google Scholar
2Chueh, K., Conley, C. and Smoller, J.. Positive invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1979), 373–92.CrossRefGoogle Scholar
3Ding, X., Chen, G.-Q. and Luo, P.. Convergence of the fractional step Lax–Friedrichs scheme and Godunov scheme for isentropic gas dynamics. Comm. Math. Phys. 121 (1989), 63–84.Google Scholar
4DiPerna, R. J.. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal 82 (1983), 27–70.CrossRefGoogle Scholar
5DiPerna, R. J.. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91(1983), 1–30.CrossRefGoogle Scholar
6Evans, L. C.. Weak Convergence Methods for Nonlinear Partial Differential Equations (Providence, RI: American Mathematical Society, 1990).CrossRefGoogle Scholar
7Fang, W. and Ito, K.. Global solutions of the time-dependent drift-diffusion semiconductor equations. J. Differential Equations 123 (1995), 523–66.CrossRefGoogle Scholar
8Fang, W. and Ito, K.. One-dimensional hydrodynamic model for semiconductors by viscosity method. In Differential Equations and Applications to Biology and to Industry, 91–9, Martelli, M., Cooke, K., Cumberbatch, E., Tang, B. and Thieme, H., eds., (World Scientific, 1996).Google Scholar
9Fang, W. and Ito, K.. Weak solutions to a hydrodynamic model of two carrier types for semiconductors. Nonlinear Anal. 28 (1997), 947–63.CrossRefGoogle Scholar
10Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N.. Linear and Quasilinear Equations of Parabolic Type (Providence, RI: American Mathematical Society, 1968).CrossRefGoogle Scholar
11Marcati, P. and Natalini, R.. Weak solutions to a hydrodynamic model for semiconductors: Cauchy problem. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 115–31.CrossRefGoogle Scholar
12Markowich, P. A., Ringhofer, C. A. and Schmeiser, C.. Semiconductor Equations (New York: Springer, 1990).CrossRefGoogle Scholar
13Smoller, J.. Shock Waves and Reaction–Diffusion Equations (New York: Springer, 1983).CrossRefGoogle Scholar
14Tartar, L.. Compensated compactness and applications to partial differential equations. Heriot Watt Symposium, Vol. IV (London: Pitman, 1979).Google Scholar
15Troianiello, G. M.. Elliptic Differential Equations and Obstacle Problems (New York: Plenum Press, 1987).CrossRefGoogle Scholar
16Zhang, B.. Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices. Comm. Math. Phys. 157 (1993), 1–22.CrossRefGoogle Scholar