Abstract
The global existence and the asymptotic behavior of an entropy weak solution with one shock discontinuity, on spatial bounded domain are investigated in the present paper. We show that, for small smooth initial data and fixed boundary with one small jump at(x t) = (0, 0), the piecewise smooth solution with one shock discontinuity exists globally in time. The shock discontinuity starts from(xt) = (0, 0), moves forward and reflects in finite time at the boundaryx = 1 to form a 1—shock, which goes backward and reflects atx = 0 also in finite time to create a new 2—shock. Such process will never stop in any finite time. As t→∞, the shock strength decays exponentially and this solution converges to a constant state determined by the initial data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Courant and K. O. Friedrich, “Supersonic flow and shock wave”, AMS Vol 21, Spring-Verlag New Yourk, 1948.
C. M. Dafermos, A system of hyperbolic conservation laws with frictional dampingZ. Angew.Math. Phys.46 (1995), Special Issue, 294–307.
L. Hsiao, “Quasilinear hyperbolic systems and dissipative mechanisms”, World Scientific, 1998.
L. Hsiao and Hailiang Li, Shock reflection for damped p-system, to appear in Q. Appl. math. 2001.
L. Hsiao and Hailiang Li, The system of compressible adiabatic flow through porous media with boundary effects, to appear in Chin. Anal Math. 2001.
L. Hsiao and T. P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with dampingComm. Math. Phys.143 (1992), 599–605.
L. Hsiao and T. Luo, Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with dampingQ. Appl. Math.56 (1998), 173–198.
L. Hsiao and S. Q. Tang, Construction and qualitative behavior of solutions for a system of nonlinear hyperbolic conservation laws with dampingQ. Appl.Math. LIII (1995), 487–505.
L. Hsiao and S. Q. Tang, Construction and qualitative behavior of solutions of perturbated Riemann problem for the system of one-dimensional isentropic flow with dampingJ. Diff. Eqns.123 (1995), 480–503.
O. A. Ladyzhenskaya, “The boundary value problems of Mathematical Physics”, Springer—Verlag new York Inc, 1985.
H. L. Li, The asymptotic behavior of solutions to the damped P-system with boundary effects, J.P.D.E. 12 (1999) No. 4, 357–368.
T. T. Li and W. C. Yu, “Boundary value problem for quasilinear hyperbolic systems”, Duke Univ. Math. Ser.V 1985.
T. Luo and T. Yang, Global existence of weak entropy solutions for damped p-system, proc.Roy. Soc. Edinburgh Sect. A 128 (1998) 797–807.
T. Luo and T. Yang, Interaction of elementary waves for compressible Euler equation with frictional damping, J. Diff. Eqns. 161 (2000) 42–86.
P. Marcati and M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, preprint 1998.
T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamicsPublicationsMathematique d’Orsay78–02, Department de Mathematique, Universite de Paris Sud, 1978.
K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with dampingJ. Diff. Eqns131 (1996), 171–188.
K. Nishihara and T. Yang, Boundary effect on asymptotic behavior of solutions to the P-system with damping, J. Diff. Eqns. to appear.
D. Serre and L. Xiao, Asymptotic behavior of large weak entropy solutions of the damped p-systemJ. P. Diff. Eqns.10 (1997), 355–368.
J.H. Wang and C. Z. Li, Global regularity and formation of singularities of solution for quasilinear hyperbolic system with dissipation, Chinese Annals Math.9A (1988), 509–523.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Hsiao, L., Li, H. (2001). Asymptotic Behavior of Entropy Weak Solution for Hyperbolic System with Damping. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. ISNM International Series of Numerical Mathematics, vol 141. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8372-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8372-6_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9538-5
Online ISBN: 978-3-0348-8372-6
eBook Packages: Springer Book Archive