Shock reflection for the damped $P$-system
Authors:
Ling Hsiao and Hailiang Li
Journal:
Quart. Appl. Math. 60 (2002), 437-460
MSC:
Primary 35L60; Secondary 35B40, 35L55, 35L67, 76L05, 76S05
DOI:
https://doi.org/10.1090/qam/1914435
MathSciNet review:
MR1914435
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Abstract: The global existence and the asymptotic behavior of the weak entropy solution, the piecewise smooth solution with one shock discontinuity, on a strip domain is investigated in the present paper. We show that, for small smooth initial data and boundary value with only one small jump at $\left ( x, t \right ) = \left ( 0, 0 \right )$, the piecewise smooth solution with one shock discontinuity exists globally in time. The shock discontinuity begins from $\left ( x, t \right ) = \left ( 0, 0 \right )$, moves forward and reflects in a finite time at the boundary $x = 1$ to form a 1-shock, which goes backward and reflects at $x = 0$ also in a finite time to create a new 2-shock. The shock strength decays exponentially and never disappears in finite time. As $t \to \infty$, this solution converges to a constant state determined by the initial and the boundary conditions.
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Y. S. Zheng, Global smooth solutions to the adiabatic gas dynamics system with dissipation terms, Proceeding of Sixth International Conference on Hyperbolic Problems, Theory, Numerical, Applications, Hong Kong, 1996
R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience Publishers, Inc., New York, NY, 1948
C. M. Dafermos, A system of hyperbolic conservation laws with fractional damping. Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids, Z. Angew. Math. Phys. 46, Special Issue, S294–S307 (1995)
L. Hsiao, Quasilinear hyperbolic systems and dissipative mechanisms, World Scientific, River Edge, NJ, 1998
L. Hsiao and Hailiang Li, The system of compressible adiabatic flow through porous media with boundary effects, preprint, 1999
L. Hsiao and T. P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143, 599–605 (1992)
L. Hsiao and T. P. Liu, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chinese Ann. of Math. 14B, 465–480 (1993)
L. Hsiao and T. Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations 125, 329–365 (1996)
L. Hsiao and T. Luo, Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping, Quart. Appl. Math. 56, 173–189 (1998)
L. Hsiao and R. H. Pan, Damped P-system on bounded domain, Contemporary Mathematics 255, 109–124 (2000)
L. Hsiao and R. H. Pan, Initial-boundary value problem for the system of compressible adiabatic flow through porous media, J. Differential Equations 159, 280–305 (1999)
L. Hsiao and D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal. 27, 70–77 (1996)
L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chinese Ann. of Math. 16B, 431–444 (1995)
L. Hsiao and S. Q. Tang, Construction and qualitative behavior of solutions for a system of nonlinear hyperbolic conservation laws with damping, Quart. Appl. Math. 53, 487–505 (1995)
L. Hsiao and S. Q. Tang, Construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping, J. Differential Equations 123, 480–503 (1995)
O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Applied Mathematical Sciences, Vol. 49, Springer-Verlag, New York, NY, 1985
H. L. Li, The asymptotic behavior of solutions to the damped P-system with boundary effects, J. Partial Differential Equations 12, 357–368 (1999)
T. T. Li and W. C. Yu, Boundary value problems for quasilinear hyperbolic systems, Duke Univ. Math. Ser. V, Durham, NC, 1985
T. Luo and T. Yang, Global weak solutions for elastic equations with damping and different end states, Proc. Roy. Soc. Edinburgh Sect. A 128, 797–807 (1998)
T. Luo and T. Yang, Interaction of elementary waves for compressible Euler equations with frictional damping, J. Differential Equations 161, 42–86 (2000)
P. Marcati and M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Quart. Appl. Math. 58, 763–784 (2000)
P. Marcati, A. Milani and P. Secchi, Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system, Manuscripta Math. 60, 49–69 (1988)
T. Nagasawa, Global asymptotics of the outer pressure problem with free boundary, Japan J. Appl. Math. 5, 205–224 (1988)
K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131, 171–188 (1996)
K. Nishihara and T. Yang, Boundary effect on asymptotic behavior of solutions to the P-system with damping, J. Differential Equations 156, 439–458 (1999)
D. Serre and L. Xiao, Asymptotic behavior of large weak entropy solutions of the damped p-system, J. Partial Differential Equations 10, 355–368 (1997)
L. A. Ying and J. H. Wang, Global solutions of the Cauchy problem for a nonhomogeneous quasilinear hyperbolic system, Comm. Pure Appl. Math. 33, 579–597 (1980)
Y. S. Zheng, Global smooth solutions to the adiabatic gas dynamics system with dissipation terms, Proceeding of Sixth International Conference on Hyperbolic Problems, Theory, Numerical, Applications, Hong Kong, 1996
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© Copyright 2002
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