Asymptotic behavior of solutions for p-system with relaxation

Changjiang Zhu

doi:10.1006/jdeq.2001.4063

Asymptotic behavior of solutions for p-system with relaxation

Changjiang Zhu

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

72 Citations (Scopus)

Abstract

In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxation v_t - u_x = 0, u_t + p(v)_x = 1/ε(f(v)-u), with initial data (v, u)(x, 0) = (v₀(x), u₀(x)) → (v_±,u_±),v_± > 0, as x → ± ∞. We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v_±, u_±) of the initial data at x = ± ∞ do not satisfy the equilibrium equation; i.e., u_± ≠ f(v_±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2 × 2 hyperbolic conservation laws with relaxation. © 2002 Elsevier Science (USA).

Original language	English
Pages (from-to)	273-306
Journal	Journal of Differential Equations
Volume	180
Issue number	2
DOIs	https://doi.org/10.1006/jdeq.2001.4063
Publication status	Published - 10 Apr 2002

Bibliographical note

Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

Funding

The research was supported by the National Natural Science Foundation of China Grant 10041001 and 10171037 and the Foundation of Liu Xue Hui Guo Ren Yuan of the Ministry of Education of China.

Research Keywords

Asymptotic behavior
Energy method
Equilibrium state
Relaxation
Stability
Sub-characteristic condition

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title = "Asymptotic behavior of solutions for p-system with relaxation",

abstract = "In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxation vt - ux = 0, ut + p(v)x = 1/ε(f(v)-u), with initial data (v, u)(x, 0) = (v0(x), u0(x)) → (v±,u±),v± > 0, as x → ± ∞. We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v±, u±) of the initial data at x = ± ∞ do not satisfy the equilibrium equation; i.e., u± ≠ f(v±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2 × 2 hyperbolic conservation laws with relaxation. {\textcopyright} 2002 Elsevier Science (USA).",

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N2 - In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxation vt - ux = 0, ut + p(v)x = 1/ε(f(v)-u), with initial data (v, u)(x, 0) = (v0(x), u0(x)) → (v±,u±),v± > 0, as x → ± ∞. We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v±, u±) of the initial data at x = ± ∞ do not satisfy the equilibrium equation; i.e., u± ≠ f(v±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2 × 2 hyperbolic conservation laws with relaxation. © 2002 Elsevier Science (USA).

AB - In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxation vt - ux = 0, ut + p(v)x = 1/ε(f(v)-u), with initial data (v, u)(x, 0) = (v0(x), u0(x)) → (v±,u±),v± > 0, as x → ± ∞. We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v±, u±) of the initial data at x = ± ∞ do not satisfy the equilibrium equation; i.e., u± ≠ f(v±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2 × 2 hyperbolic conservation laws with relaxation. © 2002 Elsevier Science (USA).

KW - Asymptotic behavior

KW - Energy method

KW - Equilibrium state

KW - Relaxation

KW - Stability

KW - Sub-characteristic condition

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DO - 10.1006/jdeq.2001.4063

M3 - RGC 21 - Publication in refereed journal

SN - 0022-0396

VL - 180

SP - 273

EP - 306

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

Asymptotic behavior of solutions for p-system with relaxation

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Research Keywords

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