## Hyperbolic Conservation Laws with a Moving Source (1997)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Lien97hyperbolicconservation,

author = {Wen-ching Lien},

title = {Hyperbolic Conservation Laws with a Moving Source},

year = {1997}

}

### OpenURL

### Abstract

The purpose of this paper is to investigate the wave behavior of hyperbolic conservation laws with a moving source. When the speed of the source is close to one of the characteristic speeds of the system, nonlinear resonance occurs and instability may result. We will study solutions with a single transonic shock wave for a general system u t + f(u) x = g(x; u). Suppose that the i-th characteristic speed is close to zero. We propose the following stability criterion: l i @g @u r i ! 0 for nonlinear stability, l i @g @u r i ? 0 for nonlinear instability Here l i and r i are the i-th normalized left and right eigenvectors of df du respectively. By using a variation of the Glimm scheme and studying the evolution of the single transonic shock wave, we prove the existence of solutions and verify the asymptotic stability (or instability). 1 Introduction In this paper, we study the time-asymptotic stability and instability of solutions to systems of conservation laws with a moving sourc...

### Citations

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550 |
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(Show Context)
Citation Context ...epresents external effects, c its speed. We assume that the systems are strictly hyperbolic and that each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax =-=[10]-=-. By the change of variables x \Gamma ct ! x; f(u) ! f(u) \Gamma cu, the system is reduced to @u @t + @f(u) @x = g(x; u); (1.1) u(x; 0) = u 0 (x): (1.2) Thus, without loss of generality, we only consi... |

506 | Shock Waves and Reaction-diffusion Equations - Smoller - 1994 |

258 |
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(Show Context)
Citation Context ...nge of the speed of a transonic shock wave are studied in Section 3. The main tool for constructing solutions to system (1.1) is the random choice method of Liu[14], which generalizes the Glimm scheme=-=[6]-=- for conservation laws. The idea is to decompose the solution into nonlinear modes for conservation laws (1.4) and steady waves (1.5). We shall describe this in Section 4. Local wave interactions are ... |

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49 |
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Citation Context ...t total variation x fu(x; t) : jxjsMgsffl : (6.20) Through (M; T ) and (\GammaM; T ) we draw generalized i-characteristic curves �� + i aand �� \Gamma i respectively,si = 1; \Delta \Delta \Del=-=ta ; n.([7]) The region bet-=-ween �� + i and �� \Gamma i is denoted by\Omega i . Due to the strict hyperbolicity of the system, �� \Sigma i do not intersect �� \Sigma j for i 6= j after some finite time T 0 ? T . ... |

48 |
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Citation Context ...stimates (6.15), (6.19) and (6.20) imply that the total amount of i-waves outside\Omega i after T 0 is O(1)ffl. Hence, in\Omega i , i-waves behave (mod O(1)ffl) like waves for scalar conservation law =-=[15]-=-. Consequently, modulo the error O(1)ffl, i-waves tend to an i-shock wave or an i-rarefaction wave if i-characteristic field is genuinely nonlinear, or a traveling wave if i-characteristic field is li... |

38 |
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Citation Context ...teractions. Once this uniform bound is established, with the aid of Helly's theorem, we can extract a convergent subsequence of u \Delta (x; t) in L 1 loc (R 2 ) , and by the consistency theorem (Liu =-=[13,14]-=-), this subsequence converges to a weak solution u(x; t) of (1.1) and (1.2). Moreover, we simultaneously show that the speed of the relatively strong 1-shock wave is bigger than or equal to the initia... |

32 |
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Citation Context ...@f(u) @x = g(x; u); (1.1) u(x; 0) = u 0 (x): (1.2) Thus, without loss of generality, we only consider systems of the form (1.1). Since (1.1) and (1.2) do not have smooth solutions in general (See Lax =-=[11]-=-), we look for weak solutons, that is, bounded measurable functions satisfying Z Z t0 / u @' @t + f(u) @' @x + g(x; u)' ! dxdt + Z 1 \Gamma1 u 0 (x)'(x; 0) dx = 0 (1.3) for any smooth function '(x; t)... |

30 |
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Citation Context ...hly on the initial data. As for system (1.1), when the characteristic speedssi (u) are non-zero and do not change signs for all u and i = 1; : : : ; n, its solution is also asymptotically stable (Liu =-=[14]-=-). Although the evolution of the solution can be complicated, it tends to a simple non-interacting wave pattern time-asymptotically. In this case, the asymptotic state consists of the aforementioned e... |

25 | Global solutions to the compressible Euler equations with geometrical structure
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Citation Context ...totic stability (or instability) in Section 6. For other works on nozzle flows, see Glimm et al.[8] and Embid et al.[5]. For conservation laws with a moving source, see Asakura [1] and Chen and Glimm =-=[2]-=-. 2 Preliminaries System (1.1) is assumed to be strictly hyperbolic, that is, df(u) du has real and distinct eigenvaluess1 (u) ! \Delta \Delta \Delta !sn (u) with right and left eigenvectors r i (u) a... |

19 |
Resonance for a quasilinear hyperbolic equation
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- 1987
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Citation Context ...ga\Gamma where l i and r i are the i-th left and right eigenvectors of df(u) du and normalized by l i \Delta r j = ffi ij . Note that this criterion is consistent with the cases of a single equation, =-=[17]-=-, and the nozzle flows mentioned above, [16]. We shall concentrate on the solutions of (1.1) containing a single transonic shock wave. In particular, we show that if a standing shock wave is perturbed... |

14 |
A generalized Riemann problem for quasi-one-dimensional gas flow
- Glimm, Marshall, et al.
- 1984
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Citation Context ...e case is also proved with the same spirit. We shall deal with the global estimates and verify the asymptotic stability (or instability) in Section 6. For other works on nozzle flows, see Glimm et al.=-=[8]-=- and Embid et al.[5]. For conservation laws with a moving source, see Asakura [1] and Chen and Glimm [2]. 2 Preliminaries System (1.1) is assumed to be strictly hyperbolic, that is, df(u) du has real ... |

8 |
Multiple steady state for 1-D transonic flow
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- 1984
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Citation Context ...d with the same spirit. We shall deal with the global estimates and verify the asymptotic stability (or instability) in Section 6. For other works on nozzle flows, see Glimm et al.[8] and Embid et al.=-=[5]-=-. For conservation laws with a moving source, see Asakura [1] and Chen and Glimm [2]. 2 Preliminaries System (1.1) is assumed to be strictly hyperbolic, that is, df(u) du has real and distinct eigenva... |

7 |
Nonlinear stability and instability of transonic gas flow through a nozzle
- Liu
- 1983
(Show Context)
Citation Context ...eometry of a duct ( i.e. the sign of a 0 (x) ) completely determines the stability of the flow: flows through an expanding duct are stable; transonic flows through a contracting duct are unstable(Liu =-=[16]-=-). A moving magnetic field for magnetohydrodynamics (MHD) can also be expressed as a moving source (Hoffman [9]). Motivated by the studies of physical models, we assume that the source has finite stre... |

5 |
Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities
- CHERN
- 1989
(Show Context)
Citation Context ...or constructing the solution of system(1.1). We basically adopt the difference scheme introduced in Liu [14]. When the single relatively strong shock wave is involved, the front tracking scheme (Chern=-=[3]-=-) is applied to trace the evolution of the relatively strong shock wave. Choose an equidistributed sequence fa k g 1 k=0 in (0; 1) and mesh lengths \Deltax = r; \Deltat = s satisfying the Courant-Frie... |

4 | Asymptotic states for hyperbolic conservation laws with a moving source - Li, Liu - 1983 |

1 |
Global solutions with a single transonic shock wave for quasilinear hyperbolic systems
- Asakura
(Show Context)
Citation Context ...es and verify the asymptotic stability (or instability) in Section 6. For other works on nozzle flows, see Glimm et al.[8] and Embid et al.[5]. For conservation laws with a moving source, see Asakura =-=[1]-=- and Chen and Glimm [2]. 2 Preliminaries System (1.1) is assumed to be strictly hyperbolic, that is, df(u) du has real and distinct eigenvaluess1 (u) ! \Delta \Delta \Delta !sn (u) with right and left... |

1 |
A single fluid model for shock formation in MHD shock tubes
- Hoffman
- 1967
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Citation Context ... expanding duct are stable; transonic flows through a contracting duct are unstable(Liu [16]). A moving magnetic field for magnetohydrodynamics (MHD) can also be expressed as a moving source (Hoffman =-=[9]-=-). Motivated by the studies of physical models, we assume that the source has finite strength, that is, g(x; u) tends to zero sufficiently fast as jxj !1 . For simplicity, g(x; u) is assumed to be pie... |