### Abstract

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation laws u_{t}+f(u)_{x}=μu_{xx} with the initial data u_{0} which tend to the constant states u_{±} as x→±∞. Stability theorems are obtained in the absence of the convexity of f and in the allowance of s (shock speed)=f′(u_{±}). Moreover, the rate of asymptotics in time is investigated. For the case f′(u_{+})<s<f′(u_{-}), if the integral of the initial disturbance over (-∞, x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate as t→∞. This rate seems to be almost optimal compared with the rate in the case f=u^{2}/2 for which an explicit form of the solution exists. The rate is also obtained in the case f′(u_{±}=s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.

Original language | English |
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Pages (from-to) | 83-96 |

Number of pages | 14 |

Journal | Communications in Mathematical Physics |

Volume | 165 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1994 Oct |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*165*(1), 83-96. https://doi.org/10.1007/BF02099739

**Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity.** / Matsumura, Akitaka; Nishihara, Kenji.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 165, no. 1, pp. 83-96. https://doi.org/10.1007/BF02099739

}

TY - JOUR

T1 - Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

AU - Matsumura, Akitaka

AU - Nishihara, Kenji

PY - 1994/10

Y1 - 1994/10

N2 - The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation laws ut+f(u)x=μuxx with the initial data u0 which tend to the constant states u± as x→±∞. Stability theorems are obtained in the absence of the convexity of f and in the allowance of s (shock speed)=f′(u±). Moreover, the rate of asymptotics in time is investigated. For the case f′(u+)-), if the integral of the initial disturbance over (-∞, x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate as t→∞. This rate seems to be almost optimal compared with the rate in the case f=u2/2 for which an explicit form of the solution exists. The rate is also obtained in the case f′(u±=s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.

AB - The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation laws ut+f(u)x=μuxx with the initial data u0 which tend to the constant states u± as x→±∞. Stability theorems are obtained in the absence of the convexity of f and in the allowance of s (shock speed)=f′(u±). Moreover, the rate of asymptotics in time is investigated. For the case f′(u+)-), if the integral of the initial disturbance over (-∞, x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate as t→∞. This rate seems to be almost optimal compared with the rate in the case f=u2/2 for which an explicit form of the solution exists. The rate is also obtained in the case f′(u±=s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.

UR - http://www.scopus.com/inward/record.url?scp=21844512198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844512198&partnerID=8YFLogxK

U2 - 10.1007/BF02099739

DO - 10.1007/BF02099739

M3 - Article

AN - SCOPUS:21844512198

VL - 165

SP - 83

EP - 96

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -