A note on the stability of the rarefaction wave of the Burgers equation

Youichi Hattori; Kenji Nishihara

doi:10.1007/BF03167186

A note on the stability of the rarefaction wave of the Burgers equation

Youichi Hattori^*, Kenji Nishihara

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

50 Citations (Scopus)

Abstract

This paper is concerned with the asymptotic behavior toward the rarefaction wave u^R(x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant state u± at x=±∞, then the solution u(x, t), roughly speaking, satisfies sup_R |u -u^R| ∼t^-1/2 as t → ∞ and, except for the "neighborhoods" of the corners, x=u±t of u^R, sup |u-u^R|∼t^-1. In the proof the exact forms of u are available.

Original language	English
Pages (from-to)	85-96
Number of pages	12
Journal	Japan Journal of Industrial and Applied Mathematics
Volume	8
Issue number	1
DOIs	https://doi.org/10.1007/BF03167186
Publication status	Published - 1991 Feb

Keywords

asymptotic behavior
Hopf transformation
rarefaction wave

ASJC Scopus subject areas

Applied Mathematics
Engineering(all)

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10.1007/BF03167186

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abstract = "This paper is concerned with the asymptotic behavior toward the rarefaction wave uR(x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant state u± at x=±∞, then the solution u(x, t), roughly speaking, satisfies supR |u -uR| ∼t-1/2 as t → ∞ and, except for the {"}neighborhoods{"} of the corners, x=u±t of uR, sup |u-uR|∼t-1. In the proof the exact forms of u are available.",

keywords = "asymptotic behavior, Hopf transformation, rarefaction wave",

author = "Youichi Hattori and Kenji Nishihara",

year = "1991",

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doi = "10.1007/BF03167186",

language = "English",

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T1 - A note on the stability of the rarefaction wave of the Burgers equation

AU - Hattori, Youichi

AU - Nishihara, Kenji

PY - 1991/2

Y1 - 1991/2

N2 - This paper is concerned with the asymptotic behavior toward the rarefaction wave uR(x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant state u± at x=±∞, then the solution u(x, t), roughly speaking, satisfies supR |u -uR| ∼t-1/2 as t → ∞ and, except for the "neighborhoods" of the corners, x=u±t of uR, sup |u-uR|∼t-1. In the proof the exact forms of u are available.

AB - This paper is concerned with the asymptotic behavior toward the rarefaction wave uR(x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant state u± at x=±∞, then the solution u(x, t), roughly speaking, satisfies supR |u -uR| ∼t-1/2 as t → ∞ and, except for the "neighborhoods" of the corners, x=u±t of uR, sup |u-uR|∼t-1. In the proof the exact forms of u are available.

KW - asymptotic behavior

KW - Hopf transformation

KW - rarefaction wave

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A note on the stability of the rarefaction wave of the Burgers equation

Abstract

Keywords

ASJC Scopus subject areas

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