Japan Journal of Industrial and Applied Mathematics

, Volume 8, Issue 1, pp 85-96

First online:

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

  • Youichi HattoriAffiliated withYamato Laboratory, IBM Japan
  • , Kenji NishiharaAffiliated withSchool of Political Science and Economics, Waseda University

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This paper is concerned with the asymptotic behavior toward the rarefaction waveu R (x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant stateu± atx=±∞, then the solutionu(x, t), roughly speaking, satisfies supR |uu R | ∼t −1/2 ast → ∞ and, except for the “neighborhoods” of the corners,x=u±t ofu R , sup |uu R |∼t −1. In the proof the exact forms ofu are available.

Key words

rarefaction wave Hopf transformation asymptotic behavior